Research Article A Probabilistic Fixed Point Result Using...
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Research Article A Probabilistic Fixed Point Result Using Altering Distance Functions Claudia Zaharia 1 and Nataša SiroviT 2 1 Department of Mathematics, West University of Timis , oara, Bulevardul V. Pˆ arvan 4, 300223 Timis , oara, Romania 2 Department of Mathematics, Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 74, 11000 Belgrade, Serbia Correspondence should be addressed to Claudia Zaharia; [email protected] Received 26 February 2015; Accepted 3 August 2015 Academic Editor: Richard I. Avery Copyright © 2015 C. Zaharia and N. ´ Cirovi´ c. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove a general fixed point theorem in Menger spaces for mappings satisfying a contractive condition of ´ Ciri´ c type, formulated by means of altering distance functions. us, we extend some recent results of Choudhury and Das, Mihet ¸, and Babaˇ cev and also clarify some aspects regarding a theorem of Choudhury, Das, and Dutta. 1. Introduction and Preliminaries In [1], Menger introduced the concept of probabilistic metric space as a generalization of metric spaces, in which the distance between points is expressed by means of distribution functions. is idea has made probabilistic metric spaces suitable for modeling phenomena when the uncertainty regarding measurements is assumed as inherent to the mea- suring process, as, for instance, in the investigation of certain physical quantities and physiological thresholds [2]. Proba- bilistic metric space theory has become a very active field of research. In particular, fixed point theory in probabilistic structures has found relevant applications in studying the existence and uniqueness of solutions of random equations [3], as well as algorithm complexity analysis [4, 5], and convergence analysis for stochastic optimization algorithms [6]. In the present paper, we establish a fixed point result for probabilistic contractions of ´ Ciri´ c type, with the contractive condition stated by means of an altering distance function. Our theorem is obtained under very weak hypotheses, and thus it generalizes or improves several known results [7–10]. We also discuss the connections with a related theorem given by Choudhury et al. in [11], in order to explain the role of our assumptions. We begin by recalling some fundamental concepts of probabilistic metric space theory. For a comprehensive exposition on this topic we refer the reader to the mono- graphs [2, 3]. Definition 1. A triangular norm (or -norm) is a mapping : [0, 1]×[0, 1]→[0, 1] which is associative, commutative, and nondecreasing in each variable and satisfies (, 1)= for all ∈[0, 1]. Some basic examples are (, ) = min(, ), (, ) = , and (, ) = max( + − 1, 0) (the minimum, product, and Łukasiewicz -norm, resp.). Another important class is that of -norms of Hadˇ zi´ c type [12], that is, -norms whose family of iterates { ()} defined by 0 () = 1, () = ( −1 (), ) ∀ ≥ 1, is equicontinuous at = 1. We will denote by D + the set of functions : R →[0, 1] which are nondecreasing and leſt continuous on R, such that (0)= 0 and lim →∞ () = 1. Definition 2. A Menger space is a triple (, , ) where is a nonempty set, is a mapping from × to D + , and is a -norm, such that the following conditions are satisfied: (i) () = 1 for all > 0 iff =. (ii) () = (), for all , ∈ , > 0. (iii) ( + ) ≥ ( (), ()), for all , , ∈ , , > 0. Hindawi Publishing Corporation Journal of Function Spaces Volume 2015, Article ID 919202, 6 pages http://dx.doi.org/10.1155/2015/919202
Transcript of Research Article A Probabilistic Fixed Point Result Using...
Research ArticleA Probabilistic Fixed Point Result UsingAltering Distance Functions
Claudia Zaharia1 and Nataša SiroviT2
1Department of Mathematics West University of Timisoara Bulevardul V Parvan 4 300223 Timisoara Romania2Department of Mathematics Faculty of Electrical Engineering University of Belgrade Bulevar Kralja Aleksandra 7411000 Belgrade Serbia
Correspondence should be addressed to Claudia Zaharia claudiazahariae-uvtro
Received 26 February 2015 Accepted 3 August 2015
Academic Editor Richard I Avery
Copyright copy 2015 C Zaharia and N Cirovic This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We prove a general fixed point theorem in Menger spaces for mappings satisfying a contractive condition of Ciric type formulatedby means of altering distance functions Thus we extend some recent results of Choudhury and Das Mihet and Babacev and alsoclarify some aspects regarding a theorem of Choudhury Das and Dutta
1 Introduction and Preliminaries
In [1] Menger introduced the concept of probabilistic metricspace as a generalization of metric spaces in which thedistance between points is expressed bymeans of distributionfunctions This idea has made probabilistic metric spacessuitable for modeling phenomena when the uncertaintyregarding measurements is assumed as inherent to the mea-suring process as for instance in the investigation of certainphysical quantities and physiological thresholds [2] Proba-bilistic metric space theory has become a very active fieldof research In particular fixed point theory in probabilisticstructures has found relevant applications in studying theexistence and uniqueness of solutions of random equations[3] as well as algorithm complexity analysis [4 5] andconvergence analysis for stochastic optimization algorithms[6]
In the present paper we establish a fixed point result forprobabilistic contractions of Ciric type with the contractivecondition stated by means of an altering distance functionOur theorem is obtained under very weak hypotheses andthus it generalizes or improves several known results [7ndash10]We also discuss the connections with a related theorem givenby Choudhury et al in [11] in order to explain the role of ourassumptions
We begin by recalling some fundamental concepts ofprobabilistic metric space theory For a comprehensive
exposition on this topic we refer the reader to the mono-graphs [2 3]
Definition 1 A triangular norm (or 119905-norm) is a mapping 119879
[0 1]times[0 1] rarr [0 1]which is associative commutative andnondecreasing in each variable and satisfies119879(119909 1) = 119909 for all119909 isin [0 1]
Some basic examples are 119879119872(119909 119910) = min(119909 119910)
119879119875(119909 119910) = 119909119910 and 119879
119871(119909 119910) = max(119909 + 119910 minus 1 0) (the
minimum product and Łukasiewicz 119905-norm resp) Anotherimportant class is that of 119905-norms of Hadzic type [12] that is119905-norms whose family of iterates 119879119899(119909)
119899defined by 119879
0(119909) =
1 119879119899(119909) = 119879(119879119899minus1
(119909) 119909) forall119899 ge 1 is equicontinuous at 119909 = 1We will denote byD+ the set of functions 119865 R rarr [0 1]
which are nondecreasing and left continuous onR such that119865(0) = 0 and lim
119905rarrinfin119865(119905) = 1
Definition 2 A Menger space is a triple (119883 119865 119879) where 119883 isa nonempty set 119865 is a mapping from119883times119883 toD+ and 119879 is a119905-norm such that the following conditions are satisfied
(i) 119865119909119910(119905) = 1 for all 119905 gt 0 iff 119909 = 119910
(ii) 119865119909119910(119905) = 119865
119910119909(119905) for all 119909 119910 isin 119883 119905 gt 0
(iii) 119865119909119910(119905 + 119904) ge 119879(119865
119909119911(119905) 119865119911119910(119904)) for all 119909 119910 119911 isin 119883 119905 119904 gt
0
Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 919202 6 pageshttpdxdoiorg1011552015919202
2 Journal of Function Spaces
(Here and in the following 119865(119909 119910) will be denoted by119865119909119910)
Let (119883 119865 119879) be a Menger space such that sup119909isin(01)
119879(119909 119909) = 1 The family 119880(120576 120582)120576gt0120582isin(01) where
119880 (120576 120582) = (119909 119910) isin119883times119883 119865119909119910 (120576) gt 1minus120582 (1)
is a base for a Hausdorff uniformity on 119883 named stronguniformity The corresponding strong topology on 119883 isintroduced by the family of neighbourhoods of 119909 isin 119883N
119909=
119873119909(120576 120582)
120576gt0120582isin(01) where119873119909(120576 120582) = 119910 isin 119883 119865119909119910(120576) gt 1minus120582
and this topology is metrizable [2]
Definition 3 Let (119883 119865 119879) be a Menger space A sequence(119909119899)119899in119883 is said to be
(i) Cauchy if for any 120576 gt 0 120582 isin (0 1) there exists 1198990 isin N
such that 119865119909119899119909119898
(120576) gt 1 minus 120582 for all 119898 119899 ge 1198990(ii) convergent to 119909 isin 119883 if for any 120576 gt 0 120582 isin (0 1) there
exists 1198990 isin N such that 119865119909119899119909(120576) gt 1 minus 120582 for all 119899 ge 1198990
119883 is said to be complete if every Cauchy sequence (119909119899)119899sub 119883
is convergent in 119883
If the 119905-norm 119879 is continuous and the sequences (119909119899)119899
(119910119899)119899
sub 119883 converge respectively to 119909 and 119910 isin 119883 then119865119909119899119910119899
(119905) converges to 119865119909119910(119905) for each continuity point 119905 of 119865
119909119910
[2]
Definition 4 Given a set119860 sub 119883 the probabilistic diameter of119860 is the mapping119863
119860 R rarr [0 1] defined by
119863119860 (119905) =
sup119904lt119905
inf119909119910isin119860
119865119909119910 (119904) 119905 gt 0
0 119905 le 0(2)
119860 is said to be probabilistically bounded if119863119860
isin D+
The notion of contraction in a Menger space was intro-duced by Sehgal in [13]
Definition 5 (see [13]) Let (119883 119865 119879) be a Menger space Amapping 119891 119883 rarr 119883 is said to be a probabilistic contraction(or Sehgal contraction) if there exists 119888 isin (0 1) such that
119865119891(119909)119891(119910) (119905) ge 119865
119909119910(119905
119888) forall119909 119910 isin 119883 119905 gt 0 (3)
Many significant contributions to the development offixed point theory in probabilistic structures can be found inmonograph [14] It should be pointed out that the triangularnorm by which the space is endowed plays a key role in theexistence of fixed points of probabilistic contractions It wasshown by Radu [15] that the largest class of continuous 119905-norms 119879 with the property that every Sehgal contraction ona complete Menger space (119883 119865 119879) has a unique fixed point isthat of 119905-norms of Hadzic type
The idea of using altering distance functions in order toobtain more general contractive conditions first appears in[16] in the setting of metric spaces The corresponding con-cept of generalized probabilistic contraction was introducedby Choudhury and Das in [8] as follows
Definition 6 (see [8]) A mapping 120593 [0infin) rarr [0infin) issaid to belong to the class Φ if it satisfies
(i) 120593(119905) = 0 iff 119905 = 0
(ii) 120593 is strictly increasing and lim119905rarrinfin
120593(119905) = infin
(iii) 120593 is continuous at 119905 = 0 and left continuous on (0infin)
The mappings 120593 isin Φ will be called altering distancefunctions
Definition 7 (cf [8]) Let (119883 119865 119879) be a Menger space Themapping 119891 119883 rarr 119883 is said to be a generalized probabilisticcontraction of Choudhury-Das type if there exist 120593 isin Φ and119888 isin (0 1) such that
119865119891(119909)119891(119910)
(120593 (119905)) ge 119865119909119910
(120593(119905
119888)) forall119909 119910 isin 119883 119905 gt 0 (4)
It was proved in [8] that such contractions on a completeMenger space endowed with the strongest 119905-norm 119879
119872have a
