Research ArticleOn Some Coincidence Best Proximity Point Results

Naeem Saleem 1 Haroon Ahmad1 Hassen Aydi 234 and Yae Ulrich Gaba 45

1Department of Mathematics University of Management and Technology Lahore Pakistan2Universite de Sousse Institut Superieur drsquoInformatique et des Techniques de Communication H Sousse 4000 Tunisia3China Medical University Hospital China Medical University Taichung 40402 Taiwan4Department of Mathematics and Applied Mathematics Sefako Makgatho Health Sciences University PO Box 60Ga-Rankuwa 0208 South Africa5Institut de Mathematiques et de Sciences Physiques (IMSPUAC)Laboratoire de Topologie Fondamentale Computationnelle et leurs Applications (Lab-ToFoCApp) Porto-Novo BP 613 Benin

Correspondence should be addressed toHassenAydi hassenaydiisimarnutn and YaeUlrichGaba yaeulrichgabagmailcom

Received 25 September 2021 Accepted 23 November 2021 Published 22 December 2021

Academic Editor Efthymios G Tsionas

Copyright copy 2021Naeem Saleem et al+is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we discuss some (coincidence) best proximity point results for generalized proximal contractions andλ minus μ-proximal Geraghty contractions in controlled metric type spaces To clarify our study various examples are given and someconclusions are drawn

1 Introduction and Preliminaries

To solve the equation Tp p (T is a mapping defined ona subset of a metric space a simplified linear space or a to-pological vector space) fixed point theory is an importanttool A nonself-mapping T J⟶ K may not have a fixedpoint From this perspective the best approximation theoremand the best proximity point are relevant A classical bestapproximation theorem was due to Fan [1] ie if J isa nonempty compact convex subset of a Hausdorff locallyconvex topological vector space X with a seminorm p andT J⟶ X is a continuousmapping then there is an elementp in J satisfying the condition that Ψ(p Tp) Ψ(Tp J)Many subsequent extensions and variations of Fanrsquos theoremhave occurred including references [2 3]

However even though the best approximation theoremsprovide an approximate solution to the equation Tp pthey do not provide an ideal approximate solution More-over the theorem of the best proximity point specifiesadequate criteria for the presence of an element p to reducethe error Ψ(p Tp) For a nonself-mapping T J⟶ KΨ(p Tp) is at least Ψ(J K) for all p in J then the bestproximity point theorem establishes a globally optimal so-lution of error Ψ(p Tp) by constraining an approximate

solution p of the equation Tp p to the condition thatΨ(p Tp) Ψ(J K) Such an ideal approximate solutionTp p is the best proximity point of the nonself-mappingT J⟶ K For sure the best proximity point hypothesesare a logical augmentation of fixed point hypotheses on thegrounds that the best proximity point is a fixed point in thelight of self-mappings

+e best proximity point hypotheses have been dem-onstrated in [4] Anuradha and Veeramani have tested theproximal pointwise contractions for the presence of a bestproximity point [2] Generally several best proximity pointtheorems were analyzed for multiple variants of contractionsin [5ndash14] A best proximity point theorem for contractionmappings was presented in [15] Some interesting commonbest proximity theorems have been discussed in [7 15]

Nadler [16] was the first who generalized the Banachcontraction principle for multivaluated mappings Laterseveral works appeared in this direction For more detailssee [17ndash20] +e best proximity point hypotheses for dif-ferent sorts of multivalued mappings have likewise beenobtained in [21 22]

Recently the authors in [23] introduced a controlledmetric type space in which the function of extended b-metricspaces was substituted by a function α(p q) depending on

HindawiJournal of MathematicsVolume 2021 Article ID 8005469 19 pageshttpsdoiorg10115520218005469

the parameters of the left-hand side of the triangular in-equality+e primary goal of this article is to include the bestproximity point theorems for generalized and modifiedproximal contractions in the context of complete controlledtype metric spaces thus providing an optimal approximatesolution to the equation Tp p It is acknowledged that theprevious best proximity point theorems include the well-known Banach contraction principle and some of itsgeneralizations

First we state the following useful definitions in thesequel

Definition 1 (see [6]) Let (XΨ) be a metric space havinga pair of nonempty subsets (J K) such that J0 is nonempty+e pair (J K) has the P-property if and only if

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(1)

where p1 p2 isin J0 and q1 q2 isin K0

Definition 2 (see [24]) Let (XΨ) be a metric space havinga pair of nonempty subsets J and K Let T J⟶ K andμ J times J⟶ [0infin) +e mapping T is said to be μ-proximaladmissible if

μ p1 p2( 1113857ge 1

Ψ u1Tp11113872 1113873 Ψ(J K)

Ψ u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎪⎪⎬

⎪⎪⎪⎭

implies μ u1 u2( 1113857ge 1 (2)

for all p1 p2 u1 u2 isin J

Definition 3 (see [25]) Let B(X) represent the closed andbounded subsets of X Let H be the PompeiundashHausdroffmetric induced by metric Ψ defined by

H(J K) max supaisinJ

D(a K) supbisinK

D(b J)1113896 1113897 (3)

for J KsubeB(X) where

D(a K) inf Ψ(a b) b isin K (4)

Definition 4 (see [23]) Let X be a nonempty set andconsider α X times X⟶ [1infin) as a function LetΨ X times X⟶ [0infin) satisfying

(1) Ψ(p1 p2) 0 if and only if p1 p2

(2) Ψ(p1 p2) Ψ(p2 p1)

(3) Ψ(p1 p2)le α(p1 p3)Ψ(p1 p3)+ α(p3 p2)Ψ(p3 p2)for all p1 p2 p3 isin X then (XΨ) is called a controlledmetric type space

From now on (XΨ) is a controlled metric type space

Definition 5 (see [23]) A sequence pn1113864 1113865 in a controlledmetric type space (XΨ) converges to some p inX if for eachpositive ε there is some positive Nε such that Ψ(pn p)lt εfor each ngeNε It can be written as

limn⟶infin

pn p (5)

Definition 6 (see [23]) +e sequence pn1113864 1113865 in a controlledmetric type space (XΨ) is said to be a Cauchy sequence iffor every εgt 0Ψ(pn pm)lt ε for all m ngeNε where Nε isin N

Definition 7 (see [23]) A controlled metric type space(XΨ) is said to be complete if every Cauchy sequence isconvergent in X

Definition 8 (see [23]) Let p isin X and εgt 0

(1) +e open ball K(p ε) is defined as follows

K(p ε) q isin XΨ(p q)lt ε1113864 1113865 (6)

(2) +e mapping T X⟶ X is said continuous atp isin X if for all εgt 0 there exists δ gt 0 such that

T(K(p δ))subeK(Tp ε) (7)

Clearly if T is continuous at p in the controlled metrictype space (XΨ) then pn⟶ p implies that Tpn⟶ Tp

as n⟶infin

Definition 9 (see [26]) Define the functionH L(X) times L(X)⟶ [0infin] by

H(J K) max sup

aisinJD(a K) sup

bisinKD(b J)1113896 1113897 if themaximumexists

infin otherwise

⎧⎪⎪⎨

⎪⎪⎩(8)

2 Journal of Mathematics

for J KsubeL(X) (it represents the set of closed subsets of X )where

D(a K) inf Ψ(a b) b isin K forK sub X (9)

Let J and K be two nonempty subsets of X Define

J0 p isin J Ψ(p q) Ψ(J K) for some q isin K1113864 1113865

K0 q isin K Ψ(p q) Ψ(J K) for somep isin J1113864 1113865(10)

where

Ψ(J K) inf Ψ(p q) p isin J q isin K1113864 1113865(distance of a set J to a setK)

(11)

and we will denote

Dlowast(p q) Ψ(p q) minus Ψ(J K) for allp isin J q isin K

(12)

Theorem 1 (see [26]) Be function H L(X) timesL(X)

⟶ [0infin] is a generalized PompeiundashHausdroff controlledmetric space on L(X)

Remark 1 (see [26]) Let (L(X) H) be a generalizedPompeiundashHausdroff-controlled metric type space +en thefollowing assertions hold (for all bounded and closed subsetsJ K C andD of X )

(1) H(C D) 0 is equivalent to C D

(2) H(C D) H(D C)

(3) H(J C)lemaxsupaisinJα(a b) α(b J)H(J K) +

max α(b C) supcisinCα(c b) H(K C)

Theorem 2 (see [26]) If (XΨ) is a complete controlledmetric space with limnm⟶infinα(pn pm)klt 1 for all pn pm

isin X where kge 1 then (L(X)H) is complete

2 Coincidence Best Proximity Points forGeneralized Proximal Contractions

In this section we will discuss some best proximity pointtheorems using the multivalued concept on a controlledmetric space (XΨ)

From now and onward J and K are nonempty subsets ofa controlled metric type space (XΨ) (until otherwisestated) Define α X times X⟶ [1infin) by αlowast(p J)

inf α(p a) for all a isin J1113864 1113865 and αlowast(J K) inf α(a b) for all

a isin J and b isin K where α X times X⟶ [1infin) and J and K

are nonempty subsets of X

Definition 10 (see [26]) A mapping T X⟶B(X) iscontinuous in a controlled metric type space (XΨ) at p isin X

if for all εgt 0 there exists δ gt 0 such thatT(K(p δ))subeK(Tp ε) (13)

where K(p ε) is given as

K(p ε) q isin XΨ(p q)lt ε1113864 1113865 (14)

Clearly if T is continuous at p then pn⟶ p impliesthat Tpn⟶ Tp as n⟶infin

We introduce the following

Definition 11 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶ K bea mapping A point p isin J is said to be a best proximity pointof the mapping T if

Ψ(p Tp) Ψ(J K) (15)

Definition 12 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K A nonempty set J issaid to be approximately compact with respect to K if everysequence pn1113864 1113865 in J satisfying the condition thatD(q pn)⟶ D(q J) for some q in K has a convergentsubsequence

Definition 13 Given T J⟶B(K) and g

J⟶ J A pairof mappings (g

T) is said to be a β-generalized proximal

contraction if there exists a real number β isin [0 1) such that

D g

u1Tp11113872 1113873 Ψ(J K)

D g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesH Tu1

Tu21113872 1113873le βH Tp1Tp21113872 1113873 (16)

Journal of Mathematics 3

for all u1 u2 p1 andp2 in J Definition 14 Amapping T J⟶B(K) is said to be a βT-generalized proximal contraction if there exists β isin [0 1)

such that

D u1Tp11113872 1113873 Ψ(J K)

D u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesH Tu1

Tu21113872 1113873le βH Tp1Tp21113872 1113873 (17)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-generalized proximal contraction will reduce toa βT-generalized proximal contraction

Definition 15 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶ K andg

J⟶ J be mappings A point p isin J is said to be a co-incidence best proximity point of the pair of mappings(g

T) if

Ψ(g

p Tp) Ψ(J K) (18)

Remark 2 If we take g

IJ (the identity mapping over J)then every coincidence best proximity point becomes a bestproximity point of the mapping T

If JcapKneempty or Ψ(J K) 0 then every best proximitypoint will reduce to a fixed point of the mapping T

Our first main result is stated as follows

Theorem 3 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶B(K)

and g

J⟶ J be one-to-one and continuous mappingsAssume that K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Further assume that the pair (g

T) is a β-generalizedproximal contraction such that

supmge1

limi⟶infin

max supqiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎨

⎩

⎫⎬

⎭max supqiisinTpi

α qi qm( 1113857 α qm Tpi1113872 1113873⎧⎨

⎩

⎫⎬

⎭ lt1k

(19)

and limn⟶infinαlowast(g

pn Tpnminus1) 1 where k isin (0 1) Benthere exists a coincidence best proximity point of the pair(g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

