Research Article Cyclic Contractions and Fixed Point...
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Hindawi Publishing Corporation International Journal of Analysis Volume 2013, Article ID 726387, 9 pages http://dx.doi.org/10.1155/2013/726387 Research Article Cyclic Contractions and Fixed Point Results via Control Functions on Partial Metric Spaces Hemant Kumar Nashine 1 and Zoran Kadelburg 2 1 Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur, Chhattisgarh 492101, India 2 Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Beograd, Serbia Correspondence should be addressed to Zoran Kadelburg; [email protected] Received 23 August 2012; Revised 31 October 2012; Accepted 14 November 2012 Academic Editor: Seenith Sivasundaram Copyright © 2013 H. K. Nashine and Z. Kadelburg. 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. Cyclic weaker-type contraction conditions involving a generalized control function (with two variables) are used for mappings on 0-complete partial metric spaces to obtain fixed point results, thus generalizing several known results. Various examples are presented showing how the obtained theorems can be used and that they are proper extensions of the known ones. 1. Introduction e celebrated Banach contraction principle has been gener- alized in several directions and widely used to obtain various fixed point results, with applications in many branches of mathematics. Cyclic representations and cyclic contractions were intro- duced by Kirk et al. [1] and further used by several authors to obtain various fixed point results. See, for example, papers [2–9]. Note that while a classical contraction has to be continuous, cyclic contractions might not be. On the other hand, Matthews [10] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks. In partial metric spaces, self- distance of an arbitrary point need not be equal to zero. Several authors obtained many useful fixed point results in these spaces—we just mention [11–27]. Several results in ordered partial metric spaces have been obtained as well [28– 36]. Some results for cyclic contractions in partial metric spaces have been very recently obtained in [37–41]. Khan et al. [42] addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function. is idea was further used in many papers, such as Choudhury [43] where generalized control functions were used. is approach has been very recently used in [44, 45] to obtain fixed point results in partial metric spaces. In this paper, we extend these results further, considering cyclic weaker-type contraction conditions involving a gen- eralized control function (with two variables) for mappings on 0-complete partial metric spaces (Romaguera [16]). We obtain fixed point theorems for such mappings, thus general- izing several known results. Various examples are presented showing how the obtained results can be used and that they are proper extensions of the known ones. 2. Preliminaries In 2003, Kirk et al. introduced the following notion of cyclic representation. Definition 1 (see [1]). Let be a nonempty set, ∈ N, and let :→ be a self-mapping. en =⋃ =1 is a cyclic representation of with respect to if (a) , =1,..., are nonempty subsets of ; (b) ( 1 ) ⊂ 2 , ( 2 ) ⊂ 3 , . . . , ( −1 ) ⊂ , ( )⊂ 1 . ey proved the following fixed point result.
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Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2013 Article ID 726387 9 pageshttpdxdoiorg1011552013726387
Research ArticleCyclic Contractions and Fixed Point Results via ControlFunctions on Partial Metric Spaces
Hemant Kumar Nashine1and Zoran Kadelburg2
1 Department of Mathematics Disha Institute of Management and Technology Satya ViharVidhansabha-Chandrakhuri Marg Mandir Hasaud Raipur Chhattisgarh 492101 India
2 Faculty of Mathematics University of Belgrade Studentski trg 16 11000 Beograd Serbia
Correspondence should be addressed to Zoran Kadelburg kadelburmatfbgacrs
Received 23 August 2012 Revised 31 October 2012 Accepted 14 November 2012
Academic Editor Seenith Sivasundaram
Copyright copy 2013 H K Nashine and Z Kadelburg This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Cyclic weaker-type contraction conditions involving a generalized control function (with two variables) are used for mappingson 0-complete partial metric spaces to obtain fixed point results thus generalizing several known results Various examples arepresented showing how the obtained theorems can be used and that they are proper extensions of the known ones
1 Introduction
The celebrated Banach contraction principle has been gener-alized in several directions and widely used to obtain variousfixed point results with applications in many branches ofmathematics
Cyclic representations and cyclic contractions were intro-duced by Kirk et al [1] and further used by several authorsto obtain various fixed point results See for example papers[2ndash9] Note that while a classical contraction has to becontinuous cyclic contractions might not be
On the other hand Matthews [10] introduced the notionof a partial metric space as a part of the study of denotationalsemantics of dataflow networks In partial metric spaces self-distance of an arbitrary point need not be equal to zeroSeveral authors obtained many useful fixed point results inthese spacesmdashwe just mention [11ndash27] Several results inordered partial metric spaces have been obtained as well [28ndash36] Some results for cyclic contractions in partial metricspaces have been very recently obtained in [37ndash41]
Khan et al [42] addressed a new category of fixed pointproblems for a single self-map with the help of a controlfunction which they called an altering distance functionThisidea was further used in many papers such as Choudhury[43] where generalized control functions were used This
approach has been very recently used in [44 45] to obtainfixed point results in partial metric spaces
In this paper we extend these results further consideringcyclic weaker-type contraction conditions involving a gen-eralized control function (with two variables) for mappingson 0-complete partial metric spaces (Romaguera [16]) Weobtain fixed point theorems for such mappings thus general-izing several known results Various examples are presentedshowing how the obtained results can be used and that theyare proper extensions of the known ones
2 Preliminaries
In 2003 Kirk et al introduced the following notion of cyclicrepresentation
Definition 1 (see [1]) Let119883 be a nonempty set119898 isin N and let119891 119883 rarr 119883 be a self-mapping Then 119883 = ⋃
119898
119894=1119860119894is a cyclic
representation of119883 with respect to 119891 if
(a) 119860119894 119894 = 1 119898 are nonempty subsets of119883
(b) 119891(1198601) sub 119860
2 119891(119860
2) sub 119860
3 119891(119860
119898minus1) sub 119860
119898
119891(119860119898) sub 119860
1
They proved the following fixed point result
2 International Journal of Analysis
Theorem 2 (see [1]) Let (119883 119889) be a complete metric space119891 119883 rarr 119883 and let 119883 = ⋃
119898
119894=1119860119894be a cyclic representation
of 119883 with respect to 119891 Suppose that 119891 satisfies the followingcondition
119889 (119891119909 119891119910) le 120595 (119889 (119909 119910))
forall119909 isin 119860119894 119910 isin 119860
119894+1 119894 isin 1 2 119898
(1)
where 119860119898+1
= 1198601and 120595 [0 1) rarr [0 1) is a function upper
semicontinuous from the right and 0 le 120595(119905) lt 119905 for 119905 gt 0Then 119891 has a fixed point 119911 isin ⋂119898
119894=1119860119894
In 2010 Pacurar and Rus introduced the following notionof cyclic weaker 120593-contraction
Definition 3 (see [2]) Let (119883 119889) be a metric space 119898 isin 119873and let 119860
1 1198602 119860
119898be closed nonempty subsets of119883 and
119883 = ⋃119898
119894=1119860119894 An operator 119891 119883 rarr 119883 is called a cyclic
weaker 120593-contraction if
(1) 119883 = ⋃119898
119894=1119860119894is a cyclic representation of 119883 with
respect to 119891(2) there exists a continuous nondecreasing function 120593
[0 1) rarr [0 1) with 120593(119905) gt 0 for 119905 isin (0 1) and 120593(0) =0 such that
119889 (119891119909 119891119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) (2)
for any 119909 isin 119860119894 119910 isin 119860
119894+1 119894 = 1 2 119898 where 119860
119898+1= 1198601
They proved the following result
Theorem 4 (see [2]) Suppose that 119891 is a cyclic weaker 120593-contraction on a complete metric space (119883 119889) Then 119891 has afixed point 119911 isin ⋂119898
119894=1119860119894
This was generalized by Karap120484nar in [3]Khan et al introduced the following notion
Definition 5 (see [42]) A function 120593 [0 +infin) rarr [0 +infin) iscalled an altering distance function if the following propertiesare satisfied
(a) 120593 is continuous and nondecreasing(b) 120593(119905) = 0 hArr 119905 = 0
Choudhury introduced a generalization of Chatterjeatype contraction as follows
Definition 6 (see [43]) A self-mapping 119879 119883 rarr 119883 on ametric space (119883 119889) is said to be a weakly 119862-contractive (or aweak Chatterjea type contraction) if for all 119909 119910 isin 119883
119889 (119879119909 119879119910) le1
2[119889 (119909 119879119910) + 119889 (119910 119879119909)]
minus 120595 (119889 (119909 119879119910) 119889 (119910 119879119909))
(3)
where 120595 [0 +infin)2
rarr [0 +infin) is a continuous functionsuch that
120595 (119909 119910) = 0 iff 119909 = 119910 = 0 (4)
In [43] the author proved that every weakChatterjea typecontraction on a complete metric space has a unique fixedpoint
The following definitions and details can be seen forexample in [10 12 13 15 16]
Definition 7 A partial metric on a nonempty set 119883 is afunction 119901 119883 times 119883 rarr R+ such that for all 119909 119910 119911 isin 119883
(p1) 119909 = 119910 hArr 119901(119909 119909) = 119901(119909 119910) = 119901(119910 119910)
(p2) 119901(119909 119909) le 119901(119909 119910)
(p3) 119901(119909 119910) = 119901(119910 119909)
(p4) 119901(119909 119910) le 119901(119909 119911) + 119901(119911 119910) minus 119901(119911 119911)
The pair (119883 119901) is called a partial metric space
It is clear that if 119901(119909 119910) = 0 then from (1199011) and (119901
2) 119909 =
119910 But if 119909 = 119910 119901(119909 119910)may not be 0Each partial metric 119901 on119883 generates a 119879
0topology 120591
119901on
119883 which has as a base the family of open 119901-balls 119861119901(119909 120576)
119909 isin 119883 120576 gt 0 where119861119901(119909 120576) = 119910 isin 119883 119901(119909 119910) lt 119901(119909 119909)+120576
for all 119909 isin 119883 and 120576 gt 0A sequence 119909
119899 in (119883 119901) converges to a point 119909 isin 119883 (in
the sense of 120591119901) if lim
119899rarrinfin119901(119909 119909
119899) = 119901(119909 119909) This will be
denoted as 119909119899rarr 119909 (119899 rarr infin) or lim
119899rarrinfin119909119899= 119909 Clearly
a limit of a sequence in a partial metric space need not beuniqueMoreover the function 119901(sdot sdot) need not be continuousin the sense that 119909
119899rarr 119909 and 119910
119899rarr 119910 imply 119901(119909
119899 119910119899) rarr
119901(119909 119910)
Example 8 (see [10]) (1) A paradigmatic example of a partialmetric space is the pair (R+ 119901) where 119901(119909 119910) = max119909 119910for all 119909 119910 isin R+
(2) Let 119883 = [119886 119887] 119886 119887 isin R 119886 le 119887 and let119901([119886 119887] [119888 119889]) = max119887 119889 minus min119886 119888 Then (119883 119901) is apartial metric space
Definition 9 Let (119883 119901) be a partial metric space Thenconsider the following
(1) A sequence 119909119899 in (119883 119901) is called a Cauchy sequence
if lim119899119898rarrinfin
119901(119909119899 119909119898) exists (and is finite) The space
(119883 119901) is said to be complete if every Cauchy sequence119909119899 in119883 converges with respect to 120591
119901 to a point 119909 isin
119883 such that 119901(119909 119909) = lim119899119898rarrinfin
119901(119909119899 119909119898)
(2) (see [16]) A sequence 119909119899 in (119883 119901) is called 0-Cauchy
if lim119899119898rarrinfin
119901(119909119899 119909119898) = 0 The space (119883 119901) is said
to be 0-complete if every 0-Cauchy sequence in 119883
converges (in 120591119901) to a point 119909 isin 119883 such that 119901(119909 119909) =
0
Lemma 10 Let (119883 119901) be a partial metric space
(a) (see [46 47]) If 119901(119909119899 119911) rarr 119901(119911 119911) = 0 as 119899 rarr infin
then 119901(119909119899 119910) rarr 119901(119911 119910) as 119899 rarr infin for each 119910 isin 119883
(b) (see [16]) If (119883 119901) is complete then it is 0-complete
The converse assertion of (b) does not hold as thefollowing easy example shows
International Journal of Analysis 3
Example 11 (see [16]) The space 119883 = [0 +infin) cap Q with thepartial metric 119901(119909 119910) = max119909 119910 is 0-complete but is notcomplete Moreover the sequence 119909
119899 with 119909
119899= 1 for each
119899 isin N is a Cauchy sequence in (119883 119901) but it is not a 0-Cauchysequence
It is easy to see that every closed subset of a 0-completepartial metric space is 0-complete
3 Main Results
In this section we will prove some fixed point theoremsfor self-mappings defined on a 0-complete partial metricspace and satisfying certain cyclic weak contractive conditioninvolving a generalized control function To achieve our goalwe introduce the new notion of a cyclic contraction
Definition 12 Let (119883 119901) be a partial metric space 119902 isin N andlet1198601 1198602 119860