unique fixed point The result was subsequently generalizedby Mihet [9] for the case of arbitrary continuous 119905-normsunder the supplementary assumption that the orbit of themapping 119891 at some 119909 isin 119883 is probabilistically bounded
Our aim is to prove a fixed point result for mappingssatisfying the more general contractive condition
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119888)) 119865119909119891(119909)
(120593(119905
119888))
119865119910119891(119910)
(120593(119905
119888))) forall119909 119910 isin 119883 119905 gt 0
(5)
for some 120593 isin Φ and 119888 isin (0 1)
2 Main Results
In order to prove our results we will need the followinglemma from [10]
Lemma 8 (see [10]) Let (119883 119865 119879) be a Menger space 120593 isin Φ
and 119888 isin (0 1) If 119909 119910 isin 119883 are such that
119865119909119910
(120593 (119905)) le 119865119909119910
(120593(119905
119888)) forall119905 gt 0 (6)
then 119909 = 119910
For each 119909 isin 119883 we will denote by119874(119891 119909) the orbit of themapping 119891 at 119909 that is 119874(119891 119909) = 119891
119899(119909) 119899 isin N
Theorem 9 Let (119883 119865 119879) be a complete Menger space with119879 a continuous 119905-norm Suppose 119891 119883 rarr 119883 is a mappingsatisfying the contractive condition (5) for some 120593 isin Φ and 119888 isin
(0 1) If there exists 119909 isin 119883 such that119874(119891 119909) is probabilisticallybounded then 119891 has a unique fixed point in 119883
Proof Let 119909 be as in the statement of the theorem We willshow that the sequence (119909
119899)119899 119909119899= 119891119899(119909) is Cauchy
Journal of Function Spaces 3
From (5) it follows that
119865119909119899119909119899+1
(120593 (119905))
ge min(119865119909119899minus1119909119899
(120593(119905
119888)) 119865119909119899119909119899+1
(120593(119905
119888)))
forall119905 gt 0
(7)
By replacing 119905 with 119905119888 above we get
119865119909119899119909119899+1
(120593(119905
119888))
ge min(119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899119909119899+1
(120593(119905
1198882 )))
forall119905 gt 0
(8)
Therefore
119865119909119899119909119899+1
(120593 (119905)) ge min(119865119909119899minus1119909119899
(120593(119905
119888))
119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899119909119899+1
(120593(119905
1198882 )))
(9)
and inductively
119865119909119899119909119899+1
(120593 (119905)) ge min(119865119909119899minus1119909119899
(120593(119905
119888))
119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899minus1119909119899
(120593(119905
119888119901))
119865119909119899119909119899+1
(120593(119905
119888119901)))
(10)
for all 119905 gt 0 and for any positive integer 119901 Since 120593 is strictlyincreasing we obtain
119865119909119899119909119899+1
(120593 (119905))
ge min(119865119909119899minus1119909119899
(120593(119905
119888)) 119865119909119899119909119899+1
(120593(119905
119888119901)))
forall119905 gt 0
(11)
for all 119901 By letting 119901 rarr infin it follows that
119865119909119899119909119899+1
(120593 (119905)) ge 119865119909119899minus1119909119899
(120593(119905
119888)) forall119905 gt 0 (12)
Consequently for all 119899 isin N and 119905 gt 0
119865119909119899119909119899+1
(120593 (119905)) ge 11986511990901199091
(120593(119905
119888119899))119899rarrinfin
997888997888997888997888997888rarr 1 (13)
Next let 119898 be a positive integer We prove by inductionon 119899 that
119865119909119899119909119899+119898
(120593 (119888119899119905)) ge min (119865
11990901199091(120593 (119905)) 1198651199090119909119898
(120593 (119905)))
forall119905 gt 0(14)
for all 119899 isin N The case 119899 = 0 is immediate Suppose now thatinequality (14) holds for some 119899 isin N Then
119865119909119899+1119909119899+119898+1
(120593 (119888119899+1
119905)) = 119865119891(119909119899)119891(119909119899+119898)(120593 (119888119899+1
119905))
ge min (119865119909119899119909119899+119898
(120593 (119888119899119905)) 119865119909119899119909119899+1
(120593 (119888119899119905))
119865119909119899+119898119909119899+119898+1
(120593 (119888119899119905))) ge min(119865
11990901199091(120593 (119905))
1198651199090119909119898
(120593 (119905)) 11986511990901199091(120593 (119905)) 11986511990901199091
(120593(119905
119888119898)))
(15)
for all 119905 gt 0 By the monotonicity of 120593 it follows that11986511990901199091
(120593(119905119888119898)) ge 119865
11990901199091(120593(119905)) for all 119905 gt 0 whence
119865119909119899+1119909119899+119898+1
(120593 (119888119899+1
119905))
ge min (11986511990901199091
(120593 (119905)) 1198651199090119909119898(120593 (119905))) forall119905 gt 0
(16)
As such we conclude that inequality (14) holds for all119898 119899 isin N or equivalently
119865119909119899119909119899+119898
(120593 (119905))
ge min(11986511990901199091
(120593(119905
119888119899)) 1198651199090119909119898
(120593(119905
119888119899)))
forall119898 119899 isin N 119905 gt 0
(17)
Now let 120576 gt 0 and 120582 isin (0 1) Given that 120593 is continuousat 0 there exists 119905 gt 0 with 120593(119905) lt 120576 Also since119863
119874(119891119909)isin D+
there exists 1198990 isin N such that 119863119874(119891119909)
(120593(119905119888119899)) gt 1 minus 120582 for all
119899 ge 1198990Thus
119865119909119899119909119899+119898
(120576) ge 119865119909119899119909119899+119898
(120593 (119905))
ge min(11986511990901199091
(120593(119905
119888119899)) 1198651199090119909119898
(120593(119905
119888119899)))
ge 119863119874(119891119909)
(120593(119905
119888119899)) gt 1minus120582
(18)
for all 119899 ge 1198990 119898 isin N and therefore (119909119899)119899is a Cauchy
sequence Accordingly there exists 119909lowast isin 119883 119909lowast = lim119899rarrinfin
119909119899
Next we will prove that 119909lowast is a fixed point of 119891
Specifically we will show that
119865119909lowast119891(119909lowast)(120593 (119905)) ge 119865
119909lowast119891(119909lowast)(120593(
119905
119888)) (19)
for all 119905 gt 0By the contractive condition (5)
119865119909119899+1119891(119909
lowast)(120593 (119905)) ge min(119865
119909119899119909119899+1
(120593(119905
119888))
119865119909lowast119891(119909lowast)(120593(
119905
119888)) 119865119909119899119909lowast (120593(
119905
119888)))
(20)
for all 119899 isin N and 119905 gt 0 If 119905 is such that 119865119909lowast119891(119909lowast)is continuous
at 120593(119905) then (19) follows by letting 119899 rarr infin in the aboveinequality and taking into account relation (13) If 119865
119909lowast119891(119909lowast)
4 Journal of Function Spaces
is not continuous at 120593(119905) let (119905119898)119898be a strictly increasing
sequence converging to 119905 such that 119865119909lowast119891(119909lowast)is continuous at
120593(119905119898) for all119898 isin N As above we infer that 119865
119909lowast119891(119909lowast)(120593(119905119898)) ge
119865119909lowast119891(119909lowast)(120593(119905119898119888)) forall 119898 isin N whence for119898 rarr infin we obtain
(19) By Lemma 8 we conclude that 119909lowast = 119891(119909lowast)
Finally we prove that 119909lowast is the only fixed point of 119891 in119883To that end let 119910 isin 119883 be such that 119891(119910) = 119910 Then using (5)we get
119865119909lowast119910(120593 (119905)) ge min(119865
119909lowast119910(120593(
119905
119888)) 119865119909lowast119909lowast (120593(
119905
119888))
119865119910119910
(120593(119905
119888))) = 119865
119909lowast119910(120593(
119905
119888)) forall119905 gt 0
(21)
Once again by Lemma 8 it follows that 119910 = 119909lowast
Corollary 10 If (119883 119865 119879) is a complete Menger space with 119879
a continuous 119905-norm of Hadzic type and 119891 119883 rarr 119883 satisfiescondition (5) for some 119888 isin (0 1) and some 120593 isin Φ such thatlim119905rarrinfin
(120593(119905)minus120593(119888119905)) = infin then 119891 has a unique fixed point in119883
Proof We will show that for every 119909 isin 119883 119874(119891 119909) isprobabilistically bounded To do so let 119909 isin 119883 be arbitraryand define (119909
119899)119899by 119909119899= 119891119899(119909) for all 119899 ge 0 We will prove by
induction on 119899 that
1198651199090119909119899
(120593 (119905)) ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0 (22)
for all 119899 ge 1 The case 119899 = 1 is trivial Suppose now that therelation holds for some 119899 ge 1 Then
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
1198651199091119909119899+1
(120593 (119888119905))) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
min (1198651199090119909119899
(120593 (119905)) 11986511990901199091(120593 (119905)) 119865119909
119899119909119899+1
(120593 (119905))))
forall119905 gt 0
(23)
Given that
119865119909119899119909119899+1
(120593 (119905)) ge 11986511990901199091
(120593(119905
119888119899)) ge 119865
11990901199091(120593 (119905))
forall119905 gt 0
11986511990901199091
(120593 (119905)) ge 11986511990901199091
(120593 (119905) minus 120593 (119888119905))
ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(24)
from the induction hypothesis we obtain
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))))
= 119879119899+1
(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(25)
which proves our claimNow since lim
119905rarrinfin11986511990901199091
(120593(119905) minus 120593(119888119905)) = 1 and the family119879119899119899is equicontinuous at 1 it follows that119863
119874(119891119909)isin D+
By setting 120593(119905) = 119905 in the above corollary we get thefollowing
Corollary 11 Let (119883 119865 119879) be a complete Menger space with 119879
being a continuous 119905-norm of Hadzic type and let 119891 119883 rarr 119883
be a mapping such that
119865119891(119909)119891(119910) (119905)
ge min(119865119909119910
(119905
119888) 119865119909119891(119909)
(119905
119888) 119865119910119891(119910)
(119905
119888))
(26)
for all 119909 119910 isin 119883 119905 gt 0 Then 119891 has a unique fixed point in 119883
Remark 12 In paper [10] Babacev proved a fixed point resultfor mappings satisfying the contractive condition
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119888))
119865119909119891(119909)
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)) 119865119909119891(119910)
(2120593(119905
119888))
119865119910119891(119909)
(2120593(119905
119888))) forall119905 gt 0
(27)
for some altering distance function 120593 and some 119888 isin (0 1) inMenger spaces with the 119905-norm119879
119872We note that by applying
the triangle inequality
119865119909119891(119910)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
119865119910119891(119909)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
(28)
so this condition essentially reduces to (5) ThereforeTheorem 9 improves the result in [10] as well as Ciricrsquos resultin [7] (which can be obtained from that of Babacev for 120593(119905) =
119905)
Also in [11] Choudhury et al gave the following relatedtheorem
Theorem 13 (see [11]) Let (119883 119865 119879) be a complete Mengerspace with continuous 119905-norm 119879 and let 119886 119887 119888 be positivenumbers with 119886 + 119887 + 119888 lt 1 and 120593 isin Φ Suppose that119891 119883 rarr 119883 satisfies the inequality
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(1199051119886))
119865119909119891(119909)
(120593(1199052119887)) 119865119910119891(119910)
(120593(1199053119888)))
(29)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Then 119891 has a unique fixed point
As indicated in [11] a mapping satisfying the contractivecondition (29) must also verify our condition (5) Namely
Journal of Function Spaces 5
suppose that (29) holds Let 120576 = (1 minus (119886 + 119887 + 119888))3 isin (0 1)and let 1199051 = (119886 + 120576)119905 1199052 = (119887 + 120576)119905 1199053 = (119888 + 120576)119905 It follows that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905 (119886 + 120576)