D g

p1Tp01113872 1113873 Ψ(J K) (20)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

D g

p2Tp11113872 1113873 Ψ(J K) (21)

+is process can be continued by selecting pn in J0satisfying the condition as follows

D g

pn Tpnminus11113872 1113873 Ψ(J K) (22)

Having selected pn1113864 1113865 satisfying the condition there existsan element pn+1 in J0 satisfying

D g

pn+1Tpn1113872 1113873 Ψ(J K) (23)

for every integer nge 0Since the pair (g

T) is a β-generalized proximal con-

traction by using equations (22) and (23) we obtain

H Tpn+1Tpn1113872 1113873le βH Tpn Tpnminus11113872 1113873 for each nge 1

(24)

We deduce that

H Tpn+1Tpn1113872 1113873le βn

H Tp1Tp01113872 1113873 for each nge 0 (25)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequencefor all natural numbers n m isin N with nltm

4 Journal of Mathematics

H Tpn Tpm1113872 1113873lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qn+2( 1113857 α qn+2Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpn+21113872 1113873

+ max s supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+2isinTpn+2

α qn+2 qm( 1113857 α qm Tpn+21113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+2

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpi

Tpi+11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpmminus1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945

mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qmminus1isinTpmminus1

α qmminus1 qm( 1113857 α qm Tpmminus11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

max supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

Journal of Mathematics 5

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

(26)

Assume that

Smminus2 1113944

mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(27)

+en we obtain

H Tpn Tpm1113872 1113873leH Tp0Tp11113872 1113873 βn max sup

qnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ Smminus1 minus Sn( 1113857

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ (28)

Using the ratio test we have

ai 1113945i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(29)

where (ai+1ai)lt (1k) Taking limit as n m⟶infin weobtain

limn⟶infin

H Tpn Tpm1113872 1113873 0 (30)

+at is Tpn1113966 1113967 is a Cauchy sequence in the completegeneralized PompeiundashHausdroff controlled metric typespace (B(X)H) hence it converges to some q in K (as theset K is closed) +erefore

Ψ(q J)leΨ q g

pn1113872 1113873le αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873D Tpnminus1 g

pn1113872 1113873

αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)(31)

Taking limn⟶infin on both sides of the above inequality wehave

6 Journal of Mathematics

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinαlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(32)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 +us

Ψ(z q) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (33)

+erefore z is a member of J0 Since J0 is contained ing

(J0) and z g

p for some p in J0 g

pnk⟶ g

p and g

isa one-to-one continuous mapping so pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp +isimplies that

D(g

p Tp) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (34)

+at is p is a coincidence best proximity point of the pair(g

T)To prove the uniqueness of the coincidence best prox-

imity point of the pair of mappings (g

T) suppose thatthere is another coincidence best proximity point qnep of thepair (g

T) We have

D(g

p Tp) Ψ(J K)

D(g

q Tq) Ψ(J K)(35)

As the mapping T is one-to-one on the set J and pne qone has H(Tp Tq)gt 0 Since the pair (g

T) is a β-gener-

alized proximal contraction one can write

(0lt )H(Tp Tq)le βH(Tp Tq)ltH(Tp Tq) (36)

It is a contradiction

Corollary 1 Let T J⟶B(K) and αlowast X times X⟶[1infin) be mappings where K is a closed subset and J isapproximately compact with respect to K with T(J0)subeK0Suppose that T is a continuous and βT-generalized proximalcontraction such that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

limn⟶infin

αlowast pn Tpnminus11113872 1113873 1 where k isin (0 1)

(37)

then there exists a unique best proximity point of T

Proof If we take identity mapping g

IJ (g is identity on J)

the remaining proof is same as in +eorem 3

Definition 16 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a β-modified proximal con-

traction if there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(38)

for all u1 u2 p1 andp2 in J

Definition 17 A mapping T J⟶ K is said to be a βT-modified proximal contraction if there exists β isin [0 1) suchthat

Ψ u1Tp11113872 1113873 Ψ(J K)

Ψ u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(39)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-modified proximal contraction is a βT-modifiedproximal contraction

Theorem 4 Let T J⟶ K and g

J⟶ J be two con-tinuous and one-to-one mappings where K is a closed subsetand J is approximately compact with respect to K withT(J0)subeK0 and J0subeg

(J0) If the pair (g

T) is a β-modified

proximal contraction and

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(40)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (41)

Since Tp1 is an element of T(J0) which is contained inK0 and J0 is contained in g

(J0) it follows that there exists an

element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (42)

By continuing this process we can construct a sequencepn1113864 1113865 in J0 satisfying the condition as follows

Ψ g

pn Tpnminus11113872 1113873 Ψ(J K) (43)

Having chosen pn1113864 1113865 in J0 there exists an element pn+1 inJ0 such that

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (44)

Journal of Mathematics 7

for every positive integer n Since the pair (g

T) is a β-modified proximal contraction from equations (43) and(44) we obtain

Ψ Tpn+1Tpn1113872 1113873le βΨ Tpn Tpnminus11113872 1113873 (45)

Recursively we have

Ψ Tpn+1Tpn1113872 1113873le βnΨ Tp1

Tp01113872 1113873 (46)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ Tpn Tpm1113872 1113873le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873Ψ Tpn+1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873Ψ Tpn+2Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873α Tpn+2Tpn+31113872 1113873Ψ Tpn+2

Tpn+31113872 1113873 + α Tpn+1Tpm1113872 1113873

α Tpn+2Tpm1113872 1113873α Tpn+3

Tpm1113872 1113873Ψ Tpn+3Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873Ψ Tpi

Tpi+11113872 1113873

+ 1113945

mminus1

kn+1α Tpk Tpm1113872 1113873Ψ Tpmminus1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873βmminus1Ψ Tp0

Tp11113872 1113873

le α TpiTpi+11113872 1113873βnΨ Tp0

Tp11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873α Tpmminus1

Tpm1113872 1113873βmminus1Ψ Tp0Tp11113872 1113873

α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

(47)

Assume that

Sl 1113944l

i01113937

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βi

(48)

It follows that

Ψ Tpn Tpm1113872 1113873leΨ Tp0Tp11113872 1113873 βnα Tpn Tpn+11113872 1113873 + Smminus1 minus Sn( 11138571113960 1113961

(49)

Using the ratio test we have

ai 1113945i

j0α Tpj

Tpm1113872 1113873α TpiTpi+11113872 1113873βi

whereai+1

ai

lt1k

(50)

By applying limit m n⟶infin in inequality (49) we get

limn⟶infinΨ Tpn Tpm1113872 1113873 0 (51)

which shows that Tpn1113966 1113967 is a Cauchy sequence hence it isconvergent to some q in K (as the set K is closed) +erefore

8 Journal of Mathematics

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

for J KsubeL(X) (it represents the set of closed subsets of X )where

D(a K) inf Ψ(a b) b isin K forK sub X (9)

Let J and K be two nonempty subsets of X Define

J0 p isin J Ψ(p q) Ψ(J K) for some q isin K1113864 1113865

K0 q isin K Ψ(p q) Ψ(J K) for somep isin J1113864 1113865(10)

where

Ψ(J K) inf Ψ(p q) p isin J q isin K1113864 1113865(distance of a set J to a setK)

(11)

and we will denote

Dlowast(p q) Ψ(p q) minus Ψ(J K) for allp isin J q isin K

(12)

Theorem 1 (see [26]) Be function H L(X) timesL(X)

⟶ [0infin] is a generalized PompeiundashHausdroff controlledmetric space on L(X)

Remark 1 (see [26]) Let (L(X) H) be a generalizedPompeiundashHausdroff-controlled metric type space +en thefollowing assertions hold (for all bounded and closed subsetsJ K C andD of X )

(1) H(C D) 0 is equivalent to C D

(2) H(C D) H(D C)

(3) H(J C)lemaxsupaisinJα(a b) α(b J)H(J K) +

max α(b C) supcisinCα(c b) H(K C)

Theorem 2 (see [26]) If (XΨ) is a complete controlledmetric space with limnm⟶infinα(pn pm)klt 1 for all pn pm

isin X where kge 1 then (L(X)H) is complete

2 Coincidence Best Proximity Points forGeneralized Proximal Contractions

In this section we will discuss some best proximity pointtheorems using the multivalued concept on a controlledmetric space (XΨ)

From now and onward J and K are nonempty subsets ofa controlled metric type space (XΨ) (until otherwisestated) Define α X times X⟶ [1infin) by αlowast(p J)

inf α(p a) for all a isin J1113864 1113865 and αlowast(J K) inf α(a b) for all

a isin J and b isin K where α X times X⟶ [1infin) and J and K

are nonempty subsets of X

Definition 10 (see [26]) A mapping T X⟶B(X) iscontinuous in a controlled metric type space (XΨ) at p isin X

if for all εgt 0 there exists δ gt 0 such thatT(K(p δ))subeK(Tp ε) (13)

where K(p ε) is given as

K(p ε) q isin XΨ(p q)lt ε1113864 1113865 (14)

Clearly if T is continuous at p then pn⟶ p impliesthat Tpn⟶ Tp as n⟶infin

We introduce the following

Definition 11 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶ K bea mapping A point p isin J is said to be a best proximity pointof the mapping T if

Ψ(p Tp) Ψ(J K) (15)

Definition 12 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K A nonempty set J issaid to be approximately compact with respect to K if everysequence pn1113864 1113865 in J satisfying the condition thatD(q pn)⟶ D(q J) for some q in K has a convergentsubsequence

Definition 13 Given T J⟶B(K) and g

J⟶ J A pairof mappings (g

T) is said to be a β-generalized proximal

contraction if there exists a real number β isin [0 1) such that

D g

u1Tp11113872 1113873 Ψ(J K)

D g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesH Tu1

Tu21113872 1113873le βH Tp1Tp21113872 1113873 (16)

Journal of Mathematics 3

for all u1 u2 p1 andp2 in J Definition 14 Amapping T J⟶B(K) is said to be a βT-generalized proximal contraction if there exists β isin [0 1)

such that

D u1Tp11113872 1113873 Ψ(J K)

D u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesH Tu1

Tu21113872 1113873le βH Tp1Tp21113872 1113873 (17)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-generalized proximal contraction will reduce toa βT-generalized proximal contraction

Definition 15 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶ K andg

J⟶ J be mappings A point p isin J is said to be a co-incidence best proximity point of the pair of mappings(g

T) if

Ψ(g

p Tp) Ψ(J K) (18)

Remark 2 If we take g

IJ (the identity mapping over J)then every coincidence best proximity point becomes a bestproximity point of the mapping T

If JcapKneempty or Ψ(J K) 0 then every best proximitypoint will reduce to a fixed point of the mapping T

Our first main result is stated as follows

Theorem 3 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶B(K)

and g

J⟶ J be one-to-one and continuous mappingsAssume that K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Further assume that the pair (g

T) is a β-generalizedproximal contraction such that

supmge1

limi⟶infin

max supqiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎨

⎩

⎫⎬

⎭max supqiisinTpi

α qi qm( 1113857 α qm Tpi1113872 1113873⎧⎨

⎩

⎫⎬

⎭ lt1k

(19)

and limn⟶infinαlowast(g

pn Tpnminus1) 1 where k isin (0 1) Benthere exists a coincidence best proximity point of the pair(g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

D g

p1Tp01113872 1113873 Ψ(J K) (20)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

D g

p2Tp11113872 1113873 Ψ(J K) (21)

+is process can be continued by selecting pn in J0satisfying the condition as follows