119902benonempty subsets of119883 and119884 = ⋃
119902
119894=1119860119894
An operator 119879 119884 rarr 119884 is called a cyclic contraction underweak contractive condition if
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with
respect to 119879(NZ2) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) (5)
where
119872(119909 119910) =max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(6)
and 120595 [0infin)2
rarr [0infin) is a lower semicontinuousmapping such that 120595(119904 119905) = 0 if and only if 119904 = 119905 = 0
Our main result is the following
Theorem 13 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of
(119883 119901) and let119884 = ⋃119902
119894=1119860119894 Suppose that119879 119884 rarr 119884 is a cyclic
contraction as defined in Definition 12 Then 119879 has a uniquefixed point 119911 isin 119884 such that 119901(119911 119911) = 0 Moreover 119911 isin ⋂119902
119894=1119860119894
Each Picard sequence 119909119899= 119879119899
1199090 1199090isin 119884 converges to 119911 in
topology 120591119901
Proof Let 1199090be an arbitrary point of 119884 Then there exists
some 1198940such that 119909
0isin 1198601198940 Now 119909
1= 119879119909
0isin 1198601198940+1
andsimilarly 119909
119899= 119879119909
119899minus1= 119879119899
1199090isin 1198601198940+119899
for 119899 isin N where119860119902+119896
= 119860119896 In the case 119901(119909
1198990 1199091198990+1
) = 0 for some 1198990isin N0 it
is clear that 1199091198990is a fixed point of 119879
Without loss of the generality we may assume that
119901 (119909119899 119909119899+1) gt 0 forall119899 isin N (7)
From the condition (NZ1) we observe that for all 119899 thereexists 119894 = 119894(119899) isin 1 2 119902 such that (119909
119899 119909119899+1) isin 119860119894times 119860119894+1
Putting 119909 = 119909119899and 119910 = 119909
119899+1in (NZ2) condition we have
119901 (119909119899+1 119909119899+2) =119901 (119879119909
119899 119879119909119899+1)
le119872(119909119899 119909119899+1) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119879119909119899))
=max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
1
2[119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1)]
minus 120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
(8)
By (1199014) we have
119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1) le 119901 (119909
119899 119909119899+1) + 119901 (119909
119899+1 119909119899+2)
(9)
Therefore
max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
1
2[119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1)]
le max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
(10)
By (8) and (10) we have
119901 (119909119899+1 119909119899+2) lemax 119901 (119909
119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
minus 120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
(11)
If max119901(119909119899 119909119899+1) 119901(119909119899+1 119909119899+2) = 119901(119909
119899+1 119909119899+2) then from
(11) we have
119901 (119909119899+1 119909119899+2)
le 119901 (119909119899+1 119909119899+2) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119909119899+1))
lt 119901 (119909119899+1 119909119899+2)
(12)
which is a contradiction (it was used that 120595(119901(119909119899 119909119899+1)
119901(119909119899 119909119899+1)) gt 0 since 119901(119909
119899 119909119899+1) gt 0) Hence 119901(119909
119899 119909119899+1) =
0 and 119909119899= 119909119899+1
which is excluded Therefore we havemax119901(119909
119899 119909119899+1) 119901(119909119899+1 119909119899+2) = 119901(119909
119899 119909119899+1) and hence
119901 (119909119899+1 119909119899+2)
le 119901 (119909119899 119909119899+1) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119909119899+1))
le 119901 (119909119899 119909119899+1)
(13)
By (13) we have that 119901(119909119899 119909119899+1) is a nonincreasing
sequence of positive real numbers Thus there exists 119903 ge 0
such that
lim119899rarrinfin
119901 (119909119899 119909119899+1) = 119903 (14)
4 International Journal of Analysis
Passing to the limit as 119899 rarr infin in (13) and using (14) andlower semicontinuity of 120595 we have
119903 le 119903 minus lim inf119899rarrinfin
120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
le 119903 minus 120595 (119903 119903)
(15)
thus 120595(119903 119903) = 0 and hence 119903 = 0 Therefore
lim119899rarrinfin
119901 (119909119899 119909119899+1) = 0 (16)
Next we claim that 119909119899 is a 0-Cauchy sequence in the
space (119883 119901) Suppose that this is not the case Then thereexists 120576 gt 0 for which we can find two sequences of positiveintegers 119898(119896) and 119899(119896) such that for all positive integers 119896
119899 (119896) gt 119898 (119896) gt 119896
119901 (119909119898(119896)
119909119899(119896)
) ge 120576
119901 (119909119898(119896)
119909119899(119896)minus1
) lt 120576
(17)
Using (17) and (1199014) we get
120576 le119901 (119909119899(119896)
119909119898(119896)
)
le119901 (119909119898(119896)
119909119899(119896)minus1
)
+ 119901 (119909119899(119896)minus1
119909119899(119896)
) minus 119901 (119909119899(119896)minus1
119909119899(119896)minus1
)
lt120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
)
(18)
Thus we have
120576 le 119901 (119909119899(119896)
119909119898(119896)
) lt 120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
) (19)
Passing to the limit as 119896 rarr infin in the above inequality andusing (16) we obtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)
) = 120576 (20)
On the other hand for all 119896 there exists 119895(119896) isin 1 119902
such that 119899(119896) minus 119898(119896) + 119895(119896) equiv 1[119902] Then 119909119898(119896)minus119895(119896)
(for 119896large enough119898(119896) gt 119895(119896)) and119909
119899(119896)lie in different adjacently
labelled sets 119860119894and 119860
119894+1for certain 119894 isin 1 119902
Using (1199014) and (20) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)
)
+ 119901 (119909119899(119896)
119909119898(119896)
) minus 119901 (119909119899(119896)
119909119899(119896)
)
le
119895(119896)minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
) + 119901 (119909119899(119896)
119909119898(119896)
)
le
119902minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
)
+ 119901 (119909119899(119896)
119909119898(119896)
) 997888rarr 120576 as 119896 997888rarr infin (from (16))
(21)
that is
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) = 120576 (22)
Using (16) we have
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
) = 0 (23)
lim119896rarrinfin
119901 (119909119899(119896)+1
119909119899(119896)
) = 0 (24)
Again using (1199014) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) le 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
+ 119901 (119909119899(119896)
119909119899(119896)+1
) minus 119901 (119909119899(119896)
119909119899(119896)
)
(25)
Passing to the limit as 119896 rarr infin in the pervious inequality andusing (24) and (22) we get
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) = 120576 (26)
Similarly we have by (1199014)
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)+1
)
+ 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) minus 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)
)
(27)
Passing to the limit as 119896 rarr infin and using (16) and (22) weobtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
) = 120576 (28)
Similarly we have by (1199014)
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
) = 120576 (29)
Using (NZ2) we obtain
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
)
le max 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
)
119901 (119909119899(119896)+1
119909119899(119896)
) 1
2119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
)
+119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
minus 120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
(30)
for all 119896 Passing to the limit as 119896 rarr infin in the last inequality(and using the lower semicontinuity of the function 120595) weobtain
120576 le 120576 minus lim inf119899rarrinfin
120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
le 120576 minus 120595 (120576 0)
(31)
International Journal of Analysis 5
which implies that 120595(120576 0) = 0 that is a contradiction since120576 gt 0 We deduce that 119909
119899 is a 0-Cauchy sequence
Since (119883 119901) is 0-complete and 119884 is closed it follows thatthe sequence 119909
119899 converges to some 119906 isin 119884 that is
lim119899rarrinfin
119909119899= 119906 (32)
We shall prove that
119906 isin
119902
⋂
119894=1
119860119894 (33)
From condition (NZ1) and since 1199090isin 1198601 we have 119909
119899119902119899ge0
sube
1198601 Since 119860
1is closed from (32) we get that 119906 isin 119860
1 Again
from the condition (NZ1) we have 119909119899119902+1
119899ge0
sube 1198602 Since
1198602is closed from (32) we get that 119906 isin 119860
2 Continuing this
process we obtain (33) and 119901(119906 119906) = 0Now we shall prove that 119906 is a fixed point of 119879 Indeed
from (33) since for all 119899 there exists 119894(119899) isin 1 2 119902 suchthat 119909
119899isin 119860119894(119899)
applying (NZ2) with 119909 = 119906 and 119910 = 119909119899 we
obtain
119901 (119879119906 119909119899+1) = 119901 (119879119906 119879119909
119899)
lemax 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus 120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
(34)
for all 119899 Passing to the limit as 119899 rarr infin in (34) and using(32) we get
119901 (119906 119879119906) le lim119899rarrinfin
max 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus lim inf119899rarrinfin
120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
lemax 0 119901 (119906 119879119906) 0 12119901 (119906 119879119906)
minus 120595 (0 119901 (119906 119879119906)) = 119901 (119906 119879119906) minus 120595 (0 119901 (119906 119879119906))
(35)
which is impossible unless 120595(0 119901(119906 119879119906)) = 0 so
119906 = 119879119906 (36)
that is 119906 is a fixed point of 119879We claim that there is a unique fixed point of 119879 Assume
on the contrary that 119879119906 = 119906 and 119879119907 = 119907 with 119901(119906 119907) gt 0
By supposition we can replace 119909 by 119906 and 119910 by 119907 in (NZ2) toobtain
119901 (119906 119907) =119901 (119879119906 119879119907)
lemax 119901 (119906 119907) 119901 (119906 119879119906) 119901 (119907 119879119907)
1
2[119901 (119906 119879119907) + 119901 (119879119906 119907)]
minus 120595 (119901 (119906 119907) 119901 (119906 119879119906))
=119901 (119906 119907) minus 120595 (119901 (119906 119907) 0) lt 119901 (119906 119907)
(37)
a contradiction Hence 119901(119906 119907) = 0 that is 119906 = 119907 Weconclude that 119879 has only one fixed point in 119883 The proof iscomplete
If we take 119902 = 1 and 1198601= 119883 in Theorem 13 then we get
the following fixed point theorem
Corollary 14 Let (119883 119901) be a 0-complete partial metric spaceand let 119879 119883 rarr 119883 satisfy the following condition there existsa lower semicontinuous mapping 120595 [0infin)
2
rarr [0infin) suchthat 120595(119905 119904) = 0 if and only if 119905 = 119904 = 0 and that
119901 (119879119909 119879119910) lemax 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119910 119879119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(38)
for all 119909 119910 isin 119883 Then 119879 has a unique fixed point 119911 isin 119883Moreover 119901(119911 119911) = 0
Corollary 14 extends and generalizes many existing fixedpoint theorems in the literature
By taking 120595(119904 119905) = (1 minus 119903)max119904 119905 where 119903 isin [0 1) inTheorem 13 we have the following result
Corollary 15 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with respect
to 119879(NZ3) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119903max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(39)
where 119903 isin [0 1) Then 119879 has a unique fixed point 119911 belongingto⋂119902119894=1119860119894 moreover 119901(119911 119911) = 0
As a special case of Corollary 15 we obtain Matthewsrsquosversion of Banach contraction principle [10]
6 International Journal of Analysis
Corollary 16 Let (119883 119901) be a 0-complete partial metric spacelet 119901 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119883 = ⋃119902
119894=1119860119894is a cyclic representation of119883with respect
to 119879(NZ4) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119899
119909 119879119899
119910) lemax 119901 (119909 119910) 119901 (119909 119879119899119909) 119901 (119910 119879119899119910)
1
2[119901 (119909 119879
119899
119910) + 119901 (119910 119879119899
119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119899
119909))
(40)
where 119899 is a positive integer and 120595 [0infin)2
rarr [0infin) is alower semi-continuous mapping such that 120595(119905 119904) = 0 if andonly if 119905 = 119904 = 0 Then 119879 has a unique fixed point belonging to⋂119902
119894=1119860119894
4 Examples
The following example shows howTheorem 13 can be used Itis adapted from [38 Example 29]
Example 17 Consider the partial metric space (119883 119889) ofExample 8 (2) It is easy to see that it is 0-complete Considerthe following closed subsets of119883
1198601= [1 minus 2
minus119899
1] 119899 isin N cup 1
1198602= [1 1 + 2
minus119899
] 119899 isin N cup 1 (41)
119884 = 1198601cup 1198602 and define a mapping 119879 119884 rarr 119884 by
119879119909 =
[1 1 + 2minus(119899+1)
] if 119909 = [1 minus 2minus119899 1]
[1 minus 2minus(119899+1)
1] if 119909 = [1 1 + 2minus119899]
1 if 119909 = 1