119886))
119865119909119891(119909)
(120593(119905 (119887 + 120576)
119887)) 119865
119910119891(119910)(120593(
119905 (119888 + 120576)
119888)))
forall119905 gt 0
(30)
Due to the monotonicity of 120593 the above relation implies that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119896))
119865119909119891(119909)
(120593(119905
119896)) 119865119910119891(119910)
(120593(119905
119896)))
(31)
for all 119905 gt 0 where 119896 = max(119886(119886 + 120576) 119887(119887 + 120576) 119888(119888 + 120576)) isin
(0 1)Note that Theorem 13 only requires that the 119905-norm by
which the space is endowed is continuous Unfortunatelywe can show that this assumption alone is not sufficient toguarantee the existence of fixed points for contractions of thistype
Specifically let (119883 119865 119879119871) be a completeMenger space and
let 119891 be a Sehgal contraction on119883 with contraction constant119896 lt 13 Then
119865119891(119909)119891(119910) (119905) ge 119865
119909119910(119905
119896)
ge min(119865119909119910
(119905
119896) 119865119909119891(119909)
(119905
119896) 119865119910119891(119910)
(119905
119896))
ge min(119865119909119910
(1199051119896) 119865119909119891(119909)
(1199052119896) 119865119910119891(119910)
(1199053119896))
(32)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Thus 119891 satisfies the conditions of Theorem 13 with 119886 = 119887 =
119888 = 119896 and 120593(119905) = 119905 However a well-known counterexampleof Sherwood ([17] Corollary 1 of Theorem 35) shows thatthere exist Sehgal contractions on complete Menger spacesendowed with the 119905-norm 119879
119871having no fixed point
It should be mentioned that a similar observation regard-ing continuity can be made with respect to Theorem 31 in[18] where the class of contractions considered also includesSehgal contractions
Finally we illustrate the applicability of Theorem 9 withthe following example
Example 14 Let 119883 = [0 1] and 119879(119886 119887) = 119886119887 Define 119865119909119910(119905) =
(119905(119905+1))|119909minus119910| for all119909 119910 isin 119883 and 119905 gt 0 (119883 119865 119879) is a completeMenger space We will only show that the triangle inequalityis verified
Assume that 119905 gt 119904 gt 0 and 119909 119910 119911 isin 119883 Since the function119905(119905 + 1) is increasing it holds that
119879 (119865119909119910 (119905) 119865119910119911 (119904)) = (
119905
119905 + 1)
|119909minus119910|
(119904
119904 + 1)
|119910minus119911|
le (119905
119905 + 1)
|119909minus119910|+|119910minus119911|
le (119905
119905 + 1)
|119909minus119911|
le (119905 + 119904
119905 + 119904 + 1)
|119909minus119911|
= 119865119909119911 (119905 + 119904)
(33)
Let 120593(119905) = 1199052(2119905 + 1) for all 119905 gt 0 119888 = 12 and
119891 (119909) =
119909
2 119909 isin [0 1)
0 119909 = 1(34)
One can easily check that 120593 is an altering distance functionand that119874(119891 119909) is probabilistically bounded for every 119909 isin 119883
We will prove that condition (5) ofTheorem 9 is satisfiedThe following three cases are possible
(1) If 119909 119910 isin [0 1) then for all 119905 gt 0 we have
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
2|119891(119909)minus119891(119910)|= (
119905
119905 + 1)
|119909minus119910|
ge (2119905
2119905 + 1)
2|119909minus119910|= 119865119909119910
(120593 (2119905))
(35)
(2) If 119909 = 119910 = 1 then
119865119891(119909)119891(119910)
(120593 (119905)) = 1 = 119865119909119910
(120593 (2119905)) (36)
for all 119905 gt 0(3) If 119909 isin [0 1) and 119910 = 1 then for all 119905 gt 0
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
119909
ge119905
119905 + 1ge (
21199052119905 + 1
)
2
= (2119905
2119905 + 1)
2|119910minus119891(119910)|= 119865119910119891(119910)
(120593 (2119905))
(37)
Thus the condition (5) is satisfied in this case as well
However note that by setting119909 = 23 and119910 = 1we obtain
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
23lt (
21199052119905 + 1
)
23
= 119865119909119910
(120593 (2119905))(38)
for all 119905 gt 0 therefore 119891 does not satisfy the strongercondition (4)
By applying Theorem 9 we conclude that the function 119891
has a unique fixed point It is easy to see that this point is119909 = 0
6 Journal of Function Spaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Menger ldquoStatistical metricsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 pp535ndash537 1942
[2] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Series in Probability and AppliedMathematics North-Holland Publishing New York NY USA 1983
[3] G Constantin and I Istratescu Elements of Probabilistic Analy-sis with Applications Kluwer Academic Publishers 1989
[4] S Romaguera A Sapena and P Tirado ldquoThe Banach fixedpoint theorem in fuzzy quasi-metric spaces with application tothe domain of wordsrdquoTopology and Its Applications vol 154 no10 pp 2196ndash2203 2007
[5] S Romaguera and P Tirado ldquoThe complexity probabilisticquasi-metric spacerdquo Journal of Mathematical Analysis andApplications vol 376 no 2 pp 732ndash740 2011
[6] J Sun X Wu V Palade W Fang C-H Lai and W XuldquoConvergence analysis and improvements of quantum-behavedparticle swarm optimizationrdquo Information Sciences vol 193 pp81ndash103 2012
[7] L B Ciric ldquoOn fixed points of generalized contractions onprobabilistic metric spacesrdquo Publications de lrsquoInstitutMathematique (Beograd) vol 18 no 32 pp 71ndash78 1975
[8] B S Choudhury and K Das ldquoA new contraction principle inMenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008
[9] D Mihet ldquoAltering distances in probabilistic Menger spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no7-8 pp 2734ndash2738 2009
[10] N A Babacev ldquoNonlinear generalized contractions on MengerPM spacesrdquo Applicable Analysis and Discrete Mathematics vol6 no 2 pp 257ndash264 2012
[11] B S Choudhury K Das and P N Dutta ldquoA fixed point resultin Menger spaces using a real functionrdquo Acta MathematicaHungarica vol 122 no 3 pp 203ndash216 2009
[12] O Hadzic ldquoOn the (120576120582)-topology of probabilistic locallyconvex spacesrdquo Glasnik Matematicki Serija III vol 13 no 332 pp 293ndash297 1978
[13] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD thesis] Wayne State University 1966
[14] O Hadzic and E Pap Fixed Point Theory in Probabilistic MetricSpaces Kluwer Academic Publishers 2001
[15] V Radu ldquoOn the t-norms of Hadzic type and fixed points inPM-spacesrdquo Review of Research (Novi Sad) vol 13 pp 81ndash861983
[16] M Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984
[17] H Sherwood ldquoComplete probabilistic metric spacesrdquoZeitschrift fur Wahrscheinlichkeitstheorie und VerwandteGebiete vol 20 no 2 pp 117ndash128 1971
[18] T Dosenovic P Kumam D Gopal D K Patel and ATakaci ldquoOn fixed point theorems involving altering distances inMenger probabilistic metric spacesrdquo Journal of Inequalities andApplications vol 2013 article 576 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
(Here and in the following 119865(119909 119910) will be denoted by119865119909119910)
Let (119883 119865 119879) be a Menger space such that sup119909isin(01)
119879(119909 119909) = 1 The family 119880(120576 120582)120576gt0120582isin(01) where
119880 (120576 120582) = (119909 119910) isin119883times119883 119865119909119910 (120576) gt 1minus120582 (1)
is a base for a Hausdorff uniformity on 119883 named stronguniformity The corresponding strong topology on 119883 isintroduced by the family of neighbourhoods of 119909 isin 119883N
119909=
119873119909(120576 120582)
120576gt0120582isin(01) where119873119909(120576 120582) = 119910 isin 119883 119865119909119910(120576) gt 1minus120582
and this topology is metrizable [2]
Definition 3 Let (119883 119865 119879) be a Menger space A sequence(119909119899)119899in119883 is said to be
(i) Cauchy if for any 120576 gt 0 120582 isin (0 1) there exists 1198990 isin N
such that 119865119909119899119909119898
(120576) gt 1 minus 120582 for all 119898 119899 ge 1198990(ii) convergent to 119909 isin 119883 if for any 120576 gt 0 120582 isin (0 1) there
exists 1198990 isin N such that 119865119909119899119909(120576) gt 1 minus 120582 for all 119899 ge 1198990
119883 is said to be complete if every Cauchy sequence (119909119899)119899sub 119883
is convergent in 119883
If the 119905-norm 119879 is continuous and the sequences (119909119899)119899
(119910119899)119899
sub 119883 converge respectively to 119909 and 119910 isin 119883 then119865119909119899119910119899
(119905) converges to 119865119909119910(119905) for each continuity point 119905 of 119865
119909119910
[2]
Definition 4 Given a set119860 sub 119883 the probabilistic diameter of119860 is the mapping119863
119860 R rarr [0 1] defined by
119863119860 (119905) =
sup119904lt119905
inf119909119910isin119860
119865119909119910 (119904) 119905 gt 0
0 119905 le 0(2)
119860 is said to be probabilistically bounded if119863119860
isin D+
The notion of contraction in a Menger space was intro-duced by Sehgal in [13]
Definition 5 (see [13]) Let (119883 119865 119879) be a Menger space Amapping 119891 119883 rarr 119883 is said to be a probabilistic contraction(or Sehgal contraction) if there exists 119888 isin (0 1) such that
119865119891(119909)119891(119910) (119905) ge 119865
119909119910(119905
119888) forall119909 119910 isin 119883 119905 gt 0 (3)
Many significant contributions to the development offixed point theory in probabilistic structures can be found inmonograph [14] It should be pointed out that the triangularnorm by which the space is endowed plays a key role in theexistence of fixed points of probabilistic contractions It wasshown by Radu [15] that the largest class of continuous 119905-norms 119879 with the property that every Sehgal contraction ona complete Menger space (119883 119865 119879) has a unique fixed point isthat of 119905-norms of Hadzic type
The idea of using altering distance functions in order toobtain more general contractive conditions first appears in[16] in the setting of metric spaces The corresponding con-cept of generalized probabilistic contraction was introducedby Choudhury and Das in [8] as follows
Definition 6 (see [8]) A mapping 120593 [0infin) rarr [0infin) issaid to belong to the class Φ if it satisfies
(i) 120593(119905) = 0 iff 119905 = 0
(ii) 120593 is strictly increasing and lim119905rarrinfin
120593(119905) = infin
(iii) 120593 is continuous at 119905 = 0 and left continuous on (0infin)
The mappings 120593 isin Φ will be called altering distancefunctions
Definition 7 (cf [8]) Let (119883 119865 119879) be a Menger space Themapping 119891 119883 rarr 119883 is said to be a generalized probabilisticcontraction of Choudhury-Das type if there exist 120593 isin Φ and119888 isin (0 1) such that
119865119891(119909)119891(119910)
(120593 (119905)) ge 119865119909119910
(120593(119905
119888)) forall119909 119910 isin 119883 119905 gt 0 (4)
It was proved in [8] that such contractions on a completeMenger space endowed with the strongest 119905-norm 119879
119872have a
unique fixed point The result was subsequently generalizedby Mihet [9] for the case of arbitrary continuous 119905-normsunder the supplementary assumption that the orbit of themapping 119891 at some 119909 isin 119883 is probabilistically bounded
Our aim is to prove a fixed point result for mappingssatisfying the more general contractive condition
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119888)) 119865119909119891(119909)
(120593(119905
119888))
119865119910119891(119910)
(120593(119905
119888))) forall119909 119910 isin 119883 119905 gt 0
(5)