D g

pn Tpnminus11113872 1113873 Ψ(J K) (22)

Having selected pn1113864 1113865 satisfying the condition there existsan element pn+1 in J0 satisfying

D g

pn+1Tpn1113872 1113873 Ψ(J K) (23)

for every integer nge 0Since the pair (g

T) is a β-generalized proximal con-

traction by using equations (22) and (23) we obtain

H Tpn+1Tpn1113872 1113873le βH Tpn Tpnminus11113872 1113873 for each nge 1

(24)

We deduce that

H Tpn+1Tpn1113872 1113873le βn

H Tp1Tp01113872 1113873 for each nge 0 (25)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequencefor all natural numbers n m isin N with nltm

4 Journal of Mathematics

H Tpn Tpm1113872 1113873lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qn+2( 1113857 α qn+2Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpn+21113872 1113873

+ max s supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+2isinTpn+2

α qn+2 qm( 1113857 α qm Tpn+21113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+2

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpi

Tpi+11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpmminus1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945

mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qmminus1isinTpmminus1

α qmminus1 qm( 1113857 α qm Tpmminus11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

max supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

Journal of Mathematics 5

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

(26)

Assume that

Smminus2 1113944

mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(27)

+en we obtain

H Tpn Tpm1113872 1113873leH Tp0Tp11113872 1113873 βn max sup

qnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ Smminus1 minus Sn( 1113857

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ (28)

Using the ratio test we have

ai 1113945i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(29)

where (ai+1ai)lt (1k) Taking limit as n m⟶infin weobtain

limn⟶infin

H Tpn Tpm1113872 1113873 0 (30)

+at is Tpn1113966 1113967 is a Cauchy sequence in the completegeneralized PompeiundashHausdroff controlled metric typespace (B(X)H) hence it converges to some q in K (as theset K is closed) +erefore

Ψ(q J)leΨ q g

pn1113872 1113873le αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873D Tpnminus1 g

pn1113872 1113873

αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)(31)

Taking limn⟶infin on both sides of the above inequality wehave

6 Journal of Mathematics

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinαlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(32)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 +us

Ψ(z q) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (33)

+erefore z is a member of J0 Since J0 is contained ing

(J0) and z g

p for some p in J0 g

pnk⟶ g

p and g

isa one-to-one continuous mapping so pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp +isimplies that

D(g

p Tp) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (34)

+at is p is a coincidence best proximity point of the pair(g

T)To prove the uniqueness of the coincidence best prox-

imity point of the pair of mappings (g

T) suppose thatthere is another coincidence best proximity point qnep of thepair (g

T) We have

D(g

p Tp) Ψ(J K)

D(g

q Tq) Ψ(J K)(35)

As the mapping T is one-to-one on the set J and pne qone has H(Tp Tq)gt 0 Since the pair (g

T) is a β-gener-

alized proximal contraction one can write

(0lt )H(Tp Tq)le βH(Tp Tq)ltH(Tp Tq) (36)

It is a contradiction

Corollary 1 Let T J⟶B(K) and αlowast X times X⟶[1infin) be mappings where K is a closed subset and J isapproximately compact with respect to K with T(J0)subeK0Suppose that T is a continuous and βT-generalized proximalcontraction such that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

limn⟶infin

αlowast pn Tpnminus11113872 1113873 1 where k isin (0 1)

(37)

then there exists a unique best proximity point of T

Proof If we take identity mapping g

IJ (g is identity on J)

the remaining proof is same as in +eorem 3

Definition 16 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a β-modified proximal con-

traction if there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(38)

for all u1 u2 p1 andp2 in J

Definition 17 A mapping T J⟶ K is said to be a βT-modified proximal contraction if there exists β isin [0 1) suchthat

Ψ u1Tp11113872 1113873 Ψ(J K)

Ψ u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(39)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-modified proximal contraction is a βT-modifiedproximal contraction

Theorem 4 Let T J⟶ K and g

J⟶ J be two con-tinuous and one-to-one mappings where K is a closed subsetand J is approximately compact with respect to K withT(J0)subeK0 and J0subeg

(J0) If the pair (g

T) is a β-modified

proximal contraction and

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(40)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (41)

Since Tp1 is an element of T(J0) which is contained inK0 and J0 is contained in g

(J0) it follows that there exists an

element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (42)

By continuing this process we can construct a sequencepn1113864 1113865 in J0 satisfying the condition as follows

Ψ g

pn Tpnminus11113872 1113873 Ψ(J K) (43)

Having chosen pn1113864 1113865 in J0 there exists an element pn+1 inJ0 such that

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (44)

Journal of Mathematics 7

for every positive integer n Since the pair (g

T) is a β-modified proximal contraction from equations (43) and(44) we obtain

Ψ Tpn+1Tpn1113872 1113873le βΨ Tpn Tpnminus11113872 1113873 (45)

Recursively we have

Ψ Tpn+1Tpn1113872 1113873le βnΨ Tp1

Tp01113872 1113873 (46)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ Tpn Tpm1113872 1113873le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873Ψ Tpn+1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873Ψ Tpn+2Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873α Tpn+2Tpn+31113872 1113873Ψ Tpn+2

Tpn+31113872 1113873 + α Tpn+1Tpm1113872 1113873

α Tpn+2Tpm1113872 1113873α Tpn+3

Tpm1113872 1113873Ψ Tpn+3Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873Ψ Tpi

Tpi+11113872 1113873

+ 1113945

mminus1

kn+1α Tpk Tpm1113872 1113873Ψ Tpmminus1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873βmminus1Ψ Tp0

Tp11113872 1113873

le α TpiTpi+11113872 1113873βnΨ Tp0

Tp11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873α Tpmminus1

Tpm1113872 1113873βmminus1Ψ Tp0Tp11113872 1113873

α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

(47)

Assume that

Sl 1113944l

i01113937

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βi

(48)

It follows that

Ψ Tpn Tpm1113872 1113873leΨ Tp0Tp11113872 1113873 βnα Tpn Tpn+11113872 1113873 + Smminus1 minus Sn( 11138571113960 1113961

(49)

Using the ratio test we have

ai 1113945i

j0α Tpj

Tpm1113872 1113873α TpiTpi+11113872 1113873βi

whereai+1

ai

lt1k

(50)

By applying limit m n⟶infin in inequality (49) we get

limn⟶infinΨ Tpn Tpm1113872 1113873 0 (51)

which shows that Tpn1113966 1113967 is a Cauchy sequence hence it isconvergent to some q in K (as the set K is closed) +erefore

8 Journal of Mathematics

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

for all u1 u2 p1 andp2 in J Definition 14 Amapping T J⟶B(K) is said to be a βT-generalized proximal contraction if there exists β isin [0 1)

such that

D u1Tp11113872 1113873 Ψ(J K)

D u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesH Tu1

Tu21113872 1113873le βH Tp1Tp21113872 1113873 (17)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-generalized proximal contraction will reduce toa βT-generalized proximal contraction

Definition 15 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶ K andg

J⟶ J be mappings A point p isin J is said to be a co-incidence best proximity point of the pair of mappings(g

T) if

Ψ(g

p Tp) Ψ(J K) (18)

Remark 2 If we take g

IJ (the identity mapping over J)then every coincidence best proximity point becomes a bestproximity point of the mapping T

If JcapKneempty or Ψ(J K) 0 then every best proximitypoint will reduce to a fixed point of the mapping T

Our first main result is stated as follows

Theorem 3 Let (XΨ) be a controlled metric type spacehaving two nonempty subsets J and K Let T J⟶B(K)

and g

J⟶ J be one-to-one and continuous mappingsAssume that K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Further assume that the pair (g

T) is a β-generalizedproximal contraction such that

supmge1

limi⟶infin

max supqiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎨

⎩

⎫⎬

⎭max supqiisinTpi

α qi qm( 1113857 α qm Tpi1113872 1113873⎧⎨

⎩

⎫⎬

⎭ lt1k

(19)

and limn⟶infinαlowast(g

pn Tpnminus1) 1 where k isin (0 1) Benthere exists a coincidence best proximity point of the pair(g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

D g

p1Tp01113872 1113873 Ψ(J K) (20)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

D g

p2Tp11113872 1113873 Ψ(J K) (21)

+is process can be continued by selecting pn in J0satisfying the condition as follows

D g

pn Tpnminus11113872 1113873 Ψ(J K) (22)

Having selected pn1113864 1113865 satisfying the condition there existsan element pn+1 in J0 satisfying

D g

pn+1Tpn1113872 1113873 Ψ(J K) (23)

for every integer nge 0Since the pair (g

T) is a β-generalized proximal con-

traction by using equations (22) and (23) we obtain

H Tpn+1Tpn1113872 1113873le βH Tpn Tpnminus11113872 1113873 for each nge 1

(24)

We deduce that

H Tpn+1Tpn1113872 1113873le βn

H Tp1Tp01113872 1113873 for each nge 0 (25)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequencefor all natural numbers n m isin N with nltm

4 Journal of Mathematics

H Tpn Tpm1113872 1113873lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qn+2( 1113857 α qn+2Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpn+21113872 1113873

+ max s supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+2isinTpn+2

α qn+2 qm( 1113857 α qm Tpn+21113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+2

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpi

Tpi+11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpmminus1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945

mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qmminus1isinTpmminus1

α qmminus1 qm( 1113857 α qm Tpmminus11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

max supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

Journal of Mathematics 5

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

(26)

Assume that

Smminus2 1113944

mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(27)

+en we obtain

H Tpn Tpm1113872 1113873leH Tp0Tp11113872 1113873 βn max sup

qnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ Smminus1 minus Sn( 1113857

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ (28)

Using the ratio test we have

ai 1113945i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(29)

where (ai+1ai)lt (1k) Taking limit as n m⟶infin weobtain

limn⟶infin

H Tpn Tpm1113872 1113873 0 (30)

+at is Tpn1113966 1113967 is a Cauchy sequence in the completegeneralized PompeiundashHausdroff controlled metric typespace (B(X)H) hence it converges to some q in K (as theset K is closed) +erefore

Ψ(q J)leΨ q g

pn1113872 1113873le αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873D Tpnminus1 g

pn1113872 1113873

αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)(31)

Taking limn⟶infin on both sides of the above inequality wehave

6 Journal of Mathematics

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinαlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(32)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 +us

Ψ(z q) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (33)

+erefore z is a member of J0 Since J0 is contained ing

(J0) and z g

p for some p in J0 g

pnk⟶ g

p and g

isa one-to-one continuous mapping so pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp +isimplies that

D(g

p Tp) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (34)

+at is p is a coincidence best proximity point of the pair(g

T)To prove the uniqueness of the coincidence best prox-

imity point of the pair of mappings (g

T) suppose thatthere is another coincidence best proximity point qnep of thepair (g

T) We have

D(g

p Tp) Ψ(J K)

D(g

q Tq) Ψ(J K)(35)

As the mapping T is one-to-one on the set J and pne qone has H(Tp Tq)gt 0 Since the pair (g

T) is a β-gener-

alized proximal contraction one can write

(0lt )H(Tp Tq)le βH(Tp Tq)ltH(Tp Tq) (36)

It is a contradiction

Corollary 1 Let T J⟶B(K) and αlowast X times X⟶[1infin) be mappings where K is a closed subset and J isapproximately compact with respect to K with T(J0)subeK0Suppose that T is a continuous and βT-generalized proximalcontraction such that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

limn⟶infin

αlowast pn Tpnminus11113872 1113873 1 where k isin (0 1)