(42)
Obviously 119884 = 1198601cup 1198602is a cyclic representation of 119884 with
respect to 119879 We will show that 119879 satisfies the contractivecondition (NZ2) of Definition 12 with the control function120595 [0 +infin)
2
rarr [0 +infin) given by 120595(119904 119905) = (12)max119904 119905Let (119909 119910) isin 119860
1times 1198602(the other possibility is treated
similarly) and consider the following cases
(1) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 lt 119896 that is119899 + 1 le 119896 Then 119901(119879119909 119879119910) = 119901([1 1 + 2
minus(119899+1)
] [1 minus
2minus(119896+1)
1]) = (12)(2minus119899
+ 2minus119896
) le (34) sdot 2minus119899
119872(119909 119910) =max 2minus119899 + 2minus119896 32sdot 2minus119899
3
2sdot 2minus119896
1
2(100381610038161003816100381610038162minus119899
minus 2minus(119896+1)
10038161003816100381610038161003816+100381610038161003816100381610038162minus119896
minus 2minus(119899+1)
10038161003816100381610038161003816)
=3
2sdot 2minus119899
(43)
and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot 2minus119899 = (34)sdot2minus119899 Hence the condition (NZ2) reduces to (34) sdot
2minus119899
le (32) sdot 2minus119899
minus (34) sdot 2minus119899 and holds true
(2) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 = 119896Then 119901(119879119909 119879119910) = 2
minus119899 119872(119909 119910) = 2 sdot 2minus119899 and
120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot 2 sdot 2minus119899
= 2minus119899 hence
(NZ2) reduces to 2minus119899 le 2 sdot 2minus119899 minus 2minus119899(3) 119909 = [1 minus 2
minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 gt 119896 that is119899 ge 119896 + 1 Then 119901(119879119909 119879119910) le (34) sdot 2
minus119896 119872(119909 119910) =
(32) sdot 2minus119896 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119896
= (34)sdot2minus119896 Hence (NZ2) reduces to (34)sdot2minus119896 le
(32) sdot 2minus119896
minus (34) sdot 2minus119896 and holds true
(4) 119909 = [1 minus 2minus119899
1] 119910 = 1 Then 119901(119879119909 119879119910) = 119901([1 1 +
2minus(119899+1)
] 1) = (12) sdot 2minus119899 119872(119909 119910) = 119901(119909 119879119909) =
(32) sdot 2minus119899 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119899
= (34) sdot 2minus119899 (NZ2) reduces to (12) sdot 2
minus119899
le
(32) sdot 2minus119899
minus (34) sdot 2minus119899
(5) The case 119909 = 1 119910 = [1 1 + 2minus119896
] is treated symmet-rically
(6) The case 119909 = 119910 = 1 is trivial
We conclude that all conditions of Theorem 13 aresatisfied The mapping 119879 has a unique fixed point 1 isin 119860
1cap
1198602
Here is another example showing the use of Theorem 13
Example 18 Let 119883 = [0 1] be equipped with the partialmetric given as
119901 (119909 119910) =
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 if 119909 119910 isin [0 1)
1 if 119909 = 1 or 119910 = 1(44)
Then (119883 119901) is a 0-complete partial metric space Let 1198601=
[0 12] 1198602= [12 1] and 119879 119883 rarr 119883 be given as
119879119909 =
1
2 if 119909 isin [0 1)
1
6 if 119909 = 1
(45)
Obviously 119883 = 1198601cup 1198602is a cyclic representation of 119883 with
respect to 119879 We will check the contractive condition (NZ2)of Definition 12 with the control function 120595 [0 +infin)
2
rarr
[0 +infin) given by120595(119904 119905) = (119904+119905)(2+119904+119905) Let (119909 119910) isin 1198601times1198602
(the other possibility is treated symmetrically) Consider thefollowing possible cases
(1) 119909 isin [0 12] 119910 isin [12 1) Then 119901(119879119909 119879119910) =
119901(12 12) = 0 119872(119909 119910) = max119910 minus 119909 12 minus
119909 119910 minus 12 (12)(12 minus 119909 + 119910 minus 12) = 119910 minus 119909120595(119901(119909 119910) 119901(119909 119879119909)) = (119910 + 12 minus 2119909)(52 + 119910 minus 2119909)It is easy to check that
119901 (119879119909 119879119910) = 0 le 119910 minus 119909 minus119910 + 12 minus 2119909
52 + 119910 minus 2119909
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(46)
holds for the given values of 119909 and 119910
International Journal of Analysis 7
(2) 119909 isin [0 12] 119910 = 1 Then 119901(119879119909 119879119910) = 119901(12 16) =
13 119872(119909 119910) = 1 and 120595(119901(119909 119910) 119901(119909 119879119909)) =
120595(1 12 minus 119909) = (32 minus 119909)(72 minus 119909) The condition(NZ2) reduces to
1
3le 1 minus
3 minus 2119909
7 minus 2119909(47)
and can be checked directlyThus all the conditions of Theorem 13 are fulfilled and
we conclude that 119879 has a unique fixed point 12 isin 1198601cap 1198602
We state amore involved example that is inspiredwith theone from [48]
Example 19 Let119883 sub ℓ1119883 ni 119909 = (119909
119899)infin
119899=1if and only if 119909
119899ge 0
for each 119899 isin N Define a partial metric 119901 on119883 by
119901 ((119909119899) (119910119899)) =
infin
sum
119899=1
max 119909119899 119910119899 (48)
(it is easy to check that axioms (1199011)ndash(1199014) hold true) Let 120572 isin
(0 1) be fixed denote 0 = (0)infin119899=1
and consider the subsets1198601
and1198602of119883 defined by119860
1= 1198601015840
cup 0 1198602= 11986010158401015840
cup 0 where
1198601015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 or 119899 = 2119896 minus 1 119896 isin N
120572119899
119899 = 2119896 ge 2119897119897 = 1 2
11986010158401015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 minus 1 or 119899 = 2119896 119896 isin N
120572119899
119899 = 2119896 minus 1 ge 2119897 minus 1119897 = 1 2
(49)
Denote 119884 = 1198601cup 1198602(obviously 119860
1cap 1198602= 0)
Consider the mapping 119879 119884 rarr 119884 given by
119879 (0) = 0
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 )
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897+1
1205722119897+2
0 1205722119897+4
0 )
(50)
Obviously 119879(1198601) sub 119860
2and 119879(119860
2) sub 119860
1 hence 119884 = 119860
1cup
1198602is a cyclic representation of 119884 with respect to 119879 Take 120595
[0 +infin)2
rarr [0 +infin) defined by 120595(119904 119905) = (1 minus 120572)max119904 119905Let us check the contractive condition (NZ2) of Theo-
rem 13 Take 119909 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ) isin 1198601and
119910 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119898
1205722119898+1
0 1205722119898+3
0 ) isin 1198602and assume for
example that 119897 le 119898 (the case 119897 gt 119898 is treated similarly aswell as the case when 119909 or 119910 is equal to 0) Then
119901 (119909 119910) = 1205722119897
+ sdot sdot sdot + 1205722119898minus2
+1205722119898
1 minus 120572
119901 (119879119909 119879119910) = 1205722119897+1
+ sdot sdot sdot + 1205722119898minus1
+1205722119898+1
1 minus 120572
119901 (119909 119879119909) =1205722119897
1 minus 120572 119901 (119910 119879119910) =
1205722119898+1
1 minus 120572
(51)
Hence
119901 (119879119909 119879119910) = 120572119901 (119909 119910) le 120572 sdot1205722119897
1 minus 120572
=1205722119897
1 minus 120572minus (1 minus 120572)
1205722119897
1 minus 120572
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(52)
Obviously 119879 has a unique fixed point 0 isin 1198601cap 1198602
Finally we present an example showing that in certainsituations the existence of a fixed point can be concludedunder partial metric conditions while the same cannot beobtained using the standard metric
Example 20 Let 119883 = R+ be equipped with the usual partialmetric 119901(119909 119910) = max119909 119910 Suppose 119860
1= [0 1] 119860
2=
[0 12] 1198603= [0 16] 119860
4= [0 142] and 119884 = ⋃
4
119894=1119860119894
Consider the mapping 119879 119884 rarr 119884 defined by
119879119909 =1199092
1 + 119909 (53)
It is clear that ⋃4119894=1119860119894is a cyclic representation of 119884 with
respect to 119879 Further consider the function 120595 [0 +infin)2
rarr
[0 +infin) given by
120595 (119904 119905) =119904 + 119905
2 + 119904 + 119905 (54)
Take an arbitrary pair (119909 119910) isin 119860119894times119860119894+1
with say 119910 le 119909 (theother possibility can be treated in a similar way) Then
119901 (119879119909 119879119910) = max 1199092
1 + 1199091199102
1 + 119910 =
1199092
1 + 119909 (55)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 2: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/2.jpg)
2 International Journal of Analysis
Theorem 2 (see [1]) Let (119883 119889) be a complete metric space119891 119883 rarr 119883 and let 119883 = ⋃
119898
119894=1119860119894be a cyclic representation
of 119883 with respect to 119891 Suppose that 119891 satisfies the followingcondition
119889 (119891119909 119891119910) le 120595 (119889 (119909 119910))
forall119909 isin 119860119894 119910 isin 119860
119894+1 119894 isin 1 2 119898
(1)
where 119860119898+1
= 1198601and 120595 [0 1) rarr [0 1) is a function upper
semicontinuous from the right and 0 le 120595(119905) lt 119905 for 119905 gt 0Then 119891 has a fixed point 119911 isin ⋂119898
119894=1119860119894
In 2010 Pacurar and Rus introduced the following notionof cyclic weaker 120593-contraction
Definition 3 (see [2]) Let (119883 119889) be a metric space 119898 isin 119873and let 119860
1 1198602 119860
119898be closed nonempty subsets of119883 and
119883 = ⋃119898
119894=1119860119894 An operator 119891 119883 rarr 119883 is called a cyclic
weaker 120593-contraction if
(1) 119883 = ⋃119898
119894=1119860119894is a cyclic representation of 119883 with
respect to 119891(2) there exists a continuous nondecreasing function 120593
[0 1) rarr [0 1) with 120593(119905) gt 0 for 119905 isin (0 1) and 120593(0) =0 such that
119889 (119891119909 119891119910) le 119889 (119909 119910) minus 120593 (119889 (119909 119910)) (2)
for any 119909 isin 119860119894 119910 isin 119860
119894+1 119894 = 1 2 119898 where 119860
119898+1= 1198601
They proved the following result
Theorem 4 (see [2]) Suppose that 119891 is a cyclic weaker 120593-contraction on a complete metric space (119883 119889) Then 119891 has afixed point 119911 isin ⋂119898
119894=1119860119894
This was generalized by Karap120484nar in [3]Khan et al introduced the following notion
Definition 5 (see [42]) A function 120593 [0 +infin) rarr [0 +infin) iscalled an altering distance function if the following propertiesare satisfied
(a) 120593 is continuous and nondecreasing(b) 120593(119905) = 0 hArr 119905 = 0
Choudhury introduced a generalization of Chatterjeatype contraction as follows
Definition 6 (see [43]) A self-mapping 119879 119883 rarr 119883 on ametric space (119883 119889) is said to be a weakly 119862-contractive (or aweak Chatterjea type contraction) if for all 119909 119910 isin 119883
119889 (119879119909 119879119910) le1
2[119889 (119909 119879119910) + 119889 (119910 119879119909)]
minus 120595 (119889 (119909 119879119910) 119889 (119910 119879119909))
(3)
where 120595 [0 +infin)2
rarr [0 +infin) is a continuous functionsuch that
120595 (119909 119910) = 0 iff 119909 = 119910 = 0 (4)
In [43] the author proved that every weakChatterjea typecontraction on a complete metric space has a unique fixedpoint
The following definitions and details can be seen forexample in [10 12 13 15 16]
Definition 7 A partial metric on a nonempty set 119883 is afunction 119901 119883 times 119883 rarr R+ such that for all 119909 119910 119911 isin 119883
(p1) 119909 = 119910 hArr 119901(119909 119909) = 119901(119909 119910) = 119901(119910 119910)
(p2) 119901(119909 119909) le 119901(119909 119910)
(p3) 119901(119909 119910) = 119901(119910 119909)
(p4) 119901(119909 119910) le 119901(119909 119911) + 119901(119911 119910) minus 119901(119911 119911)
The pair (119883 119901) is called a partial metric space
It is clear that if 119901(119909 119910) = 0 then from (1199011) and (119901
2) 119909 =
119910 But if 119909 = 119910 119901(119909 119910)may not be 0Each partial metric 119901 on119883 generates a 119879
0topology 120591
119901on
119883 which has as a base the family of open 119901-balls 119861119901(119909 120576)
119909 isin 119883 120576 gt 0 where119861119901(119909 120576) = 119910 isin 119883 119901(119909 119910) lt 119901(119909 119909)+120576
for all 119909 isin 119883 and 120576 gt 0A sequence 119909
119899 in (119883 119901) converges to a point 119909 isin 119883 (in
the sense of 120591119901) if lim
119899rarrinfin119901(119909 119909
119899) = 119901(119909 119909) This will be
denoted as 119909119899rarr 119909 (119899 rarr infin) or lim
119899rarrinfin119909119899= 119909 Clearly
a limit of a sequence in a partial metric space need not beuniqueMoreover the function 119901(sdot sdot) need not be continuousin the sense that 119909
119899rarr 119909 and 119910
119899rarr 119910 imply 119901(119909
119899 119910119899) rarr
119901(119909 119910)
Example 8 (see [10]) (1) A paradigmatic example of a partialmetric space is the pair (R+ 119901) where 119901(119909 119910) = max119909 119910for all 119909 119910 isin R+
(2) Let 119883 = [119886 119887] 119886 119887 isin R 119886 le 119887 and let119901([119886 119887] [119888 119889]) = max119887 119889 minus min119886 119888 Then (119883 119901) is apartial metric space
Definition 9 Let (119883 119901) be a partial metric space Thenconsider the following
(1) A sequence 119909119899 in (119883 119901) is called a Cauchy sequence
if lim119899119898rarrinfin
119901(119909119899 119909119898) exists (and is finite) The space
(119883 119901) is said to be complete if every Cauchy sequence119909119899 in119883 converges with respect to 120591
119901 to a point 119909 isin
119883 such that 119901(119909 119909) = lim119899119898rarrinfin
119901(119909119899 119909119898)