for some 120593 isin Φ and 119888 isin (0 1)
2 Main Results
In order to prove our results we will need the followinglemma from [10]
Lemma 8 (see [10]) Let (119883 119865 119879) be a Menger space 120593 isin Φ
and 119888 isin (0 1) If 119909 119910 isin 119883 are such that
119865119909119910
(120593 (119905)) le 119865119909119910
(120593(119905
119888)) forall119905 gt 0 (6)
then 119909 = 119910
For each 119909 isin 119883 we will denote by119874(119891 119909) the orbit of themapping 119891 at 119909 that is 119874(119891 119909) = 119891
119899(119909) 119899 isin N
Theorem 9 Let (119883 119865 119879) be a complete Menger space with119879 a continuous 119905-norm Suppose 119891 119883 rarr 119883 is a mappingsatisfying the contractive condition (5) for some 120593 isin Φ and 119888 isin
(0 1) If there exists 119909 isin 119883 such that119874(119891 119909) is probabilisticallybounded then 119891 has a unique fixed point in 119883
Proof Let 119909 be as in the statement of the theorem We willshow that the sequence (119909
119899)119899 119909119899= 119891119899(119909) is Cauchy
Journal of Function Spaces 3
From (5) it follows that
119865119909119899119909119899+1
(120593 (119905))
ge min(119865119909119899minus1119909119899
(120593(119905
119888)) 119865119909119899119909119899+1
(120593(119905
119888)))
forall119905 gt 0
(7)
By replacing 119905 with 119905119888 above we get
119865119909119899119909119899+1
(120593(119905
119888))
ge min(119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899119909119899+1
(120593(119905
1198882 )))
forall119905 gt 0
(8)
Therefore
119865119909119899119909119899+1
(120593 (119905)) ge min(119865119909119899minus1119909119899
(120593(119905
119888))
119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899119909119899+1
(120593(119905
1198882 )))
(9)
and inductively
119865119909119899119909119899+1
(120593 (119905)) ge min(119865119909119899minus1119909119899
(120593(119905
119888))
119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899minus1119909119899
(120593(119905
119888119901))
119865119909119899119909119899+1
(120593(119905
119888119901)))
(10)
for all 119905 gt 0 and for any positive integer 119901 Since 120593 is strictlyincreasing we obtain
119865119909119899119909119899+1
(120593 (119905))
ge min(119865119909119899minus1119909119899
(120593(119905
119888)) 119865119909119899119909119899+1
(120593(119905
119888119901)))
forall119905 gt 0
(11)
for all 119901 By letting 119901 rarr infin it follows that
119865119909119899119909119899+1
(120593 (119905)) ge 119865119909119899minus1119909119899
(120593(119905
119888)) forall119905 gt 0 (12)
Consequently for all 119899 isin N and 119905 gt 0
119865119909119899119909119899+1
(120593 (119905)) ge 11986511990901199091
(120593(119905
119888119899))119899rarrinfin
997888997888997888997888997888rarr 1 (13)
Next let 119898 be a positive integer We prove by inductionon 119899 that
119865119909119899119909119899+119898
(120593 (119888119899119905)) ge min (119865
11990901199091(120593 (119905)) 1198651199090119909119898
(120593 (119905)))
forall119905 gt 0(14)
for all 119899 isin N The case 119899 = 0 is immediate Suppose now thatinequality (14) holds for some 119899 isin N Then
119865119909119899+1119909119899+119898+1
(120593 (119888119899+1
119905)) = 119865119891(119909119899)119891(119909119899+119898)(120593 (119888119899+1
119905))
ge min (119865119909119899119909119899+119898
(120593 (119888119899119905)) 119865119909119899119909119899+1
(120593 (119888119899119905))
119865119909119899+119898119909119899+119898+1
(120593 (119888119899119905))) ge min(119865
11990901199091(120593 (119905))
1198651199090119909119898
(120593 (119905)) 11986511990901199091(120593 (119905)) 11986511990901199091
(120593(119905
119888119898)))
(15)
for all 119905 gt 0 By the monotonicity of 120593 it follows that11986511990901199091
(120593(119905119888119898)) ge 119865
11990901199091(120593(119905)) for all 119905 gt 0 whence
119865119909119899+1119909119899+119898+1
(120593 (119888119899+1
119905))
ge min (11986511990901199091
(120593 (119905)) 1198651199090119909119898(120593 (119905))) forall119905 gt 0
(16)
As such we conclude that inequality (14) holds for all119898 119899 isin N or equivalently
119865119909119899119909119899+119898
(120593 (119905))
ge min(11986511990901199091
(120593(119905
119888119899)) 1198651199090119909119898
(120593(119905
119888119899)))
forall119898 119899 isin N 119905 gt 0
(17)
Now let 120576 gt 0 and 120582 isin (0 1) Given that 120593 is continuousat 0 there exists 119905 gt 0 with 120593(119905) lt 120576 Also since119863
119874(119891119909)isin D+
there exists 1198990 isin N such that 119863119874(119891119909)
(120593(119905119888119899)) gt 1 minus 120582 for all
119899 ge 1198990Thus
119865119909119899119909119899+119898
(120576) ge 119865119909119899119909119899+119898
(120593 (119905))
ge min(11986511990901199091
(120593(119905
119888119899)) 1198651199090119909119898
(120593(119905
119888119899)))
ge 119863119874(119891119909)
(120593(119905
119888119899)) gt 1minus120582
(18)
for all 119899 ge 1198990 119898 isin N and therefore (119909119899)119899is a Cauchy
sequence Accordingly there exists 119909lowast isin 119883 119909lowast = lim119899rarrinfin
119909119899
Next we will prove that 119909lowast is a fixed point of 119891
Specifically we will show that
119865119909lowast119891(119909lowast)(120593 (119905)) ge 119865
119909lowast119891(119909lowast)(120593(
119905
119888)) (19)
for all 119905 gt 0By the contractive condition (5)
119865119909119899+1119891(119909
lowast)(120593 (119905)) ge min(119865
119909119899119909119899+1
(120593(119905
119888))
119865119909lowast119891(119909lowast)(120593(
119905
119888)) 119865119909119899119909lowast (120593(
119905
119888)))
(20)
for all 119899 isin N and 119905 gt 0 If 119905 is such that 119865119909lowast119891(119909lowast)is continuous
at 120593(119905) then (19) follows by letting 119899 rarr infin in the aboveinequality and taking into account relation (13) If 119865
119909lowast119891(119909lowast)
4 Journal of Function Spaces
is not continuous at 120593(119905) let (119905119898)119898be a strictly increasing
sequence converging to 119905 such that 119865119909lowast119891(119909lowast)is continuous at
120593(119905119898) for all119898 isin N As above we infer that 119865
119909lowast119891(119909lowast)(120593(119905119898)) ge
119865119909lowast119891(119909lowast)(120593(119905119898119888)) forall 119898 isin N whence for119898 rarr infin we obtain
(19) By Lemma 8 we conclude that 119909lowast = 119891(119909lowast)
Finally we prove that 119909lowast is the only fixed point of 119891 in119883To that end let 119910 isin 119883 be such that 119891(119910) = 119910 Then using (5)we get
119865119909lowast119910(120593 (119905)) ge min(119865
119909lowast119910(120593(
119905
119888)) 119865119909lowast119909lowast (120593(
119905
119888))
119865119910119910
(120593(119905
119888))) = 119865
119909lowast119910(120593(
119905
119888)) forall119905 gt 0
(21)
Once again by Lemma 8 it follows that 119910 = 119909lowast
Corollary 10 If (119883 119865 119879) is a complete Menger space with 119879
a continuous 119905-norm of Hadzic type and 119891 119883 rarr 119883 satisfiescondition (5) for some 119888 isin (0 1) and some 120593 isin Φ such thatlim119905rarrinfin
(120593(119905)minus120593(119888119905)) = infin then 119891 has a unique fixed point in119883
Proof We will show that for every 119909 isin 119883 119874(119891 119909) isprobabilistically bounded To do so let 119909 isin 119883 be arbitraryand define (119909
119899)119899by 119909119899= 119891119899(119909) for all 119899 ge 0 We will prove by
induction on 119899 that
1198651199090119909119899
(120593 (119905)) ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0 (22)
for all 119899 ge 1 The case 119899 = 1 is trivial Suppose now that therelation holds for some 119899 ge 1 Then
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
1198651199091119909119899+1
(120593 (119888119905))) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
min (1198651199090119909119899
(120593 (119905)) 11986511990901199091(120593 (119905)) 119865119909
119899119909119899+1
(120593 (119905))))
forall119905 gt 0
(23)
Given that
119865119909119899119909119899+1
(120593 (119905)) ge 11986511990901199091
(120593(119905
119888119899)) ge 119865
11990901199091(120593 (119905))
forall119905 gt 0
11986511990901199091
(120593 (119905)) ge 11986511990901199091
(120593 (119905) minus 120593 (119888119905))
ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(24)
from the induction hypothesis we obtain
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))))
= 119879119899+1
(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(25)
which proves our claimNow since lim
119905rarrinfin11986511990901199091
(120593(119905) minus 120593(119888119905)) = 1 and the family119879119899119899is equicontinuous at 1 it follows that119863
119874(119891119909)isin D+
By setting 120593(119905) = 119905 in the above corollary we get thefollowing
Corollary 11 Let (119883 119865 119879) be a complete Menger space with 119879
being a continuous 119905-norm of Hadzic type and let 119891 119883 rarr 119883
be a mapping such that
119865119891(119909)119891(119910) (119905)
ge min(119865119909119910
(119905
119888) 119865119909119891(119909)
(119905
119888) 119865119910119891(119910)
(119905
119888))
(26)
for all 119909 119910 isin 119883 119905 gt 0 Then 119891 has a unique fixed point in 119883
Remark 12 In paper [10] Babacev proved a fixed point resultfor mappings satisfying the contractive condition
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119888))
119865119909119891(119909)
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)) 119865119909119891(119910)
(2120593(119905
119888))
119865119910119891(119909)
(2120593(119905
119888))) forall119905 gt 0
(27)
for some altering distance function 120593 and some 119888 isin (0 1) inMenger spaces with the 119905-norm119879
119872We note that by applying
the triangle inequality
119865119909119891(119910)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
119865119910119891(119909)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
(28)
so this condition essentially reduces to (5) ThereforeTheorem 9 improves the result in [10] as well as Ciricrsquos resultin [7] (which can be obtained from that of Babacev for 120593(119905) =
119905)
Also in [11] Choudhury et al gave the following relatedtheorem
Theorem 13 (see [11]) Let (119883 119865 119879) be a complete Mengerspace with continuous 119905-norm 119879 and let 119886 119887 119888 be positivenumbers with 119886 + 119887 + 119888 lt 1 and 120593 isin Φ Suppose that119891 119883 rarr 119883 satisfies the inequality
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(1199051119886))
119865119909119891(119909)
(120593(1199052119887)) 119865119910119891(119910)
(120593(1199053119888)))
(29)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Then 119891 has a unique fixed point
As indicated in [11] a mapping satisfying the contractivecondition (29) must also verify our condition (5) Namely
Journal of Function Spaces 5
suppose that (29) holds Let 120576 = (1 minus (119886 + 119887 + 119888))3 isin (0 1)and let 1199051 = (119886 + 120576)119905 1199052 = (119887 + 120576)119905 1199053 = (119888 + 120576)119905 It follows that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905 (119886 + 120576)
119886))
119865119909119891(119909)
(120593(119905 (119887 + 120576)
119887)) 119865
119910119891(119910)(120593(
119905 (119888 + 120576)
119888)))
forall119905 gt 0
(30)
Due to the monotonicity of 120593 the above relation implies that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119896))
119865119909119891(119909)
(120593(119905