(37)

then there exists a unique best proximity point of T

Proof If we take identity mapping g

IJ (g is identity on J)

the remaining proof is same as in +eorem 3

Definition 16 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a β-modified proximal con-

traction if there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(38)

for all u1 u2 p1 andp2 in J

Definition 17 A mapping T J⟶ K is said to be a βT-modified proximal contraction if there exists β isin [0 1) suchthat

Ψ u1Tp11113872 1113873 Ψ(J K)

Ψ u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(39)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-modified proximal contraction is a βT-modifiedproximal contraction

Theorem 4 Let T J⟶ K and g

J⟶ J be two con-tinuous and one-to-one mappings where K is a closed subsetand J is approximately compact with respect to K withT(J0)subeK0 and J0subeg

(J0) If the pair (g

T) is a β-modified

proximal contraction and

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(40)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (41)

Since Tp1 is an element of T(J0) which is contained inK0 and J0 is contained in g

(J0) it follows that there exists an

element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (42)

By continuing this process we can construct a sequencepn1113864 1113865 in J0 satisfying the condition as follows

Ψ g

pn Tpnminus11113872 1113873 Ψ(J K) (43)

Having chosen pn1113864 1113865 in J0 there exists an element pn+1 inJ0 such that

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (44)

Journal of Mathematics 7

for every positive integer n Since the pair (g

T) is a β-modified proximal contraction from equations (43) and(44) we obtain

Ψ Tpn+1Tpn1113872 1113873le βΨ Tpn Tpnminus11113872 1113873 (45)

Recursively we have

Ψ Tpn+1Tpn1113872 1113873le βnΨ Tp1

Tp01113872 1113873 (46)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ Tpn Tpm1113872 1113873le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873Ψ Tpn+1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873Ψ Tpn+2Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873α Tpn+2Tpn+31113872 1113873Ψ Tpn+2

Tpn+31113872 1113873 + α Tpn+1Tpm1113872 1113873

α Tpn+2Tpm1113872 1113873α Tpn+3

Tpm1113872 1113873Ψ Tpn+3Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873Ψ Tpi

Tpi+11113872 1113873

+ 1113945

mminus1

kn+1α Tpk Tpm1113872 1113873Ψ Tpmminus1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873βmminus1Ψ Tp0

Tp11113872 1113873

le α TpiTpi+11113872 1113873βnΨ Tp0

Tp11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873α Tpmminus1

Tpm1113872 1113873βmminus1Ψ Tp0Tp11113872 1113873

α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

(47)

Assume that

Sl 1113944l

i01113937

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βi

(48)

It follows that

Ψ Tpn Tpm1113872 1113873leΨ Tp0Tp11113872 1113873 βnα Tpn Tpn+11113872 1113873 + Smminus1 minus Sn( 11138571113960 1113961

(49)

Using the ratio test we have

ai 1113945i

j0α Tpj

Tpm1113872 1113873α TpiTpi+11113872 1113873βi

whereai+1

ai

lt1k

(50)

By applying limit m n⟶infin in inequality (49) we get

limn⟶infinΨ Tpn Tpm1113872 1113873 0 (51)

which shows that Tpn1113966 1113967 is a Cauchy sequence hence it isconvergent to some q in K (as the set K is closed) +erefore

8 Journal of Mathematics

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

H Tpn Tpm1113872 1113873lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ max supqn+1isinTpn+1

α qn+1 qn+2( 1113857 α qn+2Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+1

Tpn+21113872 1113873

+ max s supqn+1isinTpn+1

α qn+1 qm( 1113857 α qm Tpn+11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qn+2isinTpn+2

α qn+2 qm( 1113857 α qm Tpn+21113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn+2

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpn Tpn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpi

Tpi+11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭H Tpmminus1

Tpm1113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945

mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1max sup

qnkisinTpk

α qk qm( 1113857 α qm Tpk1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qmminus1isinTpmminus1

α qmminus1 qm( 1113857 α qm Tpmminus11113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βmminus1

H Tp0Tp11113872 1113873

max supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

Journal of Mathematics 5

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

(26)

Assume that

Smminus2 1113944

mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(27)

+en we obtain

H Tpn Tpm1113872 1113873leH Tp0Tp11113872 1113873 βn max sup

qnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ Smminus1 minus Sn( 1113857

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ (28)

Using the ratio test we have

ai 1113945i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(29)

where (ai+1ai)lt (1k) Taking limit as n m⟶infin weobtain

limn⟶infin

H Tpn Tpm1113872 1113873 0 (30)

+at is Tpn1113966 1113967 is a Cauchy sequence in the completegeneralized PompeiundashHausdroff controlled metric typespace (B(X)H) hence it converges to some q in K (as theset K is closed) +erefore

Ψ(q J)leΨ q g

pn1113872 1113873le αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873D Tpnminus1 g

pn1113872 1113873

αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)(31)

Taking limn⟶infin on both sides of the above inequality wehave

6 Journal of Mathematics

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinαlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(32)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 +us

Ψ(z q) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (33)

+erefore z is a member of J0 Since J0 is contained ing

(J0) and z g

p for some p in J0 g

pnk⟶ g

p and g

isa one-to-one continuous mapping so pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp +isimplies that

D(g

p Tp) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (34)

+at is p is a coincidence best proximity point of the pair(g

T)To prove the uniqueness of the coincidence best prox-

imity point of the pair of mappings (g

T) suppose thatthere is another coincidence best proximity point qnep of thepair (g

T) We have

D(g

p Tp) Ψ(J K)

D(g

q Tq) Ψ(J K)(35)

As the mapping T is one-to-one on the set J and pne qone has H(Tp Tq)gt 0 Since the pair (g

T) is a β-gener-

alized proximal contraction one can write

(0lt )H(Tp Tq)le βH(Tp Tq)ltH(Tp Tq) (36)

It is a contradiction

Corollary 1 Let T J⟶B(K) and αlowast X times X⟶[1infin) be mappings where K is a closed subset and J isapproximately compact with respect to K with T(J0)subeK0Suppose that T is a continuous and βT-generalized proximalcontraction such that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

limn⟶infin

αlowast pn Tpnminus11113872 1113873 1 where k isin (0 1)

(37)

then there exists a unique best proximity point of T

Proof If we take identity mapping g

IJ (g is identity on J)

the remaining proof is same as in +eorem 3

Definition 16 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a β-modified proximal con-

traction if there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(38)

for all u1 u2 p1 andp2 in J

Definition 17 A mapping T J⟶ K is said to be a βT-modified proximal contraction if there exists β isin [0 1) suchthat

Ψ u1Tp11113872 1113873 Ψ(J K)

Ψ u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(39)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-modified proximal contraction is a βT-modifiedproximal contraction

Theorem 4 Let T J⟶ K and g

J⟶ J be two con-tinuous and one-to-one mappings where K is a closed subsetand J is approximately compact with respect to K withT(J0)subeK0 and J0subeg

(J0) If the pair (g

T) is a β-modified

proximal contraction and

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(40)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (41)

Since Tp1 is an element of T(J0) which is contained inK0 and J0 is contained in g

(J0) it follows that there exists an

element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (42)

By continuing this process we can construct a sequencepn1113864 1113865 in J0 satisfying the condition as follows

Ψ g

pn Tpnminus11113872 1113873 Ψ(J K) (43)

Having chosen pn1113864 1113865 in J0 there exists an element pn+1 inJ0 such that

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (44)

Journal of Mathematics 7

for every positive integer n Since the pair (g

T) is a β-modified proximal contraction from equations (43) and(44) we obtain

Ψ Tpn+1Tpn1113872 1113873le βΨ Tpn Tpnminus11113872 1113873 (45)

Recursively we have

Ψ Tpn+1Tpn1113872 1113873le βnΨ Tp1

Tp01113872 1113873 (46)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ Tpn Tpm1113872 1113873le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873Ψ Tpn+1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873Ψ Tpn+2Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873α Tpn+2Tpn+31113872 1113873Ψ Tpn+2

Tpn+31113872 1113873 + α Tpn+1Tpm1113872 1113873

α Tpn+2Tpm1113872 1113873α Tpn+3

Tpm1113872 1113873Ψ Tpn+3Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873Ψ Tpi

Tpi+11113872 1113873

+ 1113945

mminus1

kn+1α Tpk Tpm1113872 1113873Ψ Tpmminus1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873βmminus1Ψ Tp0

Tp11113872 1113873

le α TpiTpi+11113872 1113873βnΨ Tp0

Tp11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873α Tpmminus1

Tpm1113872 1113873βmminus1Ψ Tp0Tp11113872 1113873

α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

(47)

Assume that

Sl 1113944l

i01113937

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βi

(48)

It follows that

Ψ Tpn Tpm1113872 1113873leΨ Tp0Tp11113872 1113873 βnα Tpn Tpn+11113872 1113873 + Smminus1 minus Sn( 11138571113960 1113961

(49)

Using the ratio test we have

ai 1113945i

j0α Tpj

Tpm1113872 1113873α TpiTpi+11113872 1113873βi

whereai+1

ai

lt1k

(50)

By applying limit m n⟶infin in inequality (49) we get

limn⟶infinΨ Tpn Tpm1113872 1113873 0 (51)

which shows that Tpn1113966 1113967 is a Cauchy sequence hence it isconvergent to some q in K (as the set K is closed) +erefore

8 Journal of Mathematics

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

+ 1113944mminus2

in+11113945

i

jn+1max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

lemax supqnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βnH Tp0

Tp11113872 1113873

+ 1113944mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βiH Tp0

Tp11113872 1113873

(26)

Assume that

Smminus2 1113944

mminus2

in+11113945

i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝ ⎞⎟⎠max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(27)

+en we obtain

H Tpn Tpm1113872 1113873leH Tp0Tp11113872 1113873 βn max sup

qnisinTpn

α qn qn+1( 1113857 α qn+1Tpn1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭+ Smminus1 minus Sn( 1113857

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ (28)

Using the ratio test we have

ai 1113945i

j0max sup

qjisinTpj

α qj qm1113872 1113873 α qm Tpj1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭max sup

qiisinTpi

α qi qi+1( 1113857 α qi+1Tpi1113872 1113873

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭βi

(29)

where (ai+1ai)lt (1k) Taking limit as n m⟶infin weobtain

limn⟶infin

H Tpn Tpm1113872 1113873 0 (30)

+at is Tpn1113966 1113967 is a Cauchy sequence in the completegeneralized PompeiundashHausdroff controlled metric typespace (B(X)H) hence it converges to some q in K (as theset K is closed) +erefore

Ψ(q J)leΨ q g

pn1113872 1113873le αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873D Tpnminus1 g

pn1113872 1113873

αlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)(31)

Taking limn⟶infin on both sides of the above inequality wehave

6 Journal of Mathematics

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinαlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(32)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 +us

Ψ(z q) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (33)

+erefore z is a member of J0 Since J0 is contained ing

(J0) and z g

p for some p in J0 g

pnk⟶ g

p and g

isa one-to-one continuous mapping so pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp +isimplies that

D(g

p Tp) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (34)

+at is p is a coincidence best proximity point of the pair(g

T)To prove the uniqueness of the coincidence best prox-

imity point of the pair of mappings (g

T) suppose thatthere is another coincidence best proximity point qnep of thepair (g

T) We have

D(g

p Tp) Ψ(J K)

D(g

q Tq) Ψ(J K)(35)

As the mapping T is one-to-one on the set J and pne qone has H(Tp Tq)gt 0 Since the pair (g

T) is a β-gener-

alized proximal contraction one can write

(0lt )H(Tp Tq)le βH(Tp Tq)ltH(Tp Tq) (36)