(2) (see [16]) A sequence 119909119899 in (119883 119901) is called 0-Cauchy
if lim119899119898rarrinfin
119901(119909119899 119909119898) = 0 The space (119883 119901) is said
to be 0-complete if every 0-Cauchy sequence in 119883
converges (in 120591119901) to a point 119909 isin 119883 such that 119901(119909 119909) =
0
Lemma 10 Let (119883 119901) be a partial metric space
(a) (see [46 47]) If 119901(119909119899 119911) rarr 119901(119911 119911) = 0 as 119899 rarr infin
then 119901(119909119899 119910) rarr 119901(119911 119910) as 119899 rarr infin for each 119910 isin 119883
(b) (see [16]) If (119883 119901) is complete then it is 0-complete
The converse assertion of (b) does not hold as thefollowing easy example shows
International Journal of Analysis 3
Example 11 (see [16]) The space 119883 = [0 +infin) cap Q with thepartial metric 119901(119909 119910) = max119909 119910 is 0-complete but is notcomplete Moreover the sequence 119909
119899 with 119909
119899= 1 for each
119899 isin N is a Cauchy sequence in (119883 119901) but it is not a 0-Cauchysequence
It is easy to see that every closed subset of a 0-completepartial metric space is 0-complete
3 Main Results
In this section we will prove some fixed point theoremsfor self-mappings defined on a 0-complete partial metricspace and satisfying certain cyclic weak contractive conditioninvolving a generalized control function To achieve our goalwe introduce the new notion of a cyclic contraction
Definition 12 Let (119883 119901) be a partial metric space 119902 isin N andlet1198601 1198602 119860
119902benonempty subsets of119883 and119884 = ⋃
119902
119894=1119860119894
An operator 119879 119884 rarr 119884 is called a cyclic contraction underweak contractive condition if
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with
respect to 119879(NZ2) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) (5)
where
119872(119909 119910) =max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(6)
and 120595 [0infin)2
rarr [0infin) is a lower semicontinuousmapping such that 120595(119904 119905) = 0 if and only if 119904 = 119905 = 0
Our main result is the following
Theorem 13 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of
(119883 119901) and let119884 = ⋃119902
119894=1119860119894 Suppose that119879 119884 rarr 119884 is a cyclic
contraction as defined in Definition 12 Then 119879 has a uniquefixed point 119911 isin 119884 such that 119901(119911 119911) = 0 Moreover 119911 isin ⋂119902
119894=1119860119894
Each Picard sequence 119909119899= 119879119899
1199090 1199090isin 119884 converges to 119911 in
topology 120591119901
Proof Let 1199090be an arbitrary point of 119884 Then there exists
some 1198940such that 119909
0isin 1198601198940 Now 119909
1= 119879119909
0isin 1198601198940+1
andsimilarly 119909
119899= 119879119909
119899minus1= 119879119899
1199090isin 1198601198940+119899
for 119899 isin N where119860119902+119896
= 119860119896 In the case 119901(119909
1198990 1199091198990+1
) = 0 for some 1198990isin N0 it
is clear that 1199091198990is a fixed point of 119879
Without loss of the generality we may assume that
119901 (119909119899 119909119899+1) gt 0 forall119899 isin N (7)
From the condition (NZ1) we observe that for all 119899 thereexists 119894 = 119894(119899) isin 1 2 119902 such that (119909
119899 119909119899+1) isin 119860119894times 119860119894+1
Putting 119909 = 119909119899and 119910 = 119909
119899+1in (NZ2) condition we have
119901 (119909119899+1 119909119899+2) =119901 (119879119909
119899 119879119909119899+1)
le119872(119909119899 119909119899+1) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119879119909119899))
=max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
1
2[119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1)]
minus 120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
(8)
By (1199014) we have
119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1) le 119901 (119909
119899 119909119899+1) + 119901 (119909
119899+1 119909119899+2)
(9)
Therefore
max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
1
2[119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1)]
le max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
(10)
By (8) and (10) we have
119901 (119909119899+1 119909119899+2) lemax 119901 (119909
119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
minus 120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
(11)
If max119901(119909119899 119909119899+1) 119901(119909119899+1 119909119899+2) = 119901(119909
119899+1 119909119899+2) then from
(11) we have
119901 (119909119899+1 119909119899+2)
le 119901 (119909119899+1 119909119899+2) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119909119899+1))
lt 119901 (119909119899+1 119909119899+2)
(12)
which is a contradiction (it was used that 120595(119901(119909119899 119909119899+1)
119901(119909119899 119909119899+1)) gt 0 since 119901(119909
119899 119909119899+1) gt 0) Hence 119901(119909
119899 119909119899+1) =
0 and 119909119899= 119909119899+1
which is excluded Therefore we havemax119901(119909
119899 119909119899+1) 119901(119909119899+1 119909119899+2) = 119901(119909
119899 119909119899+1) and hence
119901 (119909119899+1 119909119899+2)
le 119901 (119909119899 119909119899+1) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119909119899+1))
le 119901 (119909119899 119909119899+1)
(13)
By (13) we have that 119901(119909119899 119909119899+1) is a nonincreasing
sequence of positive real numbers Thus there exists 119903 ge 0
such that
lim119899rarrinfin
119901 (119909119899 119909119899+1) = 119903 (14)
4 International Journal of Analysis
Passing to the limit as 119899 rarr infin in (13) and using (14) andlower semicontinuity of 120595 we have
119903 le 119903 minus lim inf119899rarrinfin
120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
le 119903 minus 120595 (119903 119903)
(15)
thus 120595(119903 119903) = 0 and hence 119903 = 0 Therefore
lim119899rarrinfin
119901 (119909119899 119909119899+1) = 0 (16)
Next we claim that 119909119899 is a 0-Cauchy sequence in the
space (119883 119901) Suppose that this is not the case Then thereexists 120576 gt 0 for which we can find two sequences of positiveintegers 119898(119896) and 119899(119896) such that for all positive integers 119896
119899 (119896) gt 119898 (119896) gt 119896
119901 (119909119898(119896)
119909119899(119896)
) ge 120576
119901 (119909119898(119896)
119909119899(119896)minus1
) lt 120576
(17)
Using (17) and (1199014) we get
120576 le119901 (119909119899(119896)
119909119898(119896)
)
le119901 (119909119898(119896)
119909119899(119896)minus1
)
+ 119901 (119909119899(119896)minus1
119909119899(119896)
) minus 119901 (119909119899(119896)minus1
119909119899(119896)minus1
)
lt120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
)
(18)
Thus we have
120576 le 119901 (119909119899(119896)
119909119898(119896)
) lt 120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
) (19)
Passing to the limit as 119896 rarr infin in the above inequality andusing (16) we obtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)
) = 120576 (20)
On the other hand for all 119896 there exists 119895(119896) isin 1 119902
such that 119899(119896) minus 119898(119896) + 119895(119896) equiv 1[119902] Then 119909119898(119896)minus119895(119896)
(for 119896large enough119898(119896) gt 119895(119896)) and119909
119899(119896)lie in different adjacently
labelled sets 119860119894and 119860
119894+1for certain 119894 isin 1 119902
Using (1199014) and (20) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)
)
+ 119901 (119909119899(119896)
119909119898(119896)
) minus 119901 (119909119899(119896)
119909119899(119896)
)
le
119895(119896)minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
) + 119901 (119909119899(119896)
119909119898(119896)
)
le
119902minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
)
+ 119901 (119909119899(119896)
119909119898(119896)
) 997888rarr 120576 as 119896 997888rarr infin (from (16))
(21)
that is
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) = 120576 (22)
Using (16) we have
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
) = 0 (23)
lim119896rarrinfin
119901 (119909119899(119896)+1
119909119899(119896)
) = 0 (24)
Again using (1199014) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) le 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
+ 119901 (119909119899(119896)
119909119899(119896)+1
) minus 119901 (119909119899(119896)
119909119899(119896)
)
(25)
Passing to the limit as 119896 rarr infin in the pervious inequality andusing (24) and (22) we get
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) = 120576 (26)
Similarly we have by (1199014)
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)+1
)
+ 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) minus 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)
)
(27)
Passing to the limit as 119896 rarr infin and using (16) and (22) weobtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
) = 120576 (28)
Similarly we have by (1199014)
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
) = 120576 (29)
Using (NZ2) we obtain
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
)
le max 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
)
119901 (119909119899(119896)+1
119909119899(119896)
) 1
2119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
)
+119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
minus 120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
(30)
for all 119896 Passing to the limit as 119896 rarr infin in the last inequality(and using the lower semicontinuity of the function 120595) weobtain
120576 le 120576 minus lim inf119899rarrinfin
120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
le 120576 minus 120595 (120576 0)
(31)
International Journal of Analysis 5
which implies that 120595(120576 0) = 0 that is a contradiction since120576 gt 0 We deduce that 119909
119899 is a 0-Cauchy sequence
Since (119883 119901) is 0-complete and 119884 is closed it follows thatthe sequence 119909
119899 converges to some 119906 isin 119884 that is
lim119899rarrinfin
119909119899= 119906 (32)
We shall prove that
119906 isin
119902
⋂
119894=1
119860119894 (33)
From condition (NZ1) and since 1199090isin 1198601 we have 119909
119899119902119899ge0
sube
1198601 Since 119860
1is closed from (32) we get that 119906 isin 119860
1 Again
from the condition (NZ1) we have 119909119899119902+1
119899ge0
sube 1198602 Since
1198602is closed from (32) we get that 119906 isin 119860
2 Continuing this
process we obtain (33) and 119901(119906 119906) = 0Now we shall prove that 119906 is a fixed point of 119879 Indeed
from (33) since for all 119899 there exists 119894(119899) isin 1 2 119902 suchthat 119909
119899isin 119860119894(119899)
applying (NZ2) with 119909 = 119906 and 119910 = 119909119899 we
obtain
119901 (119879119906 119909119899+1) = 119901 (119879119906 119879119909
119899)
lemax 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus 120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
(34)
for all 119899 Passing to the limit as 119899 rarr infin in (34) and using(32) we get
119901 (119906 119879119906) le lim119899rarrinfin
max 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus lim inf119899rarrinfin
120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
lemax 0 119901 (119906 119879119906) 0 12119901 (119906 119879119906)
minus 120595 (0 119901 (119906 119879119906)) = 119901 (119906 119879119906) minus 120595 (0 119901 (119906 119879119906))
(35)
which is impossible unless 120595(0 119901(119906 119879119906)) = 0 so
119906 = 119879119906 (36)
that is 119906 is a fixed point of 119879We claim that there is a unique fixed point of 119879 Assume
on the contrary that 119879119906 = 119906 and 119879119907 = 119907 with 119901(119906 119907) gt 0
By supposition we can replace 119909 by 119906 and 119910 by 119907 in (NZ2) toobtain
119901 (119906 119907) =119901 (119879119906 119879119907)
lemax 119901 (119906 119907) 119901 (119906 119879119906) 119901 (119907 119879119907)
1
2[119901 (119906 119879119907) + 119901 (119879119906 119907)]
minus 120595 (119901 (119906 119907) 119901 (119906 119879119906))
=119901 (119906 119907) minus 120595 (119901 (119906 119907) 0) lt 119901 (119906 119907)
(37)
a contradiction Hence 119901(119906 119907) = 0 that is 119906 = 119907 Weconclude that 119879 has only one fixed point in 119883 The proof iscomplete
If we take 119902 = 1 and 1198601= 119883 in Theorem 13 then we get
the following fixed point theorem
Corollary 14 Let (119883 119901) be a 0-complete partial metric spaceand let 119879 119883 rarr 119883 satisfy the following condition there existsa lower semicontinuous mapping 120595 [0infin)
2
rarr [0infin) suchthat 120595(119905 119904) = 0 if and only if 119905 = 119904 = 0 and that
119901 (119879119909 119879119910) lemax 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119910 119879119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(38)
for all 119909 119910 isin 119883 Then 119879 has a unique fixed point 119911 isin 119883Moreover 119901(119911 119911) = 0
Corollary 14 extends and generalizes many existing fixedpoint theorems in the literature
By taking 120595(119904 119905) = (1 minus 119903)max119904 119905 where 119903 isin [0 1) inTheorem 13 we have the following result
Corollary 15 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with respect
to 119879(NZ3) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119903max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(39)
where 119903 isin [0 1) Then 119879 has a unique fixed point 119911 belongingto⋂119902119894=1119860119894 moreover 119901(119911 119911) = 0