119896)) 119865119910119891(119910)
(120593(119905
119896)))
(31)
for all 119905 gt 0 where 119896 = max(119886(119886 + 120576) 119887(119887 + 120576) 119888(119888 + 120576)) isin
(0 1)Note that Theorem 13 only requires that the 119905-norm by
which the space is endowed is continuous Unfortunatelywe can show that this assumption alone is not sufficient toguarantee the existence of fixed points for contractions of thistype
Specifically let (119883 119865 119879119871) be a completeMenger space and
let 119891 be a Sehgal contraction on119883 with contraction constant119896 lt 13 Then
119865119891(119909)119891(119910) (119905) ge 119865
119909119910(119905
119896)
ge min(119865119909119910
(119905
119896) 119865119909119891(119909)
(119905
119896) 119865119910119891(119910)
(119905
119896))
ge min(119865119909119910
(1199051119896) 119865119909119891(119909)
(1199052119896) 119865119910119891(119910)
(1199053119896))
(32)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Thus 119891 satisfies the conditions of Theorem 13 with 119886 = 119887 =
119888 = 119896 and 120593(119905) = 119905 However a well-known counterexampleof Sherwood ([17] Corollary 1 of Theorem 35) shows thatthere exist Sehgal contractions on complete Menger spacesendowed with the 119905-norm 119879
119871having no fixed point
It should be mentioned that a similar observation regard-ing continuity can be made with respect to Theorem 31 in[18] where the class of contractions considered also includesSehgal contractions
Finally we illustrate the applicability of Theorem 9 withthe following example
Example 14 Let 119883 = [0 1] and 119879(119886 119887) = 119886119887 Define 119865119909119910(119905) =
(119905(119905+1))|119909minus119910| for all119909 119910 isin 119883 and 119905 gt 0 (119883 119865 119879) is a completeMenger space We will only show that the triangle inequalityis verified
Assume that 119905 gt 119904 gt 0 and 119909 119910 119911 isin 119883 Since the function119905(119905 + 1) is increasing it holds that
119879 (119865119909119910 (119905) 119865119910119911 (119904)) = (
119905
119905 + 1)
|119909minus119910|
(119904
119904 + 1)
|119910minus119911|
le (119905
119905 + 1)
|119909minus119910|+|119910minus119911|
le (119905
119905 + 1)
|119909minus119911|
le (119905 + 119904
119905 + 119904 + 1)
|119909minus119911|
= 119865119909119911 (119905 + 119904)
(33)
Let 120593(119905) = 1199052(2119905 + 1) for all 119905 gt 0 119888 = 12 and
119891 (119909) =
119909
2 119909 isin [0 1)
0 119909 = 1(34)
One can easily check that 120593 is an altering distance functionand that119874(119891 119909) is probabilistically bounded for every 119909 isin 119883
We will prove that condition (5) ofTheorem 9 is satisfiedThe following three cases are possible
(1) If 119909 119910 isin [0 1) then for all 119905 gt 0 we have
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
2|119891(119909)minus119891(119910)|= (
119905
119905 + 1)
|119909minus119910|
ge (2119905
2119905 + 1)
2|119909minus119910|= 119865119909119910
(120593 (2119905))
(35)
(2) If 119909 = 119910 = 1 then
119865119891(119909)119891(119910)
(120593 (119905)) = 1 = 119865119909119910
(120593 (2119905)) (36)
for all 119905 gt 0(3) If 119909 isin [0 1) and 119910 = 1 then for all 119905 gt 0
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
119909
ge119905
119905 + 1ge (
21199052119905 + 1
)
2
= (2119905
2119905 + 1)
2|119910minus119891(119910)|= 119865119910119891(119910)
(120593 (2119905))
(37)
Thus the condition (5) is satisfied in this case as well
However note that by setting119909 = 23 and119910 = 1we obtain
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
23lt (
21199052119905 + 1
)
23
= 119865119909119910
(120593 (2119905))(38)
for all 119905 gt 0 therefore 119891 does not satisfy the strongercondition (4)
By applying Theorem 9 we conclude that the function 119891
has a unique fixed point It is easy to see that this point is119909 = 0
6 Journal of Function Spaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Menger ldquoStatistical metricsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 pp535ndash537 1942
[2] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Series in Probability and AppliedMathematics North-Holland Publishing New York NY USA 1983
[3] G Constantin and I Istratescu Elements of Probabilistic Analy-sis with Applications Kluwer Academic Publishers 1989
[4] S Romaguera A Sapena and P Tirado ldquoThe Banach fixedpoint theorem in fuzzy quasi-metric spaces with application tothe domain of wordsrdquoTopology and Its Applications vol 154 no10 pp 2196ndash2203 2007
[5] S Romaguera and P Tirado ldquoThe complexity probabilisticquasi-metric spacerdquo Journal of Mathematical Analysis andApplications vol 376 no 2 pp 732ndash740 2011
[6] J Sun X Wu V Palade W Fang C-H Lai and W XuldquoConvergence analysis and improvements of quantum-behavedparticle swarm optimizationrdquo Information Sciences vol 193 pp81ndash103 2012
[7] L B Ciric ldquoOn fixed points of generalized contractions onprobabilistic metric spacesrdquo Publications de lrsquoInstitutMathematique (Beograd) vol 18 no 32 pp 71ndash78 1975
[8] B S Choudhury and K Das ldquoA new contraction principle inMenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008
[9] D Mihet ldquoAltering distances in probabilistic Menger spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no7-8 pp 2734ndash2738 2009
[10] N A Babacev ldquoNonlinear generalized contractions on MengerPM spacesrdquo Applicable Analysis and Discrete Mathematics vol6 no 2 pp 257ndash264 2012
[11] B S Choudhury K Das and P N Dutta ldquoA fixed point resultin Menger spaces using a real functionrdquo Acta MathematicaHungarica vol 122 no 3 pp 203ndash216 2009
[12] O Hadzic ldquoOn the (120576120582)-topology of probabilistic locallyconvex spacesrdquo Glasnik Matematicki Serija III vol 13 no 332 pp 293ndash297 1978
[13] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD thesis] Wayne State University 1966
[14] O Hadzic and E Pap Fixed Point Theory in Probabilistic MetricSpaces Kluwer Academic Publishers 2001
[15] V Radu ldquoOn the t-norms of Hadzic type and fixed points inPM-spacesrdquo Review of Research (Novi Sad) vol 13 pp 81ndash861983
[16] M Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984
[17] H Sherwood ldquoComplete probabilistic metric spacesrdquoZeitschrift fur Wahrscheinlichkeitstheorie und VerwandteGebiete vol 20 no 2 pp 117ndash128 1971
[18] T Dosenovic P Kumam D Gopal D K Patel and ATakaci ldquoOn fixed point theorems involving altering distances inMenger probabilistic metric spacesrdquo Journal of Inequalities andApplications vol 2013 article 576 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 3
From (5) it follows that
119865119909119899119909119899+1
(120593 (119905))
ge min(119865119909119899minus1119909119899
(120593(119905
119888)) 119865119909119899119909119899+1
(120593(119905
119888)))
forall119905 gt 0
(7)
By replacing 119905 with 119905119888 above we get
119865119909119899119909119899+1
(120593(119905
119888))
ge min(119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899119909119899+1
(120593(119905
1198882 )))
forall119905 gt 0
(8)
Therefore
119865119909119899119909119899+1
(120593 (119905)) ge min(119865119909119899minus1119909119899
(120593(119905
119888))
119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899119909119899+1
(120593(119905
1198882 )))
(9)
and inductively
119865119909119899119909119899+1
(120593 (119905)) ge min(119865119909119899minus1119909119899
(120593(119905
119888))
119865119909119899minus1119909119899
(120593(119905
1198882 )) 119865
119909119899minus1119909119899
(120593(119905
119888119901))
119865119909119899119909119899+1
(120593(119905
119888119901)))
(10)
for all 119905 gt 0 and for any positive integer 119901 Since 120593 is strictlyincreasing we obtain
119865119909119899119909119899+1
(120593 (119905))
ge min(119865119909119899minus1119909119899
(120593(119905
119888)) 119865119909119899119909119899+1
(120593(119905
119888119901)))
forall119905 gt 0
(11)
for all 119901 By letting 119901 rarr infin it follows that
119865119909119899119909119899+1
(120593 (119905)) ge 119865119909119899minus1119909119899
(120593(119905
119888)) forall119905 gt 0 (12)
Consequently for all 119899 isin N and 119905 gt 0
119865119909119899119909119899+1
(120593 (119905)) ge 11986511990901199091
(120593(119905
119888119899))119899rarrinfin
997888997888997888997888997888rarr 1 (13)
Next let 119898 be a positive integer We prove by inductionon 119899 that
119865119909119899119909119899+119898
(120593 (119888119899119905)) ge min (119865
11990901199091(120593 (119905)) 1198651199090119909119898
(120593 (119905)))
forall119905 gt 0(14)
for all 119899 isin N The case 119899 = 0 is immediate Suppose now thatinequality (14) holds for some 119899 isin N Then
119865119909119899+1119909119899+119898+1
(120593 (119888119899+1
119905)) = 119865119891(119909119899)119891(119909119899+119898)(120593 (119888119899+1
119905))
ge min (119865119909119899119909119899+119898
(120593 (119888119899119905)) 119865119909119899119909119899+1
(120593 (119888119899119905))
119865119909119899+119898119909119899+119898+1
(120593 (119888119899119905))) ge min(119865
11990901199091(120593 (119905))
1198651199090119909119898
(120593 (119905)) 11986511990901199091(120593 (119905)) 11986511990901199091
(120593(119905
119888119898)))
(15)
for all 119905 gt 0 By the monotonicity of 120593 it follows that11986511990901199091
(120593(119905119888119898)) ge 119865
11990901199091(120593(119905)) for all 119905 gt 0 whence
119865119909119899+1119909119899+119898+1
(120593 (119888119899+1
119905))
ge min (11986511990901199091
(120593 (119905)) 1198651199090119909119898(120593 (119905))) forall119905 gt 0
(16)
As such we conclude that inequality (14) holds for all119898 119899 isin N or equivalently
119865119909119899119909119899+119898
(120593 (119905))
ge min(11986511990901199091
(120593(119905
119888119899)) 1198651199090119909119898
(120593(119905
119888119899)))
forall119898 119899 isin N 119905 gt 0
(17)
Now let 120576 gt 0 and 120582 isin (0 1) Given that 120593 is continuousat 0 there exists 119905 gt 0 with 120593(119905) lt 120576 Also since119863
119874(119891119909)isin D+
there exists 1198990 isin N such that 119863119874(119891119909)
(120593(119905119888119899)) gt 1 minus 120582 for all
119899 ge 1198990Thus
119865119909119899119909119899+119898
(120576) ge 119865119909119899119909119899+119898
(120593 (119905))
ge min(11986511990901199091
(120593(119905
119888119899)) 1198651199090119909119898
(120593(119905
119888119899)))
ge 119863119874(119891119909)
(120593(119905
119888119899)) gt 1minus120582
(18)
for all 119899 ge 1198990 119898 isin N and therefore (119909119899)119899is a Cauchy
sequence Accordingly there exists 119909lowast isin 119883 119909lowast = lim119899rarrinfin
119909119899
Next we will prove that 119909lowast is a fixed point of 119891
Specifically we will show that
119865119909lowast119891(119909lowast)(120593 (119905)) ge 119865
119909lowast119891(119909lowast)(120593(
119905
119888)) (19)
for all 119905 gt 0By the contractive condition (5)
119865119909119899+1119891(119909
lowast)(120593 (119905)) ge min(119865
119909119899119909119899+1
(120593(119905
119888))
119865119909lowast119891(119909lowast)(120593(
119905
119888)) 119865119909119899119909lowast (120593(
119905
119888)))
(20)
for all 119899 isin N and 119905 gt 0 If 119905 is such that 119865119909lowast119891(119909lowast)is continuous
at 120593(119905) then (19) follows by letting 119899 rarr infin in the aboveinequality and taking into account relation (13) If 119865
119909lowast119891(119909lowast)
4 Journal of Function Spaces
is not continuous at 120593(119905) let (119905119898)119898be a strictly increasing