It is a contradiction

Corollary 1 Let T J⟶B(K) and αlowast X times X⟶[1infin) be mappings where K is a closed subset and J isapproximately compact with respect to K with T(J0)subeK0Suppose that T is a continuous and βT-generalized proximalcontraction such that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

limn⟶infin

αlowast pn Tpnminus11113872 1113873 1 where k isin (0 1)

(37)

then there exists a unique best proximity point of T

Proof If we take identity mapping g

IJ (g is identity on J)

the remaining proof is same as in +eorem 3

Definition 16 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a β-modified proximal con-

traction if there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(38)

for all u1 u2 p1 andp2 in J

Definition 17 A mapping T J⟶ K is said to be a βT-modified proximal contraction if there exists β isin [0 1) suchthat

Ψ u1Tp11113872 1113873 Ψ(J K)

Ψ u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(39)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-modified proximal contraction is a βT-modifiedproximal contraction

Theorem 4 Let T J⟶ K and g

J⟶ J be two con-tinuous and one-to-one mappings where K is a closed subsetand J is approximately compact with respect to K withT(J0)subeK0 and J0subeg

(J0) If the pair (g

T) is a β-modified

proximal contraction and

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(40)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (41)

Since Tp1 is an element of T(J0) which is contained inK0 and J0 is contained in g

(J0) it follows that there exists an

element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (42)

By continuing this process we can construct a sequencepn1113864 1113865 in J0 satisfying the condition as follows

Ψ g

pn Tpnminus11113872 1113873 Ψ(J K) (43)

Having chosen pn1113864 1113865 in J0 there exists an element pn+1 inJ0 such that

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (44)

Journal of Mathematics 7

for every positive integer n Since the pair (g

T) is a β-modified proximal contraction from equations (43) and(44) we obtain

Ψ Tpn+1Tpn1113872 1113873le βΨ Tpn Tpnminus11113872 1113873 (45)

Recursively we have

Ψ Tpn+1Tpn1113872 1113873le βnΨ Tp1

Tp01113872 1113873 (46)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ Tpn Tpm1113872 1113873le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873Ψ Tpn+1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873Ψ Tpn+2Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873α Tpn+2Tpn+31113872 1113873Ψ Tpn+2

Tpn+31113872 1113873 + α Tpn+1Tpm1113872 1113873

α Tpn+2Tpm1113872 1113873α Tpn+3

Tpm1113872 1113873Ψ Tpn+3Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873Ψ Tpi

Tpi+11113872 1113873

+ 1113945

mminus1

kn+1α Tpk Tpm1113872 1113873Ψ Tpmminus1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873βmminus1Ψ Tp0

Tp11113872 1113873

le α TpiTpi+11113872 1113873βnΨ Tp0

Tp11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873α Tpmminus1

Tpm1113872 1113873βmminus1Ψ Tp0Tp11113872 1113873

α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

(47)

Assume that

Sl 1113944l

i01113937

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βi

(48)

It follows that

Ψ Tpn Tpm1113872 1113873leΨ Tp0Tp11113872 1113873 βnα Tpn Tpn+11113872 1113873 + Smminus1 minus Sn( 11138571113960 1113961

(49)

Using the ratio test we have

ai 1113945i

j0α Tpj

Tpm1113872 1113873α TpiTpi+11113872 1113873βi

whereai+1

ai

lt1k

(50)

By applying limit m n⟶infin in inequality (49) we get

limn⟶infinΨ Tpn Tpm1113872 1113873 0 (51)

which shows that Tpn1113966 1113967 is a Cauchy sequence hence it isconvergent to some q in K (as the set K is closed) +erefore

8 Journal of Mathematics

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinαlowast q Tpnminus11113872 1113873D q Tpnminus11113872 1113873 + αlowast Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(32)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 +us

Ψ(z q) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (33)

+erefore z is a member of J0 Since J0 is contained ing

(J0) and z g

p for some p in J0 g

pnk⟶ g

p and g

isa one-to-one continuous mapping so pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp +isimplies that

D(g

p Tp) limk⟶infin

D g

pnk Tpnkminus1

1113872 1113873 Ψ(J K) (34)

+at is p is a coincidence best proximity point of the pair(g

T)To prove the uniqueness of the coincidence best prox-

imity point of the pair of mappings (g

T) suppose thatthere is another coincidence best proximity point qnep of thepair (g

T) We have

D(g

p Tp) Ψ(J K)

D(g

q Tq) Ψ(J K)(35)

As the mapping T is one-to-one on the set J and pne qone has H(Tp Tq)gt 0 Since the pair (g

T) is a β-gener-

alized proximal contraction one can write

(0lt )H(Tp Tq)le βH(Tp Tq)ltH(Tp Tq) (36)

It is a contradiction

Corollary 1 Let T J⟶B(K) and αlowast X times X⟶[1infin) be mappings where K is a closed subset and J isapproximately compact with respect to K with T(J0)subeK0Suppose that T is a continuous and βT-generalized proximalcontraction such that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

limn⟶infin

αlowast pn Tpnminus11113872 1113873 1 where k isin (0 1)

(37)

then there exists a unique best proximity point of T

Proof If we take identity mapping g

IJ (g is identity on J)

the remaining proof is same as in +eorem 3

Definition 16 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a β-modified proximal con-

traction if there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(38)

for all u1 u2 p1 andp2 in J

Definition 17 A mapping T J⟶ K is said to be a βT-modified proximal contraction if there exists β isin [0 1) suchthat

Ψ u1Tp11113872 1113873 Ψ(J K)

Ψ u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ Tu1

Tu21113872 1113873le βΨ Tp1Tp21113872 1113873

(39)

for all u1 u2 p1 andp2 in JNote that if we take g

IJ (the identity mapping on J)

then every β-modified proximal contraction is a βT-modifiedproximal contraction

Theorem 4 Let T J⟶ K and g

J⟶ J be two con-tinuous and one-to-one mappings where K is a closed subsetand J is approximately compact with respect to K withT(J0)subeK0 and J0subeg

(J0) If the pair (g

T) is a β-modified

proximal contraction and

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(40)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (41)

Since Tp1 is an element of T(J0) which is contained inK0 and J0 is contained in g

(J0) it follows that there exists an

element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (42)

By continuing this process we can construct a sequencepn1113864 1113865 in J0 satisfying the condition as follows

Ψ g

pn Tpnminus11113872 1113873 Ψ(J K) (43)

Having chosen pn1113864 1113865 in J0 there exists an element pn+1 inJ0 such that

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (44)

Journal of Mathematics 7

for every positive integer n Since the pair (g

T) is a β-modified proximal contraction from equations (43) and(44) we obtain

Ψ Tpn+1Tpn1113872 1113873le βΨ Tpn Tpnminus11113872 1113873 (45)

Recursively we have

Ψ Tpn+1Tpn1113872 1113873le βnΨ Tp1

Tp01113872 1113873 (46)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ Tpn Tpm1113872 1113873le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873Ψ Tpn+1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873Ψ Tpn+2Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873α Tpn+2Tpn+31113872 1113873Ψ Tpn+2

Tpn+31113872 1113873 + α Tpn+1Tpm1113872 1113873

α Tpn+2Tpm1113872 1113873α Tpn+3

Tpm1113872 1113873Ψ Tpn+3Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873Ψ Tpi

Tpi+11113872 1113873

+ 1113945

mminus1

kn+1α Tpk Tpm1113872 1113873Ψ Tpmminus1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873βmminus1Ψ Tp0

Tp11113872 1113873

le α TpiTpi+11113872 1113873βnΨ Tp0

Tp11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873α Tpmminus1

Tpm1113872 1113873βmminus1Ψ Tp0Tp11113872 1113873

α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

(47)

Assume that

Sl 1113944l

i01113937

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βi

(48)

It follows that

Ψ Tpn Tpm1113872 1113873leΨ Tp0Tp11113872 1113873 βnα Tpn Tpn+11113872 1113873 + Smminus1 minus Sn( 11138571113960 1113961

(49)

Using the ratio test we have

ai 1113945i

j0α Tpj

Tpm1113872 1113873α TpiTpi+11113872 1113873βi

whereai+1

ai

lt1k

(50)

By applying limit m n⟶infin in inequality (49) we get

limn⟶infinΨ Tpn Tpm1113872 1113873 0 (51)

which shows that Tpn1113966 1113967 is a Cauchy sequence hence it isconvergent to some q in K (as the set K is closed) +erefore

8 Journal of Mathematics

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

for every positive integer n Since the pair (g

T) is a β-modified proximal contraction from equations (43) and(44) we obtain

Ψ Tpn+1Tpn1113872 1113873le βΨ Tpn Tpnminus11113872 1113873 (45)

Recursively we have

Ψ Tpn+1Tpn1113872 1113873le βnΨ Tp1

Tp01113872 1113873 (46)

Now we have to prove that Tpn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ Tpn Tpm1113872 1113873le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873Ψ Tpn+1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873Ψ Tpn+2Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + α Tpn+1Tpm1113872 1113873α Tpn+1

Tpn+21113872 1113873Ψ Tpn+1Tpn+21113872 1113873

+ α Tpn+1Tpm1113872 1113873α Tpn+2

Tpm1113872 1113873α Tpn+2Tpn+31113872 1113873Ψ Tpn+2

Tpn+31113872 1113873 + α Tpn+1Tpm1113872 1113873

α Tpn+2Tpm1113872 1113873α Tpn+3

Tpm1113872 1113873Ψ Tpn+3Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873Ψ Tpn Tpn+11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873Ψ Tpi

Tpi+11113872 1113873

+ 1113945

mminus1

kn+1α Tpk Tpm1113872 1113873Ψ Tpmminus1

Tpm1113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873βmminus1Ψ Tp0

Tp11113872 1113873

le α TpiTpi+11113872 1113873βnΨ Tp0

Tp11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

+ 1113945mminus1

kn+1α Tpk Tpm1113872 1113873α Tpmminus1

Tpm1113872 1113873βmminus1Ψ Tp0Tp11113872 1113873

α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

jn+1α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

le α Tpn Tpn+11113872 1113873βnΨ Tp0Tp11113872 1113873 + 1113944

mminus1

in+11113945

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βiΨ Tp0

Tp11113872 1113873

(47)

Assume that

Sl 1113944l

i01113937

i

j0α Tpj

Tpm1113872 1113873⎛⎝ ⎞⎠α TpiTpi+11113872 1113873βi

(48)

It follows that

Ψ Tpn Tpm1113872 1113873leΨ Tp0Tp11113872 1113873 βnα Tpn Tpn+11113872 1113873 + Smminus1 minus Sn( 11138571113960 1113961

(49)

Using the ratio test we have

ai 1113945i

j0α Tpj

Tpm1113872 1113873α TpiTpi+11113872 1113873βi

whereai+1

ai

lt1k

(50)

By applying limit m n⟶infin in inequality (49) we get

limn⟶infinΨ Tpn Tpm1113872 1113873 0 (51)

which shows that Tpn1113966 1113967 is a Cauchy sequence hence it isconvergent to some q in K (as the set K is closed) +erefore

8 Journal of Mathematics

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

Ψ(q J)leΨ q g

pn1113872 1113873le α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ Tpnminus1 g

pn1113872 1113873

α q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)(52)

Taking limit n⟶infin on both sides of the above in-equality we have

limn⟶infinΨ q g

pn1113872 1113873le lim

n⟶infinα q Tpnminus11113872 1113873Ψ q Tpnminus11113872 1113873 + α Tpnminus1 g

pn1113872 1113873Ψ(J K)1113960 1113961

leΨ(J K)

leΨ(q J)

(53)