As a special case of Corollary 15 we obtain Matthewsrsquosversion of Banach contraction principle [10]
6 International Journal of Analysis
Corollary 16 Let (119883 119901) be a 0-complete partial metric spacelet 119901 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119883 = ⋃119902
119894=1119860119894is a cyclic representation of119883with respect
to 119879(NZ4) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119899
119909 119879119899
119910) lemax 119901 (119909 119910) 119901 (119909 119879119899119909) 119901 (119910 119879119899119910)
1
2[119901 (119909 119879
119899
119910) + 119901 (119910 119879119899
119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119899
119909))
(40)
where 119899 is a positive integer and 120595 [0infin)2
rarr [0infin) is alower semi-continuous mapping such that 120595(119905 119904) = 0 if andonly if 119905 = 119904 = 0 Then 119879 has a unique fixed point belonging to⋂119902
119894=1119860119894
4 Examples
The following example shows howTheorem 13 can be used Itis adapted from [38 Example 29]
Example 17 Consider the partial metric space (119883 119889) ofExample 8 (2) It is easy to see that it is 0-complete Considerthe following closed subsets of119883
1198601= [1 minus 2
minus119899
1] 119899 isin N cup 1
1198602= [1 1 + 2
minus119899
] 119899 isin N cup 1 (41)
119884 = 1198601cup 1198602 and define a mapping 119879 119884 rarr 119884 by
119879119909 =
[1 1 + 2minus(119899+1)
] if 119909 = [1 minus 2minus119899 1]
[1 minus 2minus(119899+1)
1] if 119909 = [1 1 + 2minus119899]
1 if 119909 = 1
(42)
Obviously 119884 = 1198601cup 1198602is a cyclic representation of 119884 with
respect to 119879 We will show that 119879 satisfies the contractivecondition (NZ2) of Definition 12 with the control function120595 [0 +infin)
2
rarr [0 +infin) given by 120595(119904 119905) = (12)max119904 119905Let (119909 119910) isin 119860
1times 1198602(the other possibility is treated
similarly) and consider the following cases
(1) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 lt 119896 that is119899 + 1 le 119896 Then 119901(119879119909 119879119910) = 119901([1 1 + 2
minus(119899+1)
] [1 minus
2minus(119896+1)
1]) = (12)(2minus119899
+ 2minus119896
) le (34) sdot 2minus119899
119872(119909 119910) =max 2minus119899 + 2minus119896 32sdot 2minus119899
3
2sdot 2minus119896
1
2(100381610038161003816100381610038162minus119899
minus 2minus(119896+1)
10038161003816100381610038161003816+100381610038161003816100381610038162minus119896
minus 2minus(119899+1)
10038161003816100381610038161003816)
=3
2sdot 2minus119899
(43)
and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot 2minus119899 = (34)sdot2minus119899 Hence the condition (NZ2) reduces to (34) sdot
2minus119899
le (32) sdot 2minus119899
minus (34) sdot 2minus119899 and holds true
(2) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 = 119896Then 119901(119879119909 119879119910) = 2
minus119899 119872(119909 119910) = 2 sdot 2minus119899 and
120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot 2 sdot 2minus119899
= 2minus119899 hence
(NZ2) reduces to 2minus119899 le 2 sdot 2minus119899 minus 2minus119899(3) 119909 = [1 minus 2
minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 gt 119896 that is119899 ge 119896 + 1 Then 119901(119879119909 119879119910) le (34) sdot 2
minus119896 119872(119909 119910) =
(32) sdot 2minus119896 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119896
= (34)sdot2minus119896 Hence (NZ2) reduces to (34)sdot2minus119896 le
(32) sdot 2minus119896
minus (34) sdot 2minus119896 and holds true
(4) 119909 = [1 minus 2minus119899
1] 119910 = 1 Then 119901(119879119909 119879119910) = 119901([1 1 +
2minus(119899+1)
] 1) = (12) sdot 2minus119899 119872(119909 119910) = 119901(119909 119879119909) =
(32) sdot 2minus119899 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119899
= (34) sdot 2minus119899 (NZ2) reduces to (12) sdot 2
minus119899
le
(32) sdot 2minus119899
minus (34) sdot 2minus119899
(5) The case 119909 = 1 119910 = [1 1 + 2minus119896
] is treated symmet-rically
(6) The case 119909 = 119910 = 1 is trivial
We conclude that all conditions of Theorem 13 aresatisfied The mapping 119879 has a unique fixed point 1 isin 119860
1cap
1198602
Here is another example showing the use of Theorem 13
Example 18 Let 119883 = [0 1] be equipped with the partialmetric given as
119901 (119909 119910) =
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 if 119909 119910 isin [0 1)
1 if 119909 = 1 or 119910 = 1(44)
Then (119883 119901) is a 0-complete partial metric space Let 1198601=
[0 12] 1198602= [12 1] and 119879 119883 rarr 119883 be given as
119879119909 =
1
2 if 119909 isin [0 1)
1
6 if 119909 = 1
(45)
Obviously 119883 = 1198601cup 1198602is a cyclic representation of 119883 with
respect to 119879 We will check the contractive condition (NZ2)of Definition 12 with the control function 120595 [0 +infin)
2
rarr
[0 +infin) given by120595(119904 119905) = (119904+119905)(2+119904+119905) Let (119909 119910) isin 1198601times1198602
(the other possibility is treated symmetrically) Consider thefollowing possible cases
(1) 119909 isin [0 12] 119910 isin [12 1) Then 119901(119879119909 119879119910) =
119901(12 12) = 0 119872(119909 119910) = max119910 minus 119909 12 minus
119909 119910 minus 12 (12)(12 minus 119909 + 119910 minus 12) = 119910 minus 119909120595(119901(119909 119910) 119901(119909 119879119909)) = (119910 + 12 minus 2119909)(52 + 119910 minus 2119909)It is easy to check that
119901 (119879119909 119879119910) = 0 le 119910 minus 119909 minus119910 + 12 minus 2119909
52 + 119910 minus 2119909
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(46)
holds for the given values of 119909 and 119910
International Journal of Analysis 7
(2) 119909 isin [0 12] 119910 = 1 Then 119901(119879119909 119879119910) = 119901(12 16) =
13 119872(119909 119910) = 1 and 120595(119901(119909 119910) 119901(119909 119879119909)) =
120595(1 12 minus 119909) = (32 minus 119909)(72 minus 119909) The condition(NZ2) reduces to
1
3le 1 minus
3 minus 2119909
7 minus 2119909(47)
and can be checked directlyThus all the conditions of Theorem 13 are fulfilled and
we conclude that 119879 has a unique fixed point 12 isin 1198601cap 1198602
We state amore involved example that is inspiredwith theone from [48]
Example 19 Let119883 sub ℓ1119883 ni 119909 = (119909
119899)infin
119899=1if and only if 119909
119899ge 0
for each 119899 isin N Define a partial metric 119901 on119883 by
119901 ((119909119899) (119910119899)) =
infin
sum
119899=1
max 119909119899 119910119899 (48)
(it is easy to check that axioms (1199011)ndash(1199014) hold true) Let 120572 isin
(0 1) be fixed denote 0 = (0)infin119899=1
and consider the subsets1198601
and1198602of119883 defined by119860
1= 1198601015840
cup 0 1198602= 11986010158401015840
cup 0 where
1198601015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 or 119899 = 2119896 minus 1 119896 isin N
120572119899
119899 = 2119896 ge 2119897119897 = 1 2
11986010158401015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 minus 1 or 119899 = 2119896 119896 isin N
120572119899
119899 = 2119896 minus 1 ge 2119897 minus 1119897 = 1 2
(49)
Denote 119884 = 1198601cup 1198602(obviously 119860
1cap 1198602= 0)
Consider the mapping 119879 119884 rarr 119884 given by
119879 (0) = 0
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 )
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897+1
1205722119897+2
0 1205722119897+4
0 )
(50)
Obviously 119879(1198601) sub 119860
2and 119879(119860
2) sub 119860
1 hence 119884 = 119860
1cup
1198602is a cyclic representation of 119884 with respect to 119879 Take 120595
[0 +infin)2
rarr [0 +infin) defined by 120595(119904 119905) = (1 minus 120572)max119904 119905Let us check the contractive condition (NZ2) of Theo-
rem 13 Take 119909 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ) isin 1198601and
119910 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119898
1205722119898+1
0 1205722119898+3
0 ) isin 1198602and assume for
example that 119897 le 119898 (the case 119897 gt 119898 is treated similarly aswell as the case when 119909 or 119910 is equal to 0) Then
119901 (119909 119910) = 1205722119897
+ sdot sdot sdot + 1205722119898minus2
+1205722119898
1 minus 120572
119901 (119879119909 119879119910) = 1205722119897+1
+ sdot sdot sdot + 1205722119898minus1
+1205722119898+1
1 minus 120572
119901 (119909 119879119909) =1205722119897
1 minus 120572 119901 (119910 119879119910) =
1205722119898+1
1 minus 120572
(51)
Hence
119901 (119879119909 119879119910) = 120572119901 (119909 119910) le 120572 sdot1205722119897
1 minus 120572
=1205722119897
1 minus 120572minus (1 minus 120572)
1205722119897
1 minus 120572
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(52)
Obviously 119879 has a unique fixed point 0 isin 1198601cap 1198602
Finally we present an example showing that in certainsituations the existence of a fixed point can be concludedunder partial metric conditions while the same cannot beobtained using the standard metric
Example 20 Let 119883 = R+ be equipped with the usual partialmetric 119901(119909 119910) = max119909 119910 Suppose 119860
1= [0 1] 119860
2=
[0 12] 1198603= [0 16] 119860
4= [0 142] and 119884 = ⋃
4
119894=1119860119894
Consider the mapping 119879 119884 rarr 119884 defined by
119879119909 =1199092
1 + 119909 (53)
It is clear that ⋃4119894=1119860119894is a cyclic representation of 119884 with
respect to 119879 Further consider the function 120595 [0 +infin)2
rarr
[0 +infin) given by
120595 (119904 119905) =119904 + 119905
2 + 119904 + 119905 (54)
Take an arbitrary pair (119909 119910) isin 119860119894times119860119894+1
with say 119910 le 119909 (theother possibility can be treated in a similar way) Then
119901 (119879119909 119879119910) = max 1199092
1 + 1199091199102
1 + 119910 =
1199092
1 + 119909 (55)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 3: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/3.jpg)
International Journal of Analysis 3
Example 11 (see [16]) The space 119883 = [0 +infin) cap Q with thepartial metric 119901(119909 119910) = max119909 119910 is 0-complete but is notcomplete Moreover the sequence 119909
119899 with 119909
119899= 1 for each
119899 isin N is a Cauchy sequence in (119883 119901) but it is not a 0-Cauchysequence
It is easy to see that every closed subset of a 0-completepartial metric space is 0-complete
3 Main Results
In this section we will prove some fixed point theoremsfor self-mappings defined on a 0-complete partial metricspace and satisfying certain cyclic weak contractive conditioninvolving a generalized control function To achieve our goalwe introduce the new notion of a cyclic contraction
Definition 12 Let (119883 119901) be a partial metric space 119902 isin N andlet1198601 1198602 119860
119902benonempty subsets of119883 and119884 = ⋃
119902
119894=1119860119894
An operator 119879 119884 rarr 119884 is called a cyclic contraction underweak contractive condition if
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with
respect to 119879(NZ2) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) (5)
where
119872(119909 119910) =max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(6)
and 120595 [0infin)2
rarr [0infin) is a lower semicontinuousmapping such that 120595(119904 119905) = 0 if and only if 119904 = 119905 = 0
Our main result is the following
Theorem 13 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of
(119883 119901) and let119884 = ⋃119902
119894=1119860119894 Suppose that119879 119884 rarr 119884 is a cyclic
contraction as defined in Definition 12 Then 119879 has a uniquefixed point 119911 isin 119884 such that 119901(119911 119911) = 0 Moreover 119911 isin ⋂119902
119894=1119860119894
Each Picard sequence 119909119899= 119879119899
1199090 1199090isin 119884 converges to 119911 in
topology 120591119901
Proof Let 1199090be an arbitrary point of 119884 Then there exists
some 1198940such that 119909
0isin 1198601198940 Now 119909
1= 119879119909
0isin 1198601198940+1
andsimilarly 119909
119899= 119879119909
119899minus1= 119879119899
1199090isin 1198601198940+119899
for 119899 isin N where119860119902+119896
= 119860119896 In the case 119901(119909
1198990 1199091198990+1
) = 0 for some 1198990isin N0 it
is clear that 1199091198990is a fixed point of 119879
Without loss of the generality we may assume that
119901 (119909119899 119909119899+1) gt 0 forall119899 isin N (7)
From the condition (NZ1) we observe that for all 119899 thereexists 119894 = 119894(119899) isin 1 2 119902 such that (119909
119899 119909119899+1) isin 119860119894times 119860119894+1
Putting 119909 = 119909119899and 119910 = 119909
119899+1in (NZ2) condition we have
119901 (119909119899+1 119909119899+2) =119901 (119879119909
119899 119879119909119899+1)
le119872(119909119899 119909119899+1) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119879119909119899))
=max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
1
2[119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1)]
minus 120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
(8)
By (1199014) we have
119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1) le 119901 (119909
119899 119909119899+1) + 119901 (119909
119899+1 119909119899+2)
(9)
Therefore
max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
1
2[119901 (119909119899 119909119899+2) + 119901 (119909
119899+1 119909119899+1)]
le max 119901 (119909119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
(10)
By (8) and (10) we have
119901 (119909119899+1 119909119899+2) lemax 119901 (119909
119899 119909119899+1) 119901 (119909
119899+1 119909119899+2)
minus 120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
(11)
If max119901(119909119899 119909119899+1) 119901(119909119899+1 119909119899+2) = 119901(119909