sequence converging to 119905 such that 119865119909lowast119891(119909lowast)is continuous at
120593(119905119898) for all119898 isin N As above we infer that 119865
119909lowast119891(119909lowast)(120593(119905119898)) ge
119865119909lowast119891(119909lowast)(120593(119905119898119888)) forall 119898 isin N whence for119898 rarr infin we obtain
(19) By Lemma 8 we conclude that 119909lowast = 119891(119909lowast)
Finally we prove that 119909lowast is the only fixed point of 119891 in119883To that end let 119910 isin 119883 be such that 119891(119910) = 119910 Then using (5)we get
119865119909lowast119910(120593 (119905)) ge min(119865
119909lowast119910(120593(
119905
119888)) 119865119909lowast119909lowast (120593(
119905
119888))
119865119910119910
(120593(119905
119888))) = 119865
119909lowast119910(120593(
119905
119888)) forall119905 gt 0
(21)
Once again by Lemma 8 it follows that 119910 = 119909lowast
Corollary 10 If (119883 119865 119879) is a complete Menger space with 119879
a continuous 119905-norm of Hadzic type and 119891 119883 rarr 119883 satisfiescondition (5) for some 119888 isin (0 1) and some 120593 isin Φ such thatlim119905rarrinfin
(120593(119905)minus120593(119888119905)) = infin then 119891 has a unique fixed point in119883
Proof We will show that for every 119909 isin 119883 119874(119891 119909) isprobabilistically bounded To do so let 119909 isin 119883 be arbitraryand define (119909
119899)119899by 119909119899= 119891119899(119909) for all 119899 ge 0 We will prove by
induction on 119899 that
1198651199090119909119899
(120593 (119905)) ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0 (22)
for all 119899 ge 1 The case 119899 = 1 is trivial Suppose now that therelation holds for some 119899 ge 1 Then
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
1198651199091119909119899+1
(120593 (119888119905))) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
min (1198651199090119909119899
(120593 (119905)) 11986511990901199091(120593 (119905)) 119865119909
119899119909119899+1
(120593 (119905))))
forall119905 gt 0
(23)
Given that
119865119909119899119909119899+1
(120593 (119905)) ge 11986511990901199091
(120593(119905
119888119899)) ge 119865
11990901199091(120593 (119905))
forall119905 gt 0
11986511990901199091
(120593 (119905)) ge 11986511990901199091
(120593 (119905) minus 120593 (119888119905))
ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(24)
from the induction hypothesis we obtain
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))))
= 119879119899+1
(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(25)
which proves our claimNow since lim
119905rarrinfin11986511990901199091
(120593(119905) minus 120593(119888119905)) = 1 and the family119879119899119899is equicontinuous at 1 it follows that119863
119874(119891119909)isin D+
By setting 120593(119905) = 119905 in the above corollary we get thefollowing
Corollary 11 Let (119883 119865 119879) be a complete Menger space with 119879
being a continuous 119905-norm of Hadzic type and let 119891 119883 rarr 119883
be a mapping such that
119865119891(119909)119891(119910) (119905)
ge min(119865119909119910
(119905
119888) 119865119909119891(119909)
(119905
119888) 119865119910119891(119910)
(119905
119888))
(26)
for all 119909 119910 isin 119883 119905 gt 0 Then 119891 has a unique fixed point in 119883
Remark 12 In paper [10] Babacev proved a fixed point resultfor mappings satisfying the contractive condition
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119888))
119865119909119891(119909)
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)) 119865119909119891(119910)
(2120593(119905
119888))
119865119910119891(119909)
(2120593(119905
119888))) forall119905 gt 0
(27)
for some altering distance function 120593 and some 119888 isin (0 1) inMenger spaces with the 119905-norm119879
119872We note that by applying
the triangle inequality
119865119909119891(119910)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
119865119910119891(119909)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
(28)
so this condition essentially reduces to (5) ThereforeTheorem 9 improves the result in [10] as well as Ciricrsquos resultin [7] (which can be obtained from that of Babacev for 120593(119905) =
119905)
Also in [11] Choudhury et al gave the following relatedtheorem
Theorem 13 (see [11]) Let (119883 119865 119879) be a complete Mengerspace with continuous 119905-norm 119879 and let 119886 119887 119888 be positivenumbers with 119886 + 119887 + 119888 lt 1 and 120593 isin Φ Suppose that119891 119883 rarr 119883 satisfies the inequality
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(1199051119886))
119865119909119891(119909)
(120593(1199052119887)) 119865119910119891(119910)
(120593(1199053119888)))
(29)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Then 119891 has a unique fixed point
As indicated in [11] a mapping satisfying the contractivecondition (29) must also verify our condition (5) Namely
Journal of Function Spaces 5
suppose that (29) holds Let 120576 = (1 minus (119886 + 119887 + 119888))3 isin (0 1)and let 1199051 = (119886 + 120576)119905 1199052 = (119887 + 120576)119905 1199053 = (119888 + 120576)119905 It follows that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905 (119886 + 120576)
119886))
119865119909119891(119909)
(120593(119905 (119887 + 120576)
119887)) 119865
119910119891(119910)(120593(
119905 (119888 + 120576)
119888)))
forall119905 gt 0
(30)
Due to the monotonicity of 120593 the above relation implies that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119896))
119865119909119891(119909)
(120593(119905
119896)) 119865119910119891(119910)
(120593(119905
119896)))
(31)
for all 119905 gt 0 where 119896 = max(119886(119886 + 120576) 119887(119887 + 120576) 119888(119888 + 120576)) isin
(0 1)Note that Theorem 13 only requires that the 119905-norm by
which the space is endowed is continuous Unfortunatelywe can show that this assumption alone is not sufficient toguarantee the existence of fixed points for contractions of thistype
Specifically let (119883 119865 119879119871) be a completeMenger space and
let 119891 be a Sehgal contraction on119883 with contraction constant119896 lt 13 Then
119865119891(119909)119891(119910) (119905) ge 119865
119909119910(119905
119896)
ge min(119865119909119910
(119905
119896) 119865119909119891(119909)
(119905
119896) 119865119910119891(119910)
(119905
119896))
ge min(119865119909119910
(1199051119896) 119865119909119891(119909)
(1199052119896) 119865119910119891(119910)
(1199053119896))
(32)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Thus 119891 satisfies the conditions of Theorem 13 with 119886 = 119887 =
119888 = 119896 and 120593(119905) = 119905 However a well-known counterexampleof Sherwood ([17] Corollary 1 of Theorem 35) shows thatthere exist Sehgal contractions on complete Menger spacesendowed with the 119905-norm 119879
119871having no fixed point
It should be mentioned that a similar observation regard-ing continuity can be made with respect to Theorem 31 in[18] where the class of contractions considered also includesSehgal contractions
Finally we illustrate the applicability of Theorem 9 withthe following example
Example 14 Let 119883 = [0 1] and 119879(119886 119887) = 119886119887 Define 119865119909119910(119905) =
(119905(119905+1))|119909minus119910| for all119909 119910 isin 119883 and 119905 gt 0 (119883 119865 119879) is a completeMenger space We will only show that the triangle inequalityis verified
Assume that 119905 gt 119904 gt 0 and 119909 119910 119911 isin 119883 Since the function119905(119905 + 1) is increasing it holds that
119879 (119865119909119910 (119905) 119865119910119911 (119904)) = (
119905
119905 + 1)
|119909minus119910|
(119904
119904 + 1)
|119910minus119911|
le (119905
119905 + 1)
|119909minus119910|+|119910minus119911|
le (119905
119905 + 1)
|119909minus119911|
le (119905 + 119904
119905 + 119904 + 1)
|119909minus119911|
= 119865119909119911 (119905 + 119904)
(33)
Let 120593(119905) = 1199052(2119905 + 1) for all 119905 gt 0 119888 = 12 and
119891 (119909) =
119909
2 119909 isin [0 1)
0 119909 = 1(34)
One can easily check that 120593 is an altering distance functionand that119874(119891 119909) is probabilistically bounded for every 119909 isin 119883
We will prove that condition (5) ofTheorem 9 is satisfiedThe following three cases are possible
(1) If 119909 119910 isin [0 1) then for all 119905 gt 0 we have
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
2|119891(119909)minus119891(119910)|= (
119905
119905 + 1)
|119909minus119910|
ge (2119905
2119905 + 1)
2|119909minus119910|= 119865119909119910
(120593 (2119905))
(35)
(2) If 119909 = 119910 = 1 then
119865119891(119909)119891(119910)
(120593 (119905)) = 1 = 119865119909119910
(120593 (2119905)) (36)
for all 119905 gt 0(3) If 119909 isin [0 1) and 119910 = 1 then for all 119905 gt 0
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
119909
ge119905
119905 + 1ge (
21199052119905 + 1
)
2
= (2119905
2119905 + 1)
2|119910minus119891(119910)|= 119865119910119891(119910)
(120593 (2119905))
(37)
Thus the condition (5) is satisfied in this case as well
However note that by setting119909 = 23 and119910 = 1we obtain
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
23lt (
21199052119905 + 1
)
23
= 119865119909119910
(120593 (2119905))(38)
for all 119905 gt 0 therefore 119891 does not satisfy the strongercondition (4)
By applying Theorem 9 we conclude that the function 119891
has a unique fixed point It is easy to see that this point is119909 = 0
6 Journal of Function Spaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Menger ldquoStatistical metricsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 pp535ndash537 1942
[2] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Series in Probability and AppliedMathematics North-Holland Publishing New York NY USA 1983
[3] G Constantin and I Istratescu Elements of Probabilistic Analy-sis with Applications Kluwer Academic Publishers 1989
[4] S Romaguera A Sapena and P Tirado ldquoThe Banach fixedpoint theorem in fuzzy quasi-metric spaces with application tothe domain of wordsrdquoTopology and Its Applications vol 154 no10 pp 2196ndash2203 2007
[5] S Romaguera and P Tirado ldquoThe complexity probabilisticquasi-metric spacerdquo Journal of Mathematical Analysis andApplications vol 376 no 2 pp 732ndash740 2011
[6] J Sun X Wu V Palade W Fang C-H Lai and W XuldquoConvergence analysis and improvements of quantum-behavedparticle swarm optimizationrdquo Information Sciences vol 193 pp81ndash103 2012
[7] L B Ciric ldquoOn fixed points of generalized contractions onprobabilistic metric spacesrdquo Publications de lrsquoInstitutMathematique (Beograd) vol 18 no 32 pp 71ndash78 1975
[8] B S Choudhury and K Das ldquoA new contraction principle inMenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008
[9] D Mihet ldquoAltering distances in probabilistic Menger spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no7-8 pp 2734ndash2738 2009
[10] N A Babacev ldquoNonlinear generalized contractions on MengerPM spacesrdquo Applicable Analysis and Discrete Mathematics vol6 no 2 pp 257ndash264 2012
[11] B S Choudhury K Das and P N Dutta ldquoA fixed point resultin Menger spaces using a real functionrdquo Acta MathematicaHungarica vol 122 no 3 pp 203ndash216 2009
[12] O Hadzic ldquoOn the (120576120582)-topology of probabilistic locallyconvex spacesrdquo Glasnik Matematicki Serija III vol 13 no 332 pp 293ndash297 1978
[13] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD thesis] Wayne State University 1966
[14] O Hadzic and E Pap Fixed Point Theory in Probabilistic MetricSpaces Kluwer Academic Publishers 2001
[15] V Radu ldquoOn the t-norms of Hadzic type and fixed points inPM-spacesrdquo Review of Research (Novi Sad) vol 13 pp 81ndash861983