+ereforeΨ(q g

pn)⟶Ψ(q J) In view of the fact thatJ is approximately compact with respect to K g

pn1113966 1113967 has

a subsequence g

pnk1113966 1113967 converging to some z g

p isin J for

some p isin J0 It follows that

Ψ(z q) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (54)

+erefore z is an element of J0 Since J0 is contained ing

(J0) we have z g

p for some p in J0 As g

pnk⟶ g

p and

g is a one-to-one continuous mapping pnk

⟶ p Since T iscontinuous it can be concluded that Tpnk

⟶ Tp Hence

Ψ(g

p Tp) limk⟶infinΨ g

pnk

Tpnkminus11113872 1113873 Ψ(J K) (55)

To prove the uniqueness suppose that q is anothercoincidence best proximity point of the pair (g

T) such that

pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(56)

Since the pair (g

T) is a β-modified proximal con-traction we have

0ltΨ(Tp Tq)le βΨ(Tp Tq)ltΨ(Tp Tq) (57)

which is a contradiction (as T is one-to-one mapping on J)Hence the pair (g

T) has a unique coincidence best

proximity point

Corollary 2 Let T J⟶ K be a given continuous mappingwhere K is a closed subset and J is approximately compactwith respect to K with T(J0)subeK0 If T is a βT-modifiedproximal contraction and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(58)

then there exists a unique best proximity point of T

Proof If we take g

IJ (the identity mapping over the setJ) the remaining proof is same as +eorem 4

Example 1 Let X 0 1 2 3 4 5 Consider the function Ψgiven as Ψ(p p) 0 and Ψ(p q) Ψ(q p) where

Ψ 0 1 2 3 4 50 0 114 113 115 112 1111 114 115 452 113 23 0 19 18 1153 115 34 19 0 78 894 112 115 18 78 0 145 111 45 115 89 14 0

34230

Take α X times X⟶ [1infin) to be symmetric which isdefined as α(p q) 19p + 21q It is easy to see that (XΨ) isa controlled metric type space Take J 0 1 2 andK 3 4 5 Obviously Ψ(J K) (115) J0 J andK0 K Now consider T J⟶ K as follows

Tp 3 if p 0 1

5 if p 21113896 (59)

Clearly T(J0)subeK0 Define g

J⟶ J by

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩(60)

We get J0subeg

(J0) Now we have to show that the pair(g

T) satisfies

Ψ(g0 T1) Ψ(0 3) Ψ(J K)

Ψ(g1 T2) Ψ(2 5) Ψ(J K)

(61)

where u1 0 u2 1 p1 1 and p2 2 Since the pair(g

T) is a β-modified proximal contraction

Ψ(T0 T1)le βΨ(T1 T2) (62)

for every β isin [0 1) the pair (g

T) is a β-modified proximalcontraction Hence 0 is the unique coincidence best prox-imity point of T and g

Definition 18 Let T J⟶ K and g

J⟶ J A pair ofmappings (g

T) is said to be a βminus proximal contraction if

there exists β isin [0 1) such that

Ψ g

u1Tp11113872 1113873 Ψ(J K)

Ψ g

u2Tp21113872 1113873 Ψ(J K)

⎫⎪⎬

⎪⎭ impliesΨ g

u1 g

u21113872 1113873le βΨ p1 p2( 1113857

(63)

for all u1 u2 p1 andp2 in J

Definition 19 A mapping T J⟶ K is said to be a βT-proximal contraction if there exists β isin [0 1) such that

Ψ u1Tp11113872 1113873 Ψ(JK)

Ψ u2Tp21113872 1113873 Ψ(JK)

⎫⎪⎬

⎪⎭ impliesΨ u1u2( 1113857leβΨ p1p2( 1113857 (64)

for all u1 u2 p1 andp2 in J

Journal of Mathematics 9

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

Note that if we take g

IJ then every β-proximalcontraction is a βT-proximal contraction

Theorem 5 Let T J⟶ K and g

J⟶ J be continuousmappings where K is a closed subset and J is approximatelycompact with respect to K with T(J0)subeK0 and J0subeg

(J0)

Suppose that g is an expansive mapping and the pair (g

T) is

a β-proximal contraction such that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α g

pn Tpnminus11113872 1113873 1 where k isin (0 1)

(65)

then there exists a unique coincidence best proximity point ofthe pair (g

T)

Proof Let p0 be an arbitrary element in J0 Since T(J0) iscontained in K0 and J0 is contained in g

(J0) there exists an

element p1 in J0 such that

Ψ g

p1Tp01113872 1113873 Ψ(J K) (66)

Again since Tp1 is an element of T(J0) which is con-tained inK0 and J0 is contained in g

(J0) it follows that there

is an element p2 in J0 such that

Ψ g

p2Tp11113872 1113873 Ψ(J K) (67)

+is process can be continued by selecting pn in J0 sothat

Ψ g

pn+1Tpn1113872 1113873 Ψ(J K) (68)

Since the pair (g

T) is a β-proximal contraction we have

Ψ g

pn+1 g

pn1113872 1113873le βΨ pn pnminus1( 1113857 (69)

As g is an expansive mapping one writes

Ψ pn+1 pn( 1113857leΨ g

pn+1 g

pn1113872 1113873le βnΨ p1 p0( 1113857 (70)

so we have

Ψ pn+1 pn( 1113857le βnΨ p1 p0( 1113857 (71)

We claim that pn1113864 1113865 is a Cauchy sequence For all naturalnumbers n m isin N with nltm we have

Ψ pn pm( 1113857le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857Ψ pn+1 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857Ψ pn+2 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + α pn+1 pm( 1113857α pn+1 pn+2( 1113857Ψ pn+1 pn+2( 1113857

+ α pn+1 pm( 1113857α pn+2 pm( 1113857α pn+2 pn+3( 1113857Ψ pn+2 pn+3( 1113857 + α pn+1 pm( 1113857

α pn+2 pm( 1113857α pn+3 pm( 1113857Ψ pn+3 pm( 1113857

le α pn pn+1( 1113857Ψ pn pn+1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857Ψ pi pi+1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857Ψ pmminus1 pm( 1113857

le α Tpn Tpn+11113872 1113873βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857βmminus 1Ψ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus2

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

+ 1113945mminus1

kn+1α pk pm( 1113857α pmminus1 pm( 1113857βmminus 1Ψ p0 p1( 1113857

α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

jn+1α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

le α pn pn+1( 1113857βnΨ p0 p1( 1113857 + 1113944mminus1

in+11113945

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βiΨ p0 p1( 1113857

(72)

10 Journal of Mathematics

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

Assume that

Sl 1113944l

i01113937

i

j0α pj pm1113872 1113873⎛⎝ ⎞⎠α pi pi+1( 1113857βi

(73)

+en we obtain

Ψ pn pm( 1113857leΨ p0 p1( 1113857 βnα pn pn+1( 1113857 + Smminus1 minus Sn( 11138571113858 1113859

(74)

Using the ratio test we have

ai 1113945i

j0α pj pm1113872 1113873α pi pi+1( 1113857βi

whereai+1

ai

lt1k

(75)

By taking limit as n m⟶infin (74) becomes

limn⟶infinΨ pn pm( 1113857 0 (76)

+erefore pn1113864 1113865 is a Cauchy sequence in the completecontrolled metric type space (XΨ) hence it is convergentto some p in J (as set J is closed) Since g

and T arecontinuous we have

Ψ(g

p Tp) limn⟶infinΨ g

pn+1

Tpn1113872 1113873 Ψ(J K) (77)

Hence p is the unique coincidence best proximity pointof the pair (g

T) To prove the uniqueness suppose that q is

another coincidence best proximity point of the pair (g

T)

such that pne q +en

Ψ(g

p Tp) Ψ(J K)

Ψ(g

q Tq) Ψ(J K)(78)

Since the pair (g

T) is a β-modified proximal con-traction we have

Ψ(p q)leΨ(g

p g

q)le βΨ(p q)ltΨ(p q) (79)

which is a contradiction Hence the pair (g

T) has a uniquecoincidence best proximity point

Corollary 3 Let T J⟶ K be a continuous mappingwhere K is closed subset and J is approximately compact withrespect to K with T(J0)subeK0 If T is a βT-proximal contractionand suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

limn⟶infin

α pn Tpnminus11113872 1113873 1 where k isin (0 1)

(80)

then there exists a best proximity point of T

Proof If we take identity mapping g

IJ the remainingproof is same as +eorem 5

3 Coincidence Best Proximity Points forGeraghty Type ProximalContractive Mappings

First we need to define a generalized Geraghty type prox-imal contractive mapping

From now and onward F is a class of all nondecreasingfunctions λ [0infin)⟶ [0 1) such that for any boundedsequence tn1113864 1113865 of positive real numbers λ tn1113864 1113865⟶ 1 impliestn⟶ 0

Definition 20 Let (XΨ) be a controlled metric type spacehaving a pair of nonempty subsets (J K) such that J0 isnonempty+en a pair (J K) has the P-property if and onlyif

Ψ p1 q1( 1113857 Ψ(J K)

Ψ p2 q2( 1113857 Ψ(J K)1113897 impliesΨ p1 p2( 1113857 Ψ q1 q2( 1113857

(81)

Definition 21 Let T J⟶B(K) g J⟶ J aremappingsA pair (g

T) is said to be a λ minus μ-proximal Geraghty con-

traction if μ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(g

u Tp) Ψ(J K)

D(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (82)

where

M(u v p q) max Ψ(g

p g

q)D(g

p Tp) minus αlowast(g

q Tp)Ψ(J K)

αlowast(g

p g

q)1113896

Dlowast(g

u Tp)

D(g

u Tq) minus αlowast(g

v Tq)Ψ(J K)

αlowast(g

u g

v)1113897

(83)

for all u v p q isin J where λ isin F

Journal of Mathematics 11

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

Definition 22 A mapping T J⟶B(K) is said to bea (λ minus μ)T-proximal Geraghty contraction ifμ J times J⟶ [0infin) is such that

μ(p q)ge 1

D(u Tp) Ψ(J K)

D(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies that μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (84)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus αlowast(q Tp)Ψ(J K)

αlowast(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus αlowast(v Tq)Ψ(J K)

αlowast(u v)1113897

(85)

for all u v p q isin J where λ isin FIf we take g

IJ (the identity mapping over J) then

every λ minus μ-proximal Geraghty contraction will reduce toa λ minus μ-generalized proximal Geraghty contraction

Theorem 6 Let T J⟶B(K) g

J⟶ J andμ J times J⟶ [0 +infin) be mappings where J is a closed subsetand the pair (J K) satisfies the P-property with T(J0)subeK0and J0subeg

(J0) If a pair of continuous mappings (g

T) is

a λ minus μ-proximal Geraghty contraction where T is μ-proxi-mal admissible then there exist elements p0 p1 isin J0 such thatD(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(86)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof From the given condition there exist p0 p1 isin J0such that D(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 AsT(J0)subeK0 there exists p2 isin J0 such thatD(g

p2

Tp1) Ψ(J K) As T is μ-proximal admissibleμ(p0 p1)ge 1

D g

p1Tp01113872 1113873 Ψ(J K)

D g

p2Tp11113872 1113873 Ψ(J K)