119899+1 119909119899+2) then from
(11) we have
119901 (119909119899+1 119909119899+2)
le 119901 (119909119899+1 119909119899+2) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119909119899+1))
lt 119901 (119909119899+1 119909119899+2)
(12)
which is a contradiction (it was used that 120595(119901(119909119899 119909119899+1)
119901(119909119899 119909119899+1)) gt 0 since 119901(119909
119899 119909119899+1) gt 0) Hence 119901(119909
119899 119909119899+1) =
0 and 119909119899= 119909119899+1
which is excluded Therefore we havemax119901(119909
119899 119909119899+1) 119901(119909119899+1 119909119899+2) = 119901(119909
119899 119909119899+1) and hence
119901 (119909119899+1 119909119899+2)
le 119901 (119909119899 119909119899+1) minus 120595 (119901 (119909
119899 119909119899+1) 119901 (119909
119899 119909119899+1))
le 119901 (119909119899 119909119899+1)
(13)
By (13) we have that 119901(119909119899 119909119899+1) is a nonincreasing
sequence of positive real numbers Thus there exists 119903 ge 0
such that
lim119899rarrinfin
119901 (119909119899 119909119899+1) = 119903 (14)
4 International Journal of Analysis
Passing to the limit as 119899 rarr infin in (13) and using (14) andlower semicontinuity of 120595 we have
119903 le 119903 minus lim inf119899rarrinfin
120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
le 119903 minus 120595 (119903 119903)
(15)
thus 120595(119903 119903) = 0 and hence 119903 = 0 Therefore
lim119899rarrinfin
119901 (119909119899 119909119899+1) = 0 (16)
Next we claim that 119909119899 is a 0-Cauchy sequence in the
space (119883 119901) Suppose that this is not the case Then thereexists 120576 gt 0 for which we can find two sequences of positiveintegers 119898(119896) and 119899(119896) such that for all positive integers 119896
119899 (119896) gt 119898 (119896) gt 119896
119901 (119909119898(119896)
119909119899(119896)
) ge 120576
119901 (119909119898(119896)
119909119899(119896)minus1
) lt 120576
(17)
Using (17) and (1199014) we get
120576 le119901 (119909119899(119896)
119909119898(119896)
)
le119901 (119909119898(119896)
119909119899(119896)minus1
)
+ 119901 (119909119899(119896)minus1
119909119899(119896)
) minus 119901 (119909119899(119896)minus1
119909119899(119896)minus1
)
lt120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
)
(18)
Thus we have
120576 le 119901 (119909119899(119896)
119909119898(119896)
) lt 120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
) (19)
Passing to the limit as 119896 rarr infin in the above inequality andusing (16) we obtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)
) = 120576 (20)
On the other hand for all 119896 there exists 119895(119896) isin 1 119902
such that 119899(119896) minus 119898(119896) + 119895(119896) equiv 1[119902] Then 119909119898(119896)minus119895(119896)
(for 119896large enough119898(119896) gt 119895(119896)) and119909
119899(119896)lie in different adjacently
labelled sets 119860119894and 119860
119894+1for certain 119894 isin 1 119902
Using (1199014) and (20) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)
)
+ 119901 (119909119899(119896)
119909119898(119896)
) minus 119901 (119909119899(119896)
119909119899(119896)
)
le
119895(119896)minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
) + 119901 (119909119899(119896)
119909119898(119896)
)
le
119902minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
)
+ 119901 (119909119899(119896)
119909119898(119896)
) 997888rarr 120576 as 119896 997888rarr infin (from (16))
(21)
that is
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) = 120576 (22)
Using (16) we have
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
) = 0 (23)
lim119896rarrinfin
119901 (119909119899(119896)+1
119909119899(119896)
) = 0 (24)
Again using (1199014) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) le 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
+ 119901 (119909119899(119896)
119909119899(119896)+1
) minus 119901 (119909119899(119896)
119909119899(119896)
)
(25)
Passing to the limit as 119896 rarr infin in the pervious inequality andusing (24) and (22) we get
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) = 120576 (26)
Similarly we have by (1199014)
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)+1
)
+ 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) minus 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)
)
(27)
Passing to the limit as 119896 rarr infin and using (16) and (22) weobtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
) = 120576 (28)
Similarly we have by (1199014)
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
) = 120576 (29)
Using (NZ2) we obtain
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
)
le max 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
)
119901 (119909119899(119896)+1
119909119899(119896)
) 1
2119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
)
+119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
minus 120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
(30)
for all 119896 Passing to the limit as 119896 rarr infin in the last inequality(and using the lower semicontinuity of the function 120595) weobtain
120576 le 120576 minus lim inf119899rarrinfin
120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
le 120576 minus 120595 (120576 0)
(31)
International Journal of Analysis 5
which implies that 120595(120576 0) = 0 that is a contradiction since120576 gt 0 We deduce that 119909
119899 is a 0-Cauchy sequence
Since (119883 119901) is 0-complete and 119884 is closed it follows thatthe sequence 119909
119899 converges to some 119906 isin 119884 that is
lim119899rarrinfin
119909119899= 119906 (32)
We shall prove that
119906 isin
119902
⋂
119894=1
119860119894 (33)
From condition (NZ1) and since 1199090isin 1198601 we have 119909
119899119902119899ge0
sube
1198601 Since 119860
1is closed from (32) we get that 119906 isin 119860
1 Again
from the condition (NZ1) we have 119909119899119902+1
119899ge0
sube 1198602 Since
1198602is closed from (32) we get that 119906 isin 119860
2 Continuing this
process we obtain (33) and 119901(119906 119906) = 0Now we shall prove that 119906 is a fixed point of 119879 Indeed
from (33) since for all 119899 there exists 119894(119899) isin 1 2 119902 suchthat 119909
119899isin 119860119894(119899)
applying (NZ2) with 119909 = 119906 and 119910 = 119909119899 we
obtain
119901 (119879119906 119909119899+1) = 119901 (119879119906 119879119909
119899)
lemax 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus 120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
(34)
for all 119899 Passing to the limit as 119899 rarr infin in (34) and using(32) we get
119901 (119906 119879119906) le lim119899rarrinfin
max 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus lim inf119899rarrinfin
120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
lemax 0 119901 (119906 119879119906) 0 12119901 (119906 119879119906)
minus 120595 (0 119901 (119906 119879119906)) = 119901 (119906 119879119906) minus 120595 (0 119901 (119906 119879119906))
(35)
which is impossible unless 120595(0 119901(119906 119879119906)) = 0 so
119906 = 119879119906 (36)
that is 119906 is a fixed point of 119879We claim that there is a unique fixed point of 119879 Assume
on the contrary that 119879119906 = 119906 and 119879119907 = 119907 with 119901(119906 119907) gt 0
By supposition we can replace 119909 by 119906 and 119910 by 119907 in (NZ2) toobtain
119901 (119906 119907) =119901 (119879119906 119879119907)
lemax 119901 (119906 119907) 119901 (119906 119879119906) 119901 (119907 119879119907)
1
2[119901 (119906 119879119907) + 119901 (119879119906 119907)]
minus 120595 (119901 (119906 119907) 119901 (119906 119879119906))
=119901 (119906 119907) minus 120595 (119901 (119906 119907) 0) lt 119901 (119906 119907)
(37)
a contradiction Hence 119901(119906 119907) = 0 that is 119906 = 119907 Weconclude that 119879 has only one fixed point in 119883 The proof iscomplete
If we take 119902 = 1 and 1198601= 119883 in Theorem 13 then we get
the following fixed point theorem
Corollary 14 Let (119883 119901) be a 0-complete partial metric spaceand let 119879 119883 rarr 119883 satisfy the following condition there existsa lower semicontinuous mapping 120595 [0infin)
2
rarr [0infin) suchthat 120595(119905 119904) = 0 if and only if 119905 = 119904 = 0 and that
119901 (119879119909 119879119910) lemax 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119910 119879119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(38)
for all 119909 119910 isin 119883 Then 119879 has a unique fixed point 119911 isin 119883Moreover 119901(119911 119911) = 0
Corollary 14 extends and generalizes many existing fixedpoint theorems in the literature
By taking 120595(119904 119905) = (1 minus 119903)max119904 119905 where 119903 isin [0 1) inTheorem 13 we have the following result
Corollary 15 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with respect
to 119879(NZ3) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119903max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(39)
where 119903 isin [0 1) Then 119879 has a unique fixed point 119911 belongingto⋂119902119894=1119860119894 moreover 119901(119911 119911) = 0
As a special case of Corollary 15 we obtain Matthewsrsquosversion of Banach contraction principle [10]
6 International Journal of Analysis
Corollary 16 Let (119883 119901) be a 0-complete partial metric spacelet 119901 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119883 = ⋃119902
119894=1119860119894is a cyclic representation of119883with respect
to 119879(NZ4) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119899
119909 119879119899
119910) lemax 119901 (119909 119910) 119901 (119909 119879119899119909) 119901 (119910 119879119899119910)
1
2[119901 (119909 119879
119899
119910) + 119901 (119910 119879119899
119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119899
119909))
(40)
where 119899 is a positive integer and 120595 [0infin)2
rarr [0infin) is alower semi-continuous mapping such that 120595(119905 119904) = 0 if andonly if 119905 = 119904 = 0 Then 119879 has a unique fixed point belonging to⋂119902
119894=1119860119894
4 Examples
The following example shows howTheorem 13 can be used Itis adapted from [38 Example 29]
Example 17 Consider the partial metric space (119883 119889) ofExample 8 (2) It is easy to see that it is 0-complete Considerthe following closed subsets of119883
1198601= [1 minus 2
minus119899
1] 119899 isin N cup 1
1198602= [1 1 + 2
minus119899
] 119899 isin N cup 1 (41)
119884 = 1198601cup 1198602 and define a mapping 119879 119884 rarr 119884 by
119879119909 =
[1 1 + 2minus(119899+1)
] if 119909 = [1 minus 2minus119899 1]
[1 minus 2minus(119899+1)
1] if 119909 = [1 1 + 2minus119899]
1 if 119909 = 1
(42)
Obviously 119884 = 1198601cup 1198602is a cyclic representation of 119884 with
respect to 119879 We will show that 119879 satisfies the contractivecondition (NZ2) of Definition 12 with the control function120595 [0 +infin)
2
rarr [0 +infin) given by 120595(119904 119905) = (12)max119904 119905Let (119909 119910) isin 119860
1times 1198602(the other possibility is treated
similarly) and consider the following cases
(1) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 lt 119896 that is119899 + 1 le 119896 Then 119901(119879119909 119879119910) = 119901([1 1 + 2
minus(119899+1)
] [1 minus
2minus(119896+1)
1]) = (12)(2minus119899
+ 2minus119896
) le (34) sdot 2minus119899
119872(119909 119910) =max 2minus119899 + 2minus119896 32sdot 2minus119899
3
2sdot 2minus119896
1
2(100381610038161003816100381610038162minus119899
minus 2minus(119896+1)
10038161003816100381610038161003816+100381610038161003816100381610038162minus119896
minus 2minus(119899+1)
10038161003816100381610038161003816)
=3
2sdot 2minus119899
(43)
and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot 2minus119899 = (34)sdot2minus119899 Hence the condition (NZ2) reduces to (34) sdot
2minus119899
le (32) sdot 2minus119899
minus (34) sdot 2minus119899 and holds true
(2) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 = 119896Then 119901(119879119909 119879119910) = 2
minus119899 119872(119909 119910) = 2 sdot 2minus119899 and
120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot 2 sdot 2minus119899
= 2minus119899 hence
(NZ2) reduces to 2minus119899 le 2 sdot 2minus119899 minus 2minus119899(3) 119909 = [1 minus 2
minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 gt 119896 that is119899 ge 119896 + 1 Then 119901(119879119909 119879119910) le (34) sdot 2
minus119896 119872(119909 119910) =
(32) sdot 2minus119896 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119896
= (34)sdot2minus119896 Hence (NZ2) reduces to (34)sdot2minus119896 le
(32) sdot 2minus119896
minus (34) sdot 2minus119896 and holds true
(4) 119909 = [1 minus 2minus119899
1] 119910 = 1 Then 119901(119879119909 119879119910) = 119901([1 1 +
2minus(119899+1)
] 1) = (12) sdot 2minus119899 119872(119909 119910) = 119901(119909 119879119909) =
(32) sdot 2minus119899 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119899
= (34) sdot 2minus119899 (NZ2) reduces to (12) sdot 2
minus119899
le
(32) sdot 2minus119899
minus (34) sdot 2minus119899
(5) The case 119909 = 1 119910 = [1 1 + 2minus119896
] is treated symmet-rically
(6) The case 119909 = 119910 = 1 is trivial
We conclude that all conditions of Theorem 13 aresatisfied The mapping 119879 has a unique fixed point 1 isin 119860
1cap
1198602
Here is another example showing the use of Theorem 13
Example 18 Let 119883 = [0 1] be equipped with the partialmetric given as
119901 (119909 119910) =
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 if 119909 119910 isin [0 1)
1 if 119909 = 1 or 119910 = 1(44)