[16] M Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984
[17] H Sherwood ldquoComplete probabilistic metric spacesrdquoZeitschrift fur Wahrscheinlichkeitstheorie und VerwandteGebiete vol 20 no 2 pp 117ndash128 1971
[18] T Dosenovic P Kumam D Gopal D K Patel and ATakaci ldquoOn fixed point theorems involving altering distances inMenger probabilistic metric spacesrdquo Journal of Inequalities andApplications vol 2013 article 576 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
is not continuous at 120593(119905) let (119905119898)119898be a strictly increasing
sequence converging to 119905 such that 119865119909lowast119891(119909lowast)is continuous at
120593(119905119898) for all119898 isin N As above we infer that 119865
119909lowast119891(119909lowast)(120593(119905119898)) ge
119865119909lowast119891(119909lowast)(120593(119905119898119888)) forall 119898 isin N whence for119898 rarr infin we obtain
(19) By Lemma 8 we conclude that 119909lowast = 119891(119909lowast)
Finally we prove that 119909lowast is the only fixed point of 119891 in119883To that end let 119910 isin 119883 be such that 119891(119910) = 119910 Then using (5)we get
119865119909lowast119910(120593 (119905)) ge min(119865
119909lowast119910(120593(
119905
119888)) 119865119909lowast119909lowast (120593(
119905
119888))
119865119910119910
(120593(119905
119888))) = 119865
119909lowast119910(120593(
119905
119888)) forall119905 gt 0
(21)
Once again by Lemma 8 it follows that 119910 = 119909lowast
Corollary 10 If (119883 119865 119879) is a complete Menger space with 119879
a continuous 119905-norm of Hadzic type and 119891 119883 rarr 119883 satisfiescondition (5) for some 119888 isin (0 1) and some 120593 isin Φ such thatlim119905rarrinfin
(120593(119905)minus120593(119888119905)) = infin then 119891 has a unique fixed point in119883
Proof We will show that for every 119909 isin 119883 119874(119891 119909) isprobabilistically bounded To do so let 119909 isin 119883 be arbitraryand define (119909
119899)119899by 119909119899= 119891119899(119909) for all 119899 ge 0 We will prove by
induction on 119899 that
1198651199090119909119899
(120593 (119905)) ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0 (22)
for all 119899 ge 1 The case 119899 = 1 is trivial Suppose now that therelation holds for some 119899 ge 1 Then
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
1198651199091119909119899+1
(120593 (119888119905))) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
min (1198651199090119909119899
(120593 (119905)) 11986511990901199091(120593 (119905)) 119865119909
119899119909119899+1
(120593 (119905))))
forall119905 gt 0
(23)
Given that
119865119909119899119909119899+1
(120593 (119905)) ge 11986511990901199091
(120593(119905
119888119899)) ge 119865
11990901199091(120593 (119905))
forall119905 gt 0
11986511990901199091
(120593 (119905)) ge 11986511990901199091
(120593 (119905) minus 120593 (119888119905))
ge 119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(24)
from the induction hypothesis we obtain
1198651199090119909119899+1
(120593 (119905)) ge 119879 (11986511990901199091
(120593 (119905) minus 120593 (119888119905))
119879119899(11986511990901199091
(120593 (119905) minus 120593 (119888119905))))
= 119879119899+1
(11986511990901199091
(120593 (119905) minus 120593 (119888119905))) forall119905 gt 0
(25)
which proves our claimNow since lim
119905rarrinfin11986511990901199091
(120593(119905) minus 120593(119888119905)) = 1 and the family119879119899119899is equicontinuous at 1 it follows that119863
119874(119891119909)isin D+
By setting 120593(119905) = 119905 in the above corollary we get thefollowing
Corollary 11 Let (119883 119865 119879) be a complete Menger space with 119879
being a continuous 119905-norm of Hadzic type and let 119891 119883 rarr 119883
be a mapping such that
119865119891(119909)119891(119910) (119905)
ge min(119865119909119910
(119905
119888) 119865119909119891(119909)
(119905
119888) 119865119910119891(119910)
(119905
119888))
(26)
for all 119909 119910 isin 119883 119905 gt 0 Then 119891 has a unique fixed point in 119883
Remark 12 In paper [10] Babacev proved a fixed point resultfor mappings satisfying the contractive condition
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119888))
119865119909119891(119909)
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)) 119865119909119891(119910)
(2120593(119905
119888))
119865119910119891(119909)
(2120593(119905
119888))) forall119905 gt 0
(27)
for some altering distance function 120593 and some 119888 isin (0 1) inMenger spaces with the 119905-norm119879
119872We note that by applying
the triangle inequality
119865119909119891(119910)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
119865119910119891(119909)
(2120593(119905
119888))
ge min(119865119909119910
(120593(119905
119888)) 119865119910119891(119910)
(120593(119905
119888)))
(28)
so this condition essentially reduces to (5) ThereforeTheorem 9 improves the result in [10] as well as Ciricrsquos resultin [7] (which can be obtained from that of Babacev for 120593(119905) =
119905)
Also in [11] Choudhury et al gave the following relatedtheorem
Theorem 13 (see [11]) Let (119883 119865 119879) be a complete Mengerspace with continuous 119905-norm 119879 and let 119886 119887 119888 be positivenumbers with 119886 + 119887 + 119888 lt 1 and 120593 isin Φ Suppose that119891 119883 rarr 119883 satisfies the inequality
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(1199051119886))
119865119909119891(119909)
(120593(1199052119887)) 119865119910119891(119910)
(120593(1199053119888)))
(29)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Then 119891 has a unique fixed point
As indicated in [11] a mapping satisfying the contractivecondition (29) must also verify our condition (5) Namely
Journal of Function Spaces 5
suppose that (29) holds Let 120576 = (1 minus (119886 + 119887 + 119888))3 isin (0 1)and let 1199051 = (119886 + 120576)119905 1199052 = (119887 + 120576)119905 1199053 = (119888 + 120576)119905 It follows that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905 (119886 + 120576)
119886))
119865119909119891(119909)
(120593(119905 (119887 + 120576)
119887)) 119865
119910119891(119910)(120593(
119905 (119888 + 120576)
119888)))
forall119905 gt 0
(30)
Due to the monotonicity of 120593 the above relation implies that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119896))
119865119909119891(119909)
(120593(119905
119896)) 119865119910119891(119910)
(120593(119905
119896)))
(31)
for all 119905 gt 0 where 119896 = max(119886(119886 + 120576) 119887(119887 + 120576) 119888(119888 + 120576)) isin
(0 1)Note that Theorem 13 only requires that the 119905-norm by
which the space is endowed is continuous Unfortunatelywe can show that this assumption alone is not sufficient toguarantee the existence of fixed points for contractions of thistype
Specifically let (119883 119865 119879119871) be a completeMenger space and
let 119891 be a Sehgal contraction on119883 with contraction constant119896 lt 13 Then
119865119891(119909)119891(119910) (119905) ge 119865
119909119910(119905
119896)
ge min(119865119909119910
(119905
119896) 119865119909119891(119909)
(119905
119896) 119865119910119891(119910)
(119905
119896))
ge min(119865119909119910
(1199051119896) 119865119909119891(119909)
(1199052119896) 119865119910119891(119910)
(1199053119896))
(32)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Thus 119891 satisfies the conditions of Theorem 13 with 119886 = 119887 =
119888 = 119896 and 120593(119905) = 119905 However a well-known counterexampleof Sherwood ([17] Corollary 1 of Theorem 35) shows thatthere exist Sehgal contractions on complete Menger spacesendowed with the 119905-norm 119879
119871having no fixed point
It should be mentioned that a similar observation regard-ing continuity can be made with respect to Theorem 31 in[18] where the class of contractions considered also includesSehgal contractions
Finally we illustrate the applicability of Theorem 9 withthe following example
Example 14 Let 119883 = [0 1] and 119879(119886 119887) = 119886119887 Define 119865119909119910(119905) =
(119905(119905+1))|119909minus119910| for all119909 119910 isin 119883 and 119905 gt 0 (119883 119865 119879) is a completeMenger space We will only show that the triangle inequalityis verified
Assume that 119905 gt 119904 gt 0 and 119909 119910 119911 isin 119883 Since the function119905(119905 + 1) is increasing it holds that
119879 (119865119909119910 (119905) 119865119910119911 (119904)) = (
119905
119905 + 1)
|119909minus119910|
(119904
119904 + 1)
|119910minus119911|
le (119905
119905 + 1)
|119909minus119910|+|119910minus119911|
le (119905
119905 + 1)
|119909minus119911|
le (119905 + 119904
119905 + 119904 + 1)
|119909minus119911|
= 119865119909119911 (119905 + 119904)
(33)
Let 120593(119905) = 1199052(2119905 + 1) for all 119905 gt 0 119888 = 12 and
119891 (119909) =
119909
2 119909 isin [0 1)
0 119909 = 1(34)
One can easily check that 120593 is an altering distance functionand that119874(119891 119909) is probabilistically bounded for every 119909 isin 119883
We will prove that condition (5) ofTheorem 9 is satisfiedThe following three cases are possible
(1) If 119909 119910 isin [0 1) then for all 119905 gt 0 we have
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
2|119891(119909)minus119891(119910)|= (
119905
119905 + 1)
|119909minus119910|
ge (2119905
2119905 + 1)
2|119909minus119910|= 119865119909119910
(120593 (2119905))
(35)
(2) If 119909 = 119910 = 1 then
119865119891(119909)119891(119910)
(120593 (119905)) = 1 = 119865119909119910
(120593 (2119905)) (36)
for all 119905 gt 0(3) If 119909 isin [0 1) and 119910 = 1 then for all 119905 gt 0
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
119909
ge119905
119905 + 1ge (
21199052119905 + 1
)
2
= (2119905
2119905 + 1)
2|119910minus119891(119910)|= 119865119910119891(119910)
(120593 (2119905))
(37)
Thus the condition (5) is satisfied in this case as well
However note that by setting119909 = 23 and119910 = 1we obtain
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
23lt (
21199052119905 + 1
)
23
= 119865119909119910
(120593 (2119905))(38)
for all 119905 gt 0 therefore 119891 does not satisfy the strongercondition (4)
By applying Theorem 9 we conclude that the function 119891
has a unique fixed point It is easy to see that this point is119909 = 0
6 Journal of Function Spaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Menger ldquoStatistical metricsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 pp535ndash537 1942
[2] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Series in Probability and AppliedMathematics North-Holland Publishing New York NY USA 1983
[3] G Constantin and I Istratescu Elements of Probabilistic Analy-sis with Applications Kluwer Academic Publishers 1989
[4] S Romaguera A Sapena and P Tirado ldquoThe Banach fixedpoint theorem in fuzzy quasi-metric spaces with application tothe domain of wordsrdquoTopology and Its Applications vol 154 no10 pp 2196ndash2203 2007
[5] S Romaguera and P Tirado ldquoThe complexity probabilisticquasi-metric spacerdquo Journal of Mathematical Analysis andApplications vol 376 no 2 pp 732ndash740 2011
[6] J Sun X Wu V Palade W Fang C-H Lai and W XuldquoConvergence analysis and improvements of quantum-behavedparticle swarm optimizationrdquo Information Sciences vol 193 pp81ndash103 2012
[7] L B Ciric ldquoOn fixed points of generalized contractions onprobabilistic metric spacesrdquo Publications de lrsquoInstitutMathematique (Beograd) vol 18 no 32 pp 71ndash78 1975
[8] B S Choudhury and K Das ldquoA new contraction principle inMenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008
[9] D Mihet ldquoAltering distances in probabilistic Menger spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no7-8 pp 2734ndash2738 2009