(87)

using the P-property Ψ(g

p1 g

p2) H(Tp0Tp1) Since

the pair (g

T) is a λ minus μ-proximal Geraghty contraction withμ(p1 p2)ge 1 we have

Ψ g

p1 g

p21113872 1113873le λ M p0 p1 p1 p2( 1113857( 1113857M p0 p1 p1 p2( 1113857 (88)

where

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873D g

p0

Tp01113872 1113873 minus αlowast g

p1Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

Dlowast

g

p1Tp01113872 1113873

D g

p1Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873αlowast g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 + αlowast g

p1

Tp01113872 1113873D g

p1Tp01113872 1113873

αlowast g

p0 g

p11113872 1113873

⎧⎨

⎩

minusαlowast g

p1

Tp01113872 1113873Ψ(J K)

αlowast g

p0 g

p11113872 1113873D g

p1

Tp01113872 1113873 minus Ψ(J K)αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873

αlowast g

p1 g

p21113872 1113873

+αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873 minus αlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎫⎬

⎭

lemax Ψ g

p0 g

p11113872 1113873Ψ g

p0 g

p11113872 1113873 0Ψ g

p1 g

p21113872 11138731113966 1113967

(89)

12 Journal of Mathematics

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

and we have

M p0 p1 p1 p2( 1113857lemax Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967

(90)

If

max Ψ g

p0 g

p11113872 1113873Ψ g

p1 g

p21113872 11138731113966 1113967 Ψ g

p1 g

p21113872 1113873 (91)

then inequality (88) becomes

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (92)

which is a contradiction So we can conclude that

Ψ g

p1 g

p21113872 1113873le λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (93)

Further by the fact that T(J0)subeK0 there exists p3 isin J0such that D(g

p3

Tp2) Ψ(J K) As T is μ-proximal ad-missible mapping where μ(p2 p3)ge 1

D g

p2Tp11113872 1113873 Ψ(J K)

D g

p3Tp21113872 1113873 Ψ(J K)

(94)

using the P-Property we haveΨ(g

p2 g

p3) H(Tp1Tp2)

Since the pair (g

T) is a λ minus μ-proximal Geraghty mappingwith μ(p2 p3)ge 1 one writes

Ψ g

p2 g

p31113872 1113873le λ M p2 p3 p1 p2( 1113857( 1113857M p2 p3 p1 p2( 1113857

(95)

where

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873D g

p1

Tp11113872 1113873 minus αlowast g

p2Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

Dlowast

g

p2Tp11113872 1113873

D g

p2Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873αlowast g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 + αlowast g

p2

Tp11113872 1113873D g

p2Tp11113872 1113873

αlowast g

p1 g

p21113872 1113873

⎧⎨

⎩

minusαlowast g

p2

Tp11113872 1113873Ψ(J K)

αlowast g

p1 g

p21113872 1113873D g

p2

Tp11113872 1113873 minus Ψ(J K)αlowast g

p2 g

p31113872 1113873Ψ g

p2 g

p31113872 1113873

αlowast g

p2 g

p31113872 1113873

+αlowast g

p3

Tp21113872 1113873D g

p3Tp21113872 1113873 minus αlowast g

p3

Tp21113872 1113873Ψ(J K)

αlowast g

p2 g

p31113872 1113873

⎫⎬

⎭

lemax Ψ g

p1 g

p21113872 1113873Ψ g

p1 g

p21113872 1113873 0Ψ g

p2 g

p31113872 11138731113966 1113967

(96)

and we have

M p2 p3 p1 p2( 1113857lemax Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967

(97)

If

max Ψ g

p1 g

p21113872 1113873Ψ g

p2 g

p31113872 11138731113966 1113967 Ψ g

p2 g

p31113872 1113873 (98)

then inequality (95) becomes

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p2 g

p31113872 11138731113872 1113873Ψ g

p2 g

p31113872 1113873 (99)

which is a contradiction +us

Ψ g

p2 g

p31113872 1113873le λ Ψ g

p1 g

p21113872 11138731113872 1113873Ψ g

p1 g

p21113872 1113873 (100)

Similarly we can construct a sequence pn1113864 1113865subeJ0 whereμ(pn pn+1)ge 1 for all n isin Ncup 0

D g

pn Tpnminus11113872 1113873 Ψ(J K)

D g

pn+1Tpn1113872 1113873 Ψ(J K)

(101)

using the P-property Ψ(g

pn g

pn+1) H(Tpn Tpnminus1)Since the pair (g

T) is a λ minus μ-proximal Geraghty con-

traction with μ(pn pn+1)ge 1 we get

Ψ g

pn g

pn+11113872 1113873le λ M pn pn+1 pnminus1 pn( 1113857( 1113857M pn pn+1 pnminus1 pn( 1113857

(102)

where

Journal of Mathematics 13

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873D g

pnminus1

Tpnminus11113872 1113873 minus αlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

Dlowast

g

pn Tpnminus11113872 1113873D g

pn Tpn1113872 1113873 minus αlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873αlowast g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 + αlowast g

pn Tpnminus11113872 1113873D g

pn Tpnminus11113872 1113873

αlowast g

pnminus1 g

pn1113872 1113873

⎧⎨

⎩

minusαlowast g

pn Tpnminus11113872 1113873Ψ(J K)

αlowast g

pnminus1 g

pn1113872 1113873D g

pn Tpnminus11113872 1113873 minus Ψ(J K)

αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast gpn+1Tpn1113872 1113873D g

pn+1

Tpn1113872 11138731113872

αlowast g

pn g

pn+11113872 1113873minusαlowast g

pn+1

Tpn1113872 1113873Ψ(J K)

αlowast g

pn g

pn+11113872 1113873

⎫⎬

⎭

lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pnminus1 g

pn1113872 1113873 0Ψ g

pn g

pn+11113872 11138731113966 1113967

(103)

After simplification we have

M pn pn+1 pnminus1 pn( 1113857lemax Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967

(104)

If

max Ψ g

pnminus1 g

pn1113872 1113873Ψ g

pn g

pn+11113872 11138731113966 1113967 Ψ g

pn g

pn+11113872 1113873

(105)

then inequality (102) becomes

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pn g

pn+11113872 11138731113872 1113873Ψ g

pn g

pn+11113872 1113873

(106)

which is a contradiction So we conclude that

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

(107)

Further

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873leΨ g

pnminus1 g

pn1113872 1113873

(108)

which shows that Ψ(g

pn g

pn+1)1113966 1113967 is a decreasing sequenceSince λ isin F from (108) we have

λ Ψ g

pn g

pn+11113872 11138731113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873(109)

Continuing on the same lines we can write

λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873le middot middot middot le λ Ψ g

p0 g

p11113872 11138731113872 1113873

(110)

Using inequality (107)

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873Ψ g

pnminus2 g

pnminus11113872 1113873

(111)

From inequalities (107) and (111) we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873Ψ g

pnminus1 g

pn1113872 1113873

le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873

middot Ψ g

pnminus2 g

pnminus11113872 1113873

(112)

Following on similar lines we have

Ψ g

pn g

pn+11113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873λ Ψ g

pnminus2 g

pnminus11113872 11138731113872 1113873 λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873(113)

We deduce

Ψ g

pn g

pn+11113872 1113873le λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 (114)

From (108) suppose that Ψ(g

pnminus1 g

pn)gt 0 so we canconclude

Ψ g

pn g

pn+11113872 1113873

Ψ g

pnminus1 g

pn1113872 1113873le λ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 for all nge 1

(115)

Let l limn⟶+infinΨ(g

pnminus1 g

pn) Using equation (108)and letting n⟶ +infin we obtain that

14 Journal of Mathematics

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

l

l 1le lim

n⟶+infinλ Ψ g

pnminus1 g

pn1113872 11138731113872 1113873le 1 (116)

+us limn⟶+infinΨ(g

pnminus1 g

pn) 1 Using the definitionof λ we conclude that

limn⟶+infinΨ g

pnminus1 g

pn1113872 1113873 0 (117)

Now we have to show that g

pn1113966 1113967 is a Cauchy sequenceFor all natural numbers n m isin N with nltm we have

Ψ g

pn g

pm1113872 1113873le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873Ψ g

pn+1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+2 g

pm1113872 1113873Ψ g

pn+2 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873αlowast g

pn+1 g

pn+21113872 1113873Ψ g

pn+1 g

pn+21113872 1113873

+ αlowast g

pn+1 g

pm1113872 1113873αlowast gpn+2 g

pm1113872 1113873αlowast1113872 g

pn+2 g

pn+31113872 1113873Ψ g

pn+2 g

pn+31113872 1113873 + αlowast g

pn+1 g

pm1113872 1113873

αlowast g

pn+2 g

pm1113872 1113873αlowast g

pn+3 g

pm1113872 1113873Ψ g

pn+3 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873Ψ g

pn g

pn+11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873Ψ g

pi g

pi+11113872 1113873 + 1113945

mminus1

kn+1αlowast g

pk g

pm1113872 1113873Ψ g

pmminus1 g

pm1113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 + 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus2

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λ Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113945mminus1

kn+1αlowast g

pk g

pm1113872 1113873αlowast g

pmminus1 g

pm1113872 1113873

λmminus 1 Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873 αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

jn+1αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

le αlowast g

pn g

pn+11113872 1113873λn Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

+ 1113944mminus1

in+11113945

i

j0αlowast g

pj g

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pi g

pi+11113872 1113873

λi Ψ g

p0 g

p11113872 11138731113872 1113873Ψ g

p0 g

p11113872 1113873

(118)

Assume that

Sl 1113944l

i01113937

i

j0αlowast g

pjg

pm1113872 1113873⎛⎝ ⎞⎠αlowast g

pig

pi+11113872 1113873λi Ψ g

p0g

p11113872 11138731113872 1113873

(119)

+en we obtain

Ψ g

pn g

pm1113872 1113873leΨ g

p0 g

p11113872 1113873 λn Ψ g

p0 g

p11113872 11138731113872 1113873αlowast g

pn g

pn+11113872 11138731113960

+ Smminus1 minus Sn( 11138571113859

(120)

Journal of Mathematics 15

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

Using the ratio test we have

ai 1113945

i

j0αlowast g

pj g

pm1113872 1113873αlowast g

pi g

pi+11113872 1113873λi Ψ g

p0 g

p11113872 11138731113872 1113873

whereai+1

ai

lt1k

(121)

Taking limit as n m⟶infin inequality (120) becomes

limnm⟶infinΨ g

pn g

pm1113872 1113873 0 (122)

+is implies that g

pn1113966 1113967 is a Cauchy sequence in thecomplete controlled metric type space (XΨ) hence it isconvergent and suppose that it converges to some plowast in J0subeJ(as set J is closed) which assures that the sequence pn1113864 1113865subeJ0since pn⟶ plowast As (g

T) is a pair of continuous mappings

one writes

D g

plowast Tplowast

1113872 1113873 Ψ(J K) (123)

+erefore plowast is a coincidence best proximity point of thepair (g

T)

For uniqueness suppose that there are two distinctcoincidence best proximity points of (g

T) such that

plowast ne qlowast +us s Ψ(plowast qlowast)gt 0 Since Ψ(g

plowast Tplowast)

Ψ(g

qlowast Tqlowast) Ψ(J K) using the P-property we concludethat s H(Tplowast Tqlowast) Since the pair (g

T) is a λ minus μ-

proximal Geraghty contraction we obtain sle λ(s)s +usλ(s)ge 1 Since λ(s)ge 1 we conclude that λ(s) 1 andtherefore s 0 which is contradiction

Example 2 Let X 0 1 2 3 4 5 be endowed with thefunction Ψ given as Ψ(p q) Ψ(q p) and Ψ(p p) 0where

Ψ 0 1 2 3 4 50 0 12 13 110 15 161 12 0 14 23 110 342 13 14 0 67 78 1103 110 23 67 0 12 134 15 110 78 12 0 145 16 34 110 13 14 0