Then (119883 119901) is a 0-complete partial metric space Let 1198601=
[0 12] 1198602= [12 1] and 119879 119883 rarr 119883 be given as
119879119909 =
1
2 if 119909 isin [0 1)
1
6 if 119909 = 1
(45)
Obviously 119883 = 1198601cup 1198602is a cyclic representation of 119883 with
respect to 119879 We will check the contractive condition (NZ2)of Definition 12 with the control function 120595 [0 +infin)
2
rarr
[0 +infin) given by120595(119904 119905) = (119904+119905)(2+119904+119905) Let (119909 119910) isin 1198601times1198602
(the other possibility is treated symmetrically) Consider thefollowing possible cases
(1) 119909 isin [0 12] 119910 isin [12 1) Then 119901(119879119909 119879119910) =
119901(12 12) = 0 119872(119909 119910) = max119910 minus 119909 12 minus
119909 119910 minus 12 (12)(12 minus 119909 + 119910 minus 12) = 119910 minus 119909120595(119901(119909 119910) 119901(119909 119879119909)) = (119910 + 12 minus 2119909)(52 + 119910 minus 2119909)It is easy to check that
119901 (119879119909 119879119910) = 0 le 119910 minus 119909 minus119910 + 12 minus 2119909
52 + 119910 minus 2119909
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(46)
holds for the given values of 119909 and 119910
International Journal of Analysis 7
(2) 119909 isin [0 12] 119910 = 1 Then 119901(119879119909 119879119910) = 119901(12 16) =
13 119872(119909 119910) = 1 and 120595(119901(119909 119910) 119901(119909 119879119909)) =
120595(1 12 minus 119909) = (32 minus 119909)(72 minus 119909) The condition(NZ2) reduces to
1
3le 1 minus
3 minus 2119909
7 minus 2119909(47)
and can be checked directlyThus all the conditions of Theorem 13 are fulfilled and
we conclude that 119879 has a unique fixed point 12 isin 1198601cap 1198602
We state amore involved example that is inspiredwith theone from [48]
Example 19 Let119883 sub ℓ1119883 ni 119909 = (119909
119899)infin
119899=1if and only if 119909
119899ge 0
for each 119899 isin N Define a partial metric 119901 on119883 by
119901 ((119909119899) (119910119899)) =
infin
sum
119899=1
max 119909119899 119910119899 (48)
(it is easy to check that axioms (1199011)ndash(1199014) hold true) Let 120572 isin
(0 1) be fixed denote 0 = (0)infin119899=1
and consider the subsets1198601
and1198602of119883 defined by119860
1= 1198601015840
cup 0 1198602= 11986010158401015840
cup 0 where
1198601015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 or 119899 = 2119896 minus 1 119896 isin N
120572119899
119899 = 2119896 ge 2119897119897 = 1 2
11986010158401015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 minus 1 or 119899 = 2119896 119896 isin N
120572119899
119899 = 2119896 minus 1 ge 2119897 minus 1119897 = 1 2
(49)
Denote 119884 = 1198601cup 1198602(obviously 119860
1cap 1198602= 0)
Consider the mapping 119879 119884 rarr 119884 given by
119879 (0) = 0
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 )
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897+1
1205722119897+2
0 1205722119897+4
0 )
(50)
Obviously 119879(1198601) sub 119860
2and 119879(119860
2) sub 119860
1 hence 119884 = 119860
1cup
1198602is a cyclic representation of 119884 with respect to 119879 Take 120595
[0 +infin)2
rarr [0 +infin) defined by 120595(119904 119905) = (1 minus 120572)max119904 119905Let us check the contractive condition (NZ2) of Theo-
rem 13 Take 119909 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ) isin 1198601and
119910 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119898
1205722119898+1
0 1205722119898+3
0 ) isin 1198602and assume for
example that 119897 le 119898 (the case 119897 gt 119898 is treated similarly aswell as the case when 119909 or 119910 is equal to 0) Then
119901 (119909 119910) = 1205722119897
+ sdot sdot sdot + 1205722119898minus2
+1205722119898
1 minus 120572
119901 (119879119909 119879119910) = 1205722119897+1
+ sdot sdot sdot + 1205722119898minus1
+1205722119898+1
1 minus 120572
119901 (119909 119879119909) =1205722119897
1 minus 120572 119901 (119910 119879119910) =
1205722119898+1
1 minus 120572
(51)
Hence
119901 (119879119909 119879119910) = 120572119901 (119909 119910) le 120572 sdot1205722119897
1 minus 120572
=1205722119897
1 minus 120572minus (1 minus 120572)
1205722119897
1 minus 120572
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(52)
Obviously 119879 has a unique fixed point 0 isin 1198601cap 1198602
Finally we present an example showing that in certainsituations the existence of a fixed point can be concludedunder partial metric conditions while the same cannot beobtained using the standard metric
Example 20 Let 119883 = R+ be equipped with the usual partialmetric 119901(119909 119910) = max119909 119910 Suppose 119860
1= [0 1] 119860
2=
[0 12] 1198603= [0 16] 119860
4= [0 142] and 119884 = ⋃
4
119894=1119860119894
Consider the mapping 119879 119884 rarr 119884 defined by
119879119909 =1199092
1 + 119909 (53)
It is clear that ⋃4119894=1119860119894is a cyclic representation of 119884 with
respect to 119879 Further consider the function 120595 [0 +infin)2
rarr
[0 +infin) given by
120595 (119904 119905) =119904 + 119905
2 + 119904 + 119905 (54)
Take an arbitrary pair (119909 119910) isin 119860119894times119860119894+1
with say 119910 le 119909 (theother possibility can be treated in a similar way) Then
119901 (119879119909 119879119910) = max 1199092
1 + 1199091199102
1 + 119910 =
1199092
1 + 119909 (55)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
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![Page 4: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/4.jpg)
4 International Journal of Analysis
Passing to the limit as 119899 rarr infin in (13) and using (14) andlower semicontinuity of 120595 we have
119903 le 119903 minus lim inf119899rarrinfin
120595 (119901 (119909119899 119909119899+1) 119901 (119909
119899 119909119899+1))
le 119903 minus 120595 (119903 119903)
(15)
thus 120595(119903 119903) = 0 and hence 119903 = 0 Therefore
lim119899rarrinfin
119901 (119909119899 119909119899+1) = 0 (16)
Next we claim that 119909119899 is a 0-Cauchy sequence in the
space (119883 119901) Suppose that this is not the case Then thereexists 120576 gt 0 for which we can find two sequences of positiveintegers 119898(119896) and 119899(119896) such that for all positive integers 119896
119899 (119896) gt 119898 (119896) gt 119896
119901 (119909119898(119896)
119909119899(119896)
) ge 120576
119901 (119909119898(119896)
119909119899(119896)minus1
) lt 120576
(17)
Using (17) and (1199014) we get
120576 le119901 (119909119899(119896)
119909119898(119896)
)
le119901 (119909119898(119896)
119909119899(119896)minus1
)
+ 119901 (119909119899(119896)minus1
119909119899(119896)
) minus 119901 (119909119899(119896)minus1
119909119899(119896)minus1
)
lt120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
)
(18)
Thus we have
120576 le 119901 (119909119899(119896)
119909119898(119896)
) lt 120576 + 119901 (119909119899(119896)
119909119899(119896)minus1
) (19)
Passing to the limit as 119896 rarr infin in the above inequality andusing (16) we obtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)
) = 120576 (20)
On the other hand for all 119896 there exists 119895(119896) isin 1 119902
such that 119899(119896) minus 119898(119896) + 119895(119896) equiv 1[119902] Then 119909119898(119896)minus119895(119896)
(for 119896large enough119898(119896) gt 119895(119896)) and119909
119899(119896)lie in different adjacently
labelled sets 119860119894and 119860
119894+1for certain 119894 isin 1 119902
Using (1199014) and (20) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)
)
+ 119901 (119909119899(119896)
119909119898(119896)
) minus 119901 (119909119899(119896)
119909119899(119896)
)
le
119895(119896)minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
) + 119901 (119909119899(119896)
119909119898(119896)
)
le
119902minus1
sum
119897=0
119901 (119909119898(119896)minus119895(119896)+119897
119909119898(119896)minus119895(119896)+119897+1
)
+ 119901 (119909119899(119896)
119909119898(119896)
) 997888rarr 120576 as 119896 997888rarr infin (from (16))
(21)
that is
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) = 120576 (22)
Using (16) we have
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
) = 0 (23)
lim119896rarrinfin
119901 (119909119899(119896)+1
119909119899(119896)
) = 0 (24)
Again using (1199014) we get
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) le 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
+ 119901 (119909119899(119896)
119909119899(119896)+1
) minus 119901 (119909119899(119896)
119909119899(119896)
)
(25)
Passing to the limit as 119896 rarr infin in the pervious inequality andusing (24) and (22) we get
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
) = 120576 (26)
Similarly we have by (1199014)
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
le 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)+1
)
+ 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) minus 119901 (119909119898(119896)minus119895(119896)
119909119898(119896)minus119895(119896)
)
(27)
Passing to the limit as 119896 rarr infin and using (16) and (22) weobtain
lim119896rarrinfin
119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
) = 120576 (28)
Similarly we have by (1199014)
lim119896rarrinfin
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
) = 120576 (29)
Using (NZ2) we obtain
119901 (119909119898(119896)minus119895(119896)+1
119909119899(119896)+1
)
le max 119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
)
119901 (119909119899(119896)+1
119909119899(119896)
) 1
2119901 (119909119898(119896)minus119895(119896)
119909119899(119896)+1
)
+119901 (119909119899(119896)
119909119898(119896)minus119895(119896)+1
)
minus 120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
) 119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
(30)
for all 119896 Passing to the limit as 119896 rarr infin in the last inequality(and using the lower semicontinuity of the function 120595) weobtain
120576 le 120576 minus lim inf119899rarrinfin
120595 (119901 (119909119898(119896)minus119895(119896)
119909119899(119896)
)
119901 (119909119898(119896)minus119895(119896)+1
119909119898(119896)minus119895(119896)
))
le 120576 minus 120595 (120576 0)
(31)
International Journal of Analysis 5
which implies that 120595(120576 0) = 0 that is a contradiction since120576 gt 0 We deduce that 119909
119899 is a 0-Cauchy sequence
Since (119883 119901) is 0-complete and 119884 is closed it follows thatthe sequence 119909
119899 converges to some 119906 isin 119884 that is
lim119899rarrinfin
119909119899= 119906 (32)
We shall prove that
119906 isin
119902
⋂
119894=1
119860119894 (33)
From condition (NZ1) and since 1199090isin 1198601 we have 119909
119899119902119899ge0
sube
1198601 Since 119860
1is closed from (32) we get that 119906 isin 119860
1 Again
from the condition (NZ1) we have 119909119899119902+1
119899ge0
sube 1198602 Since
1198602is closed from (32) we get that 119906 isin 119860
2 Continuing this
process we obtain (33) and 119901(119906 119906) = 0Now we shall prove that 119906 is a fixed point of 119879 Indeed
from (33) since for all 119899 there exists 119894(119899) isin 1 2 119902 suchthat 119909
119899isin 119860119894(119899)
applying (NZ2) with 119909 = 119906 and 119910 = 119909119899 we
obtain
119901 (119879119906 119909119899+1) = 119901 (119879119906 119879119909
119899)
lemax 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus 120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
(34)
for all 119899 Passing to the limit as 119899 rarr infin in (34) and using(32) we get
119901 (119906 119879119906) le lim119899rarrinfin
max 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus lim inf119899rarrinfin
120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
lemax 0 119901 (119906 119879119906) 0 12119901 (119906 119879119906)
minus 120595 (0 119901 (119906 119879119906)) = 119901 (119906 119879119906) minus 120595 (0 119901 (119906 119879119906))
(35)
which is impossible unless 120595(0 119901(119906 119879119906)) = 0 so
119906 = 119879119906 (36)
that is 119906 is a fixed point of 119879We claim that there is a unique fixed point of 119879 Assume
on the contrary that 119879119906 = 119906 and 119879119907 = 119907 with 119901(119906 119907) gt 0
By supposition we can replace 119909 by 119906 and 119910 by 119907 in (NZ2) toobtain
119901 (119906 119907) =119901 (119879119906 119879119907)
lemax 119901 (119906 119907) 119901 (119906 119879119906) 119901 (119907 119879119907)
1
2[119901 (119906 119879119907) + 119901 (119879119906 119907)]
minus 120595 (119901 (119906 119907) 119901 (119906 119879119906))
=119901 (119906 119907) minus 120595 (119901 (119906 119907) 0) lt 119901 (119906 119907)
(37)
a contradiction Hence 119901(119906 119907) = 0 that is 119906 = 119907 Weconclude that 119879 has only one fixed point in 119883 The proof iscomplete
If we take 119902 = 1 and 1198601= 119883 in Theorem 13 then we get
the following fixed point theorem
Corollary 14 Let (119883 119901) be a 0-complete partial metric spaceand let 119879 119883 rarr 119883 satisfy the following condition there existsa lower semicontinuous mapping 120595 [0infin)
2
rarr [0infin) suchthat 120595(119905 119904) = 0 if and only if 119905 = 119904 = 0 and that
119901 (119879119909 119879119910) lemax 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119910 119879119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(38)
for all 119909 119910 isin 119883 Then 119879 has a unique fixed point 119911 isin 119883Moreover 119901(119911 119911) = 0
Corollary 14 extends and generalizes many existing fixedpoint theorems in the literature
By taking 120595(119904 119905) = (1 minus 119903)max119904 119905 where 119903 isin [0 1) inTheorem 13 we have the following result