[10] N A Babacev ldquoNonlinear generalized contractions on MengerPM spacesrdquo Applicable Analysis and Discrete Mathematics vol6 no 2 pp 257ndash264 2012
[11] B S Choudhury K Das and P N Dutta ldquoA fixed point resultin Menger spaces using a real functionrdquo Acta MathematicaHungarica vol 122 no 3 pp 203ndash216 2009
[12] O Hadzic ldquoOn the (120576120582)-topology of probabilistic locallyconvex spacesrdquo Glasnik Matematicki Serija III vol 13 no 332 pp 293ndash297 1978
[13] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD thesis] Wayne State University 1966
[14] O Hadzic and E Pap Fixed Point Theory in Probabilistic MetricSpaces Kluwer Academic Publishers 2001
[15] V Radu ldquoOn the t-norms of Hadzic type and fixed points inPM-spacesrdquo Review of Research (Novi Sad) vol 13 pp 81ndash861983
[16] M Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984
[17] H Sherwood ldquoComplete probabilistic metric spacesrdquoZeitschrift fur Wahrscheinlichkeitstheorie und VerwandteGebiete vol 20 no 2 pp 117ndash128 1971
[18] T Dosenovic P Kumam D Gopal D K Patel and ATakaci ldquoOn fixed point theorems involving altering distances inMenger probabilistic metric spacesrdquo Journal of Inequalities andApplications vol 2013 article 576 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 5
suppose that (29) holds Let 120576 = (1 minus (119886 + 119887 + 119888))3 isin (0 1)and let 1199051 = (119886 + 120576)119905 1199052 = (119887 + 120576)119905 1199053 = (119888 + 120576)119905 It follows that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905 (119886 + 120576)
119886))
119865119909119891(119909)
(120593(119905 (119887 + 120576)
119887)) 119865
119910119891(119910)(120593(
119905 (119888 + 120576)
119888)))
forall119905 gt 0
(30)
Due to the monotonicity of 120593 the above relation implies that
119865119891(119909)119891(119910)
(120593 (119905)) ge min(119865119909119910
(120593(119905
119896))
119865119909119891(119909)
(120593(119905
119896)) 119865119910119891(119910)
(120593(119905
119896)))
(31)
for all 119905 gt 0 where 119896 = max(119886(119886 + 120576) 119887(119887 + 120576) 119888(119888 + 120576)) isin
(0 1)Note that Theorem 13 only requires that the 119905-norm by
which the space is endowed is continuous Unfortunatelywe can show that this assumption alone is not sufficient toguarantee the existence of fixed points for contractions of thistype
Specifically let (119883 119865 119879119871) be a completeMenger space and
let 119891 be a Sehgal contraction on119883 with contraction constant119896 lt 13 Then
119865119891(119909)119891(119910) (119905) ge 119865
119909119910(119905
119896)
ge min(119865119909119910
(119905
119896) 119865119909119891(119909)
(119905
119896) 119865119910119891(119910)
(119905
119896))
ge min(119865119909119910
(1199051119896) 119865119909119891(119909)
(1199052119896) 119865119910119891(119910)
(1199053119896))
(32)
for all 119909 119910 isin 119883 119905 gt 0 and 1199051 1199052 1199053 gt 0 with 1199051 + 1199052 + 1199053 = 119905Thus 119891 satisfies the conditions of Theorem 13 with 119886 = 119887 =
119888 = 119896 and 120593(119905) = 119905 However a well-known counterexampleof Sherwood ([17] Corollary 1 of Theorem 35) shows thatthere exist Sehgal contractions on complete Menger spacesendowed with the 119905-norm 119879
119871having no fixed point
It should be mentioned that a similar observation regard-ing continuity can be made with respect to Theorem 31 in[18] where the class of contractions considered also includesSehgal contractions
Finally we illustrate the applicability of Theorem 9 withthe following example
Example 14 Let 119883 = [0 1] and 119879(119886 119887) = 119886119887 Define 119865119909119910(119905) =
(119905(119905+1))|119909minus119910| for all119909 119910 isin 119883 and 119905 gt 0 (119883 119865 119879) is a completeMenger space We will only show that the triangle inequalityis verified
Assume that 119905 gt 119904 gt 0 and 119909 119910 119911 isin 119883 Since the function119905(119905 + 1) is increasing it holds that
119879 (119865119909119910 (119905) 119865119910119911 (119904)) = (
119905
119905 + 1)
|119909minus119910|
(119904
119904 + 1)
|119910minus119911|
le (119905
119905 + 1)
|119909minus119910|+|119910minus119911|
le (119905
119905 + 1)
|119909minus119911|
le (119905 + 119904
119905 + 119904 + 1)
|119909minus119911|
= 119865119909119911 (119905 + 119904)
(33)
Let 120593(119905) = 1199052(2119905 + 1) for all 119905 gt 0 119888 = 12 and
119891 (119909) =
119909
2 119909 isin [0 1)
0 119909 = 1(34)
One can easily check that 120593 is an altering distance functionand that119874(119891 119909) is probabilistically bounded for every 119909 isin 119883
We will prove that condition (5) ofTheorem 9 is satisfiedThe following three cases are possible
(1) If 119909 119910 isin [0 1) then for all 119905 gt 0 we have
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
2|119891(119909)minus119891(119910)|= (
119905
119905 + 1)
|119909minus119910|
ge (2119905
2119905 + 1)
2|119909minus119910|= 119865119909119910
(120593 (2119905))
(35)
(2) If 119909 = 119910 = 1 then
119865119891(119909)119891(119910)
(120593 (119905)) = 1 = 119865119909119910
(120593 (2119905)) (36)
for all 119905 gt 0(3) If 119909 isin [0 1) and 119910 = 1 then for all 119905 gt 0
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
119909
ge119905
119905 + 1ge (
21199052119905 + 1
)
2
= (2119905
2119905 + 1)
2|119910minus119891(119910)|= 119865119910119891(119910)
(120593 (2119905))
(37)
Thus the condition (5) is satisfied in this case as well
However note that by setting119909 = 23 and119910 = 1we obtain
119865119891(119909)119891(119910)
(120593 (119905)) = (119905
119905 + 1)
23lt (
21199052119905 + 1
)
23
= 119865119909119910
(120593 (2119905))(38)
for all 119905 gt 0 therefore 119891 does not satisfy the strongercondition (4)
By applying Theorem 9 we conclude that the function 119891
has a unique fixed point It is easy to see that this point is119909 = 0
6 Journal of Function Spaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Menger ldquoStatistical metricsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 pp535ndash537 1942
[2] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Series in Probability and AppliedMathematics North-Holland Publishing New York NY USA 1983
[3] G Constantin and I Istratescu Elements of Probabilistic Analy-sis with Applications Kluwer Academic Publishers 1989
[4] S Romaguera A Sapena and P Tirado ldquoThe Banach fixedpoint theorem in fuzzy quasi-metric spaces with application tothe domain of wordsrdquoTopology and Its Applications vol 154 no10 pp 2196ndash2203 2007
[5] S Romaguera and P Tirado ldquoThe complexity probabilisticquasi-metric spacerdquo Journal of Mathematical Analysis andApplications vol 376 no 2 pp 732ndash740 2011
[6] J Sun X Wu V Palade W Fang C-H Lai and W XuldquoConvergence analysis and improvements of quantum-behavedparticle swarm optimizationrdquo Information Sciences vol 193 pp81ndash103 2012
[7] L B Ciric ldquoOn fixed points of generalized contractions onprobabilistic metric spacesrdquo Publications de lrsquoInstitutMathematique (Beograd) vol 18 no 32 pp 71ndash78 1975
[8] B S Choudhury and K Das ldquoA new contraction principle inMenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008
[9] D Mihet ldquoAltering distances in probabilistic Menger spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no7-8 pp 2734ndash2738 2009
[10] N A Babacev ldquoNonlinear generalized contractions on MengerPM spacesrdquo Applicable Analysis and Discrete Mathematics vol6 no 2 pp 257ndash264 2012
[11] B S Choudhury K Das and P N Dutta ldquoA fixed point resultin Menger spaces using a real functionrdquo Acta MathematicaHungarica vol 122 no 3 pp 203ndash216 2009
[12] O Hadzic ldquoOn the (120576120582)-topology of probabilistic locallyconvex spacesrdquo Glasnik Matematicki Serija III vol 13 no 332 pp 293ndash297 1978
[13] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD thesis] Wayne State University 1966
[14] O Hadzic and E Pap Fixed Point Theory in Probabilistic MetricSpaces Kluwer Academic Publishers 2001
[15] V Radu ldquoOn the t-norms of Hadzic type and fixed points inPM-spacesrdquo Review of Research (Novi Sad) vol 13 pp 81ndash861983
[16] M Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984
[17] H Sherwood ldquoComplete probabilistic metric spacesrdquoZeitschrift fur Wahrscheinlichkeitstheorie und VerwandteGebiete vol 20 no 2 pp 117ndash128 1971
[18] T Dosenovic P Kumam D Gopal D K Patel and ATakaci ldquoOn fixed point theorems involving altering distances inMenger probabilistic metric spacesrdquo Journal of Inequalities andApplications vol 2013 article 576 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Function Spaces
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K Menger ldquoStatistical metricsrdquo Proceedings of the NationalAcademy of Sciences of the United States of America vol 28 pp535ndash537 1942
[2] B Schweizer and A Sklar Probabilistic Metric Spaces North-Holland Series in Probability and AppliedMathematics North-Holland Publishing New York NY USA 1983
[3] G Constantin and I Istratescu Elements of Probabilistic Analy-sis with Applications Kluwer Academic Publishers 1989
[4] S Romaguera A Sapena and P Tirado ldquoThe Banach fixedpoint theorem in fuzzy quasi-metric spaces with application tothe domain of wordsrdquoTopology and Its Applications vol 154 no10 pp 2196ndash2203 2007
[5] S Romaguera and P Tirado ldquoThe complexity probabilisticquasi-metric spacerdquo Journal of Mathematical Analysis andApplications vol 376 no 2 pp 732ndash740 2011
[6] J Sun X Wu V Palade W Fang C-H Lai and W XuldquoConvergence analysis and improvements of quantum-behavedparticle swarm optimizationrdquo Information Sciences vol 193 pp81ndash103 2012
[7] L B Ciric ldquoOn fixed points of generalized contractions onprobabilistic metric spacesrdquo Publications de lrsquoInstitutMathematique (Beograd) vol 18 no 32 pp 71ndash78 1975
[8] B S Choudhury and K Das ldquoA new contraction principle inMenger spacesrdquo Acta Mathematica Sinica vol 24 no 8 pp1379ndash1386 2008
[9] D Mihet ldquoAltering distances in probabilistic Menger spacesrdquoNonlinear Analysis Theory Methods amp Applications vol 71 no7-8 pp 2734ndash2738 2009
[10] N A Babacev ldquoNonlinear generalized contractions on MengerPM spacesrdquo Applicable Analysis and Discrete Mathematics vol6 no 2 pp 257ndash264 2012
[11] B S Choudhury K Das and P N Dutta ldquoA fixed point resultin Menger spaces using a real functionrdquo Acta MathematicaHungarica vol 122 no 3 pp 203ndash216 2009
[12] O Hadzic ldquoOn the (120576120582)-topology of probabilistic locallyconvex spacesrdquo Glasnik Matematicki Serija III vol 13 no 332 pp 293ndash297 1978
[13] V M Sehgal Some fixed point theorems in functional analysisand probability [PhD thesis] Wayne State University 1966
[14] O Hadzic and E Pap Fixed Point Theory in Probabilistic MetricSpaces Kluwer Academic Publishers 2001
[15] V Radu ldquoOn the t-norms of Hadzic type and fixed points inPM-spacesrdquo Review of Research (Novi Sad) vol 13 pp 81ndash861983
[16] M Khan M Swaleh and S Sessa ldquoFixed point theorems byaltering distances between the pointsrdquo Bulletin of the AustralianMathematical Society vol 30 no 1 pp 1ndash9 1984
[17] H Sherwood ldquoComplete probabilistic metric spacesrdquoZeitschrift fur Wahrscheinlichkeitstheorie und VerwandteGebiete vol 20 no 2 pp 117ndash128 1971
[18] T Dosenovic P Kumam D Gopal D K Patel and ATakaci ldquoOn fixed point theorems involving altering distances inMenger probabilistic metric spacesrdquo Journal of Inequalities andApplications vol 2013 article 576 10 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