Take α X times X⟶ [1infin) to be symmetric and definedas α(p q) 16p + 18q It is easy to see that (XΨ) iscontrolled type metric space Suppose J 0 1 2 andK 3 4 5 After a simple calculation Ψ(J K) (110)the P-property is satisfied J0 J and K0 K Consider

Tp 3 if p 2

3 4 if p 0 1 1113896

g

p

0 if p 0

1 if p 2

2 if p 1

⎧⎪⎪⎨

⎪⎪⎩

(124)

Clearly T(J0)subeK0 and J0subeg

(J0) Now we have to showthat the pair (g

T) is a λ minus μ-proximal Geraghty contraction

μ(p q)H(Tp Tq)le λ(M(u v p q))M(u v p q) (125)

for all u v p q isin J and for the function μ J times J⟶ [0 +infin)

is defined by

μ(p q) Ψ(p q) + 1 (126)

Hence

D(g0 T2) D(1 3) Ψ(J K)

D(g2 T1) D(1 3 4 ) Ψ(J K)

(127)

After simple calculationsH(Tp Tq) H(3 3 4 ) 0μ(p q) Ψ(3 3 4 ) + 1 1 and

M(0 2 2 1) max Ψ(g2 g

1)D(g

2 T2) minus αlowast(g1 T2)(110)

αlowast(g2 g

1)1113896

Dlowast(g

0 T2)D(g

0 T1) minus αlowast(g2 T1)(110)

αlowast(g0 g

2)1113897

max Ψ(1 2)D(1 3) minus αlowast(2 3)(110)

αlowast(1 2)1113896

Dlowast(0 3)

D(0 3 4 ) minus αlowast(1 3 4 )(110)

αlowast(0 1)1113897

max14minus2381560

0minus69180

1113882 1113883 14

(128)

16 Journal of Mathematics

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

Now we have to show that the pair (g

T) is a λ minus μ-proximal Geraghty contraction

(1)(0)le λ(M(u v p q))14

1113874 1113875

le λ(M(u v p q))14

1113874 1113875

(129)

and for every λ [0infin)⟶ [0 1] the pair (g

T) is a λ minus μ-proximal Geraghty contraction Hence 0 is the uniquecoincidence point of the pair of mappings (g

T)

Corollary 4 Let T J⟶B(K) be a continuous mappingwhere J is a closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-proximalGeraghty contraction where T is μ-proximal admissible then

there exist elements p0 p1 isin J0 such thatD(p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

αlowast pi pi+1( 1113857αlowast pi pm( 1113857lt1k

where k isin (0 1)

(130)

then T has a unique best proximity point plowast isin J

Proof If we take g

IJ (an identity mapping over J) theremaining proof is same as +eorem 6

Definition 23 Let T J⟶ K g J⟶ J and μ J times J⟶[0 +infin) be mappings A pair of mappings (g

T) is said to be

a λ minus μ-modified proximal Geraghty contraction if

μ(p q)ge 1

Ψ(g

u Tp) Ψ(J K)

Ψ(g

v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (131)

where

M(u v p q) max Ψ(g

p g

q)Ψ(g

p Tp) minus α(g

q Tp)Ψ(J K)

α(g

p g

q)1113896

Ψlowast(g

u Tp)Ψ(g

u Tq) minus α(g

v Tq)Ψ(J K)

α(g

u g

v)1113897

(132)

for all u v p q isin J where λ isin F Definition 24 Let T J⟶ K and μ J times J⟶ [0 +infin) bemappings A mapping T is said to be a (λ minus μ)T-modifiedproximal Geraghty contraction if

μ(p q)ge 1

Ψ(u Tp) Ψ(J K)

Ψ(v Tq) Ψ(J K)

⎫⎪⎪⎬

⎪⎪⎭ implies μ(p q)Ψ(Tp Tq)le λ(M(u v p q))M(u v p q) (133)

where

M(u v p q) max Ψ(p q)Ψ(p Tp) minus α(q Tp)Ψ(J K)

α(p q)1113896

Ψlowast(u Tp)Ψ(u Tq) minus α(v Tq)Ψ(J K)

α(u v)1113897

(134)

Journal of Mathematics 17

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

for all u v p q isin J where λ isin FNote that if we take g

IJ (an identity mapping over J)

then every λ minus μ-modified proximal Geraghty contractionwill reduce to a (λ minus μ)T-modified proximal Geraghtycontraction

Theorem 7 Let T J⟶ K and g

J⟶ J be continuousmappings where J is closed subset and the pair (J K) satisfiesthe P-property with T(J0)subeK0 and J0subeg

(J0) If the pair of

mappings (g

T) is a λ minus μ-modified proximal Geraghtycontraction where T is a μ-proximal admissible then thereexist elements p0 p1 isin J0 such that Ψ(g

p1

Tp0) Ψ(J K)

and μ(p0 p1)ge 1 If pn1113864 1113865 is a sequence in J such thatμ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(135)

then the pair (g

T) has a unique coincidence best proximitypoint plowast isin J

Proof It is a simple consequence of +eorem 6

Corollary 5 Let T J⟶ K be a continuous mappingwhere J is closed subset and the pair (J K) satisfies theP-property with T(J0)subeK0 If T is a (λ minus μ)T-modifiedproximal Geraghty contraction where T is a μ-proximaladmissible then there exist elements p0 p1 isin J0 such thatΨ(g

p1

Tp0) Ψ(J K) and μ(p0 p1)ge 1 If pn1113864 1113865 is a se-quence in J such that μ(pn pn+1)ge 1 and suppose that

supmge1

limi⟶infin

α pi pi+1( 1113857α pi pm( 1113857lt1k

where k isin (0 1)

(136)

then the pair (g

T) has a unique best proximity point plowast isin J

Proof If we take g

IJ the remaining proof is same as+eorem 7

4 Conclusion

In our paper we ensured the existence of some bestproximity point results via the multivalued concept incontrolled metric spaces To our knowledge we are the firstwho worked on best proximity points for the class ofmultivalued mappings in this setting We open the door fornew perspectives when dealing with new generalized mul-tivalued (proximal) contractions

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interestconcerning the publication of this article

Authorsrsquo Contributions

All authors contributed equally and significantly for writingthis article All authors read and approved the finalmanuscript

References

[1] K Fan ldquoExtensions of two fixed point theorems of F EBrowderrdquo Mathematische Zeitschrift vol 112 no 3pp 234ndash240 1969

[2] J Anuradha and P Veeramani ldquoProximal pointwise con-tractionrdquo Topology and Its Applications vol 156 no 18pp 2942ndash2948 2009

[3] N Saleem I Iqbal B Iqbal and S Radenoviacutec ldquoCoincidenceand fixed points of multivalued F-contractions in generalizedmetric space with applicationrdquo Journal of Fixed Point Beoryand Applications vol 22 2020

[4] S Sadiq Basha ldquoExtensions of banachrsquos contraction princi-plerdquoNumerical Functional Analysis and Optimization vol 31no 5 pp 569ndash576 2010

[5] W A Kirk S Reich and P Veeramani ldquoProximinal retractsand best proximity pair theoremsrdquo Numerical FunctionalAnalysis and Optimization vol 24 no 7-8 pp 851ndash862 2003

[6] V Sankar Raj ldquoA best proximity point theorem for weaklycontractive non-self-mappingsrdquo Nonlinear Analysis BeoryMethods amp Applications vol 74 no 14 pp 4804ndash4808 2011

[7] P S Srinivasan ldquoAnalysis-Best proximity pair theoremsrdquoActa Scientiarum Mathematicarum vol 67 no 1-2pp 421ndash430 2001

[8] T Suzuki M Kikkawa and C Vetro ldquo+e existence of bestproximity points in metric spaces with the property UCNonlinear Analysis +eoryrdquo Methods amp Applications vol 71no 7-8 pp 2918ndash2926 2009

[9] C Vetro and T Suzuki ldquo+ree existence theorems for weakcontractions of Matkowski typerdquo International Journal ofMathematics and Statistics pp 110ndash220 2010

[10] M Simkhah D Turkoglu S Sedghi and N Shobe ldquoSuzukitype fixed point results in p-metric spacesrdquo Communicationsin Optimization Beory Article ID 13 2019

[11] E Naraghirad ldquoBregman best proximity points for Bregmanasymptotic cyclic contraction mappings in Banach spacesrdquoJournal of Nonlinear and Variational Analysis vol 3pp 27ndash44 2019

[12] V Parvaneh M R Haddadi and H Aydi ldquoOn best proximitypoint results for some type of mappingsrdquo Journal of FunctionSpaces vol 2020 Article ID 6298138 6 pages 2020

[13] H Aydi H Lakzian Z D Mitrovic and S Radenovic ldquoBestproximity points of MT-cyclic contractions with propertyUCrdquo Numerical Functional Analysis and Optimizationvol 41 no 7 pp 871ndash882 2020

[14] F Gu and W Shatanawi ldquoSome new results on commoncoupled fixed points of two hybrid pairs of mappings in partialmetric spacesrdquo Journal of Nonlinear Functional AnalysisArticle ID 13 2019

[15] S Sadiq Basha N Shahzad and R Jeyaraj ldquoCommon bestproximity points global optimization of multi-objectivefunctionsrdquo Applied Mathematics Letters vol 24 no 6pp 883ndash886 2011

[16] S Nadler ldquoMulti-valued contraction mappingsrdquo PacificJournal of Mathematics vol 30 no 2 pp 475ndash488 1969

[17] H Kaneko ldquoSingle and multivalued contractionsrdquo Bolletinodell Unione Matematica Italiana vol 6 pp 29ndash33 1985

18 Journal of Mathematics

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19

[18] E Ameer H Aydi M Arshad H Alsamir and M NooranildquoHybrid multivalued type contraction mappings in αk-complete partial b-metric spaces and applicationsrdquo Symmetryvol 11 no 1 Article ID 86 2019

[19] S Czerwik ldquoNonlinear set-valued contraction mappings in b-metric spacesrdquo Atti del Seminario Matematico e Fisico del-lrsquoUniversita di Modena vol 46 no 2 pp 263ndash276 1998

[20] P Patle D Patel H Aydi and S Radenovic ldquoOn H+Typemultivalued contractions and applications in symmetric andprobabilistic spacesrdquo Mathematics vol 7 no 2 Article ID144 2019

[21] S Basha and P Veeramani ldquoBest approximations and bestproximity pairsrdquo Acta Scientiarum Mathematicarum vol 63pp 289ndash300 1997

[22] S Sadiq Basha and P Veeramani ldquoBest proximity pair the-orems for multifunctions with open fibresrdquo Journal of Ap-proximation Beory vol 103 no 1 pp 119ndash129 2000

[23] N Mlaiki H Aydi N Souayah and T Abdeljawad ldquoCon-trolled metric type spaces and the related contraction prin-ciplerdquo Mathematics vol 6 no 10 Article ID 194 2018

[24] M Jleli and B Samet ldquoBest proximity points for α-ψ-proximalcontractive type mappings and applicationsrdquo Bulletin desSciences Mathematiques vol 137 no 8 pp 977ndash995 2013

[25] T R Rockafellar and R J V Wets Variational AnalysisSpringer Berlin Germany 2005

[26] N Alamgir Q Kiran H Isik and H Aydi ldquoFixed pointresults via a Hausdorff controlled type metricrdquo Advances inDifference Equations vol 24 pp 1ndash20 2020

[27] M Jleli B Samet C Vetro and F Vetro ldquoFixed points formultivalued mappings in b-metric spacesrdquo Abstract andApplied Analysis vol 2015 Article ID 718074 7 pages 2015

Journal of Mathematics 19