Corollary 15 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with respect
to 119879(NZ3) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119903max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(39)
where 119903 isin [0 1) Then 119879 has a unique fixed point 119911 belongingto⋂119902119894=1119860119894 moreover 119901(119911 119911) = 0
As a special case of Corollary 15 we obtain Matthewsrsquosversion of Banach contraction principle [10]
6 International Journal of Analysis
Corollary 16 Let (119883 119901) be a 0-complete partial metric spacelet 119901 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119883 = ⋃119902
119894=1119860119894is a cyclic representation of119883with respect
to 119879(NZ4) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119899
119909 119879119899
119910) lemax 119901 (119909 119910) 119901 (119909 119879119899119909) 119901 (119910 119879119899119910)
1
2[119901 (119909 119879
119899
119910) + 119901 (119910 119879119899
119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119899
119909))
(40)
where 119899 is a positive integer and 120595 [0infin)2
rarr [0infin) is alower semi-continuous mapping such that 120595(119905 119904) = 0 if andonly if 119905 = 119904 = 0 Then 119879 has a unique fixed point belonging to⋂119902
119894=1119860119894
4 Examples
The following example shows howTheorem 13 can be used Itis adapted from [38 Example 29]
Example 17 Consider the partial metric space (119883 119889) ofExample 8 (2) It is easy to see that it is 0-complete Considerthe following closed subsets of119883
1198601= [1 minus 2
minus119899
1] 119899 isin N cup 1
1198602= [1 1 + 2
minus119899
] 119899 isin N cup 1 (41)
119884 = 1198601cup 1198602 and define a mapping 119879 119884 rarr 119884 by
119879119909 =
[1 1 + 2minus(119899+1)
] if 119909 = [1 minus 2minus119899 1]
[1 minus 2minus(119899+1)
1] if 119909 = [1 1 + 2minus119899]
1 if 119909 = 1
(42)
Obviously 119884 = 1198601cup 1198602is a cyclic representation of 119884 with
respect to 119879 We will show that 119879 satisfies the contractivecondition (NZ2) of Definition 12 with the control function120595 [0 +infin)
2
rarr [0 +infin) given by 120595(119904 119905) = (12)max119904 119905Let (119909 119910) isin 119860
1times 1198602(the other possibility is treated
similarly) and consider the following cases
(1) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 lt 119896 that is119899 + 1 le 119896 Then 119901(119879119909 119879119910) = 119901([1 1 + 2
minus(119899+1)
] [1 minus
2minus(119896+1)
1]) = (12)(2minus119899
+ 2minus119896
) le (34) sdot 2minus119899
119872(119909 119910) =max 2minus119899 + 2minus119896 32sdot 2minus119899
3
2sdot 2minus119896
1
2(100381610038161003816100381610038162minus119899
minus 2minus(119896+1)
10038161003816100381610038161003816+100381610038161003816100381610038162minus119896
minus 2minus(119899+1)
10038161003816100381610038161003816)
=3
2sdot 2minus119899
(43)
and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot 2minus119899 = (34)sdot2minus119899 Hence the condition (NZ2) reduces to (34) sdot
2minus119899
le (32) sdot 2minus119899
minus (34) sdot 2minus119899 and holds true
(2) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 = 119896Then 119901(119879119909 119879119910) = 2
minus119899 119872(119909 119910) = 2 sdot 2minus119899 and
120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot 2 sdot 2minus119899
= 2minus119899 hence
(NZ2) reduces to 2minus119899 le 2 sdot 2minus119899 minus 2minus119899(3) 119909 = [1 minus 2
minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 gt 119896 that is119899 ge 119896 + 1 Then 119901(119879119909 119879119910) le (34) sdot 2
minus119896 119872(119909 119910) =
(32) sdot 2minus119896 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119896
= (34)sdot2minus119896 Hence (NZ2) reduces to (34)sdot2minus119896 le
(32) sdot 2minus119896
minus (34) sdot 2minus119896 and holds true
(4) 119909 = [1 minus 2minus119899
1] 119910 = 1 Then 119901(119879119909 119879119910) = 119901([1 1 +
2minus(119899+1)
] 1) = (12) sdot 2minus119899 119872(119909 119910) = 119901(119909 119879119909) =
(32) sdot 2minus119899 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119899
= (34) sdot 2minus119899 (NZ2) reduces to (12) sdot 2
minus119899
le
(32) sdot 2minus119899
minus (34) sdot 2minus119899
(5) The case 119909 = 1 119910 = [1 1 + 2minus119896
] is treated symmet-rically
(6) The case 119909 = 119910 = 1 is trivial
We conclude that all conditions of Theorem 13 aresatisfied The mapping 119879 has a unique fixed point 1 isin 119860
1cap
1198602
Here is another example showing the use of Theorem 13
Example 18 Let 119883 = [0 1] be equipped with the partialmetric given as
119901 (119909 119910) =
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 if 119909 119910 isin [0 1)
1 if 119909 = 1 or 119910 = 1(44)
Then (119883 119901) is a 0-complete partial metric space Let 1198601=
[0 12] 1198602= [12 1] and 119879 119883 rarr 119883 be given as
119879119909 =
1
2 if 119909 isin [0 1)
1
6 if 119909 = 1
(45)
Obviously 119883 = 1198601cup 1198602is a cyclic representation of 119883 with
respect to 119879 We will check the contractive condition (NZ2)of Definition 12 with the control function 120595 [0 +infin)
2
rarr
[0 +infin) given by120595(119904 119905) = (119904+119905)(2+119904+119905) Let (119909 119910) isin 1198601times1198602
(the other possibility is treated symmetrically) Consider thefollowing possible cases
(1) 119909 isin [0 12] 119910 isin [12 1) Then 119901(119879119909 119879119910) =
119901(12 12) = 0 119872(119909 119910) = max119910 minus 119909 12 minus
119909 119910 minus 12 (12)(12 minus 119909 + 119910 minus 12) = 119910 minus 119909120595(119901(119909 119910) 119901(119909 119879119909)) = (119910 + 12 minus 2119909)(52 + 119910 minus 2119909)It is easy to check that
119901 (119879119909 119879119910) = 0 le 119910 minus 119909 minus119910 + 12 minus 2119909
52 + 119910 minus 2119909
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(46)
holds for the given values of 119909 and 119910
International Journal of Analysis 7
(2) 119909 isin [0 12] 119910 = 1 Then 119901(119879119909 119879119910) = 119901(12 16) =
13 119872(119909 119910) = 1 and 120595(119901(119909 119910) 119901(119909 119879119909)) =
120595(1 12 minus 119909) = (32 minus 119909)(72 minus 119909) The condition(NZ2) reduces to
1
3le 1 minus
3 minus 2119909
7 minus 2119909(47)
and can be checked directlyThus all the conditions of Theorem 13 are fulfilled and
we conclude that 119879 has a unique fixed point 12 isin 1198601cap 1198602
We state amore involved example that is inspiredwith theone from [48]
Example 19 Let119883 sub ℓ1119883 ni 119909 = (119909
119899)infin
119899=1if and only if 119909
119899ge 0
for each 119899 isin N Define a partial metric 119901 on119883 by
119901 ((119909119899) (119910119899)) =
infin
sum
119899=1
max 119909119899 119910119899 (48)
(it is easy to check that axioms (1199011)ndash(1199014) hold true) Let 120572 isin
(0 1) be fixed denote 0 = (0)infin119899=1
and consider the subsets1198601
and1198602of119883 defined by119860
1= 1198601015840
cup 0 1198602= 11986010158401015840
cup 0 where
1198601015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 or 119899 = 2119896 minus 1 119896 isin N
120572119899
119899 = 2119896 ge 2119897119897 = 1 2
11986010158401015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 minus 1 or 119899 = 2119896 119896 isin N
120572119899
119899 = 2119896 minus 1 ge 2119897 minus 1119897 = 1 2
(49)
Denote 119884 = 1198601cup 1198602(obviously 119860
1cap 1198602= 0)
Consider the mapping 119879 119884 rarr 119884 given by
119879 (0) = 0
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 )
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897+1
1205722119897+2
0 1205722119897+4
0 )
(50)
Obviously 119879(1198601) sub 119860
2and 119879(119860
2) sub 119860
1 hence 119884 = 119860
1cup
1198602is a cyclic representation of 119884 with respect to 119879 Take 120595
[0 +infin)2
rarr [0 +infin) defined by 120595(119904 119905) = (1 minus 120572)max119904 119905Let us check the contractive condition (NZ2) of Theo-
rem 13 Take 119909 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ) isin 1198601and
119910 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119898
1205722119898+1
0 1205722119898+3
0 ) isin 1198602and assume for
example that 119897 le 119898 (the case 119897 gt 119898 is treated similarly aswell as the case when 119909 or 119910 is equal to 0) Then
119901 (119909 119910) = 1205722119897
+ sdot sdot sdot + 1205722119898minus2
+1205722119898
1 minus 120572
119901 (119879119909 119879119910) = 1205722119897+1
+ sdot sdot sdot + 1205722119898minus1
+1205722119898+1
1 minus 120572
119901 (119909 119879119909) =1205722119897
1 minus 120572 119901 (119910 119879119910) =
1205722119898+1
1 minus 120572
(51)
Hence
119901 (119879119909 119879119910) = 120572119901 (119909 119910) le 120572 sdot1205722119897
1 minus 120572
=1205722119897
1 minus 120572minus (1 minus 120572)
1205722119897
1 minus 120572
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(52)
Obviously 119879 has a unique fixed point 0 isin 1198601cap 1198602
Finally we present an example showing that in certainsituations the existence of a fixed point can be concludedunder partial metric conditions while the same cannot beobtained using the standard metric
Example 20 Let 119883 = R+ be equipped with the usual partialmetric 119901(119909 119910) = max119909 119910 Suppose 119860
1= [0 1] 119860
2=
[0 12] 1198603= [0 16] 119860
4= [0 142] and 119884 = ⋃
4
119894=1119860119894
Consider the mapping 119879 119884 rarr 119884 defined by
119879119909 =1199092
1 + 119909 (53)
It is clear that ⋃4119894=1119860119894is a cyclic representation of 119884 with
respect to 119879 Further consider the function 120595 [0 +infin)2
rarr
[0 +infin) given by
120595 (119904 119905) =119904 + 119905
2 + 119904 + 119905 (54)
Take an arbitrary pair (119909 119910) isin 119860119894times119860119894+1
with say 119910 le 119909 (theother possibility can be treated in a similar way) Then
119901 (119879119909 119879119910) = max 1199092
1 + 1199091199102
1 + 119910 =
1199092
1 + 119909 (55)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 5: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/5.jpg)
International Journal of Analysis 5
which implies that 120595(120576 0) = 0 that is a contradiction since120576 gt 0 We deduce that 119909
119899 is a 0-Cauchy sequence
Since (119883 119901) is 0-complete and 119884 is closed it follows thatthe sequence 119909
119899 converges to some 119906 isin 119884 that is
lim119899rarrinfin
119909119899= 119906 (32)
We shall prove that
119906 isin
119902
⋂
119894=1
119860119894 (33)
From condition (NZ1) and since 1199090isin 1198601 we have 119909
119899119902119899ge0
sube
1198601 Since 119860
1is closed from (32) we get that 119906 isin 119860
1 Again
from the condition (NZ1) we have 119909119899119902+1
119899ge0
sube 1198602 Since
1198602is closed from (32) we get that 119906 isin 119860
2 Continuing this
process we obtain (33) and 119901(119906 119906) = 0Now we shall prove that 119906 is a fixed point of 119879 Indeed
from (33) since for all 119899 there exists 119894(119899) isin 1 2 119902 suchthat 119909
119899isin 119860119894(119899)
applying (NZ2) with 119909 = 119906 and 119910 = 119909119899 we
obtain
119901 (119879119906 119909119899+1) = 119901 (119879119906 119879119909
119899)
lemax 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus 120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
(34)
for all 119899 Passing to the limit as 119899 rarr infin in (34) and using(32) we get
119901 (119906 119879119906) le lim119899rarrinfin
max 119901 (119906 119909119899) 119901 (119906 119879119906) 119901 (119909
119899 119909119899+1)
1
2119901 (119906 119909
119899+1) + 119901 (119909
119899 119879119906)
minus lim inf119899rarrinfin
120595 (119901 (119906 119909119899) 119901 (119906 119879119906))
lemax 0 119901 (119906 119879119906) 0 12119901 (119906 119879119906)
minus 120595 (0 119901 (119906 119879119906)) = 119901 (119906 119879119906) minus 120595 (0 119901 (119906 119879119906))
(35)
which is impossible unless 120595(0 119901(119906 119879119906)) = 0 so
119906 = 119879119906 (36)
that is 119906 is a fixed point of 119879We claim that there is a unique fixed point of 119879 Assume
on the contrary that 119879119906 = 119906 and 119879119907 = 119907 with 119901(119906 119907) gt 0
By supposition we can replace 119909 by 119906 and 119910 by 119907 in (NZ2) toobtain
119901 (119906 119907) =119901 (119879119906 119879119907)
lemax 119901 (119906 119907) 119901 (119906 119879119906) 119901 (119907 119879119907)
1
2[119901 (119906 119879119907) + 119901 (119879119906 119907)]
minus 120595 (119901 (119906 119907) 119901 (119906 119879119906))
=119901 (119906 119907) minus 120595 (119901 (119906 119907) 0) lt 119901 (119906 119907)
(37)
a contradiction Hence 119901(119906 119907) = 0 that is 119906 = 119907 Weconclude that 119879 has only one fixed point in 119883 The proof iscomplete
If we take 119902 = 1 and 1198601= 119883 in Theorem 13 then we get
the following fixed point theorem
Corollary 14 Let (119883 119901) be a 0-complete partial metric spaceand let 119879 119883 rarr 119883 satisfy the following condition there existsa lower semicontinuous mapping 120595 [0infin)
2
rarr [0infin) suchthat 120595(119905 119904) = 0 if and only if 119905 = 119904 = 0 and that
119901 (119879119909 119879119910) lemax 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119910 119879119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(38)
for all 119909 119910 isin 119883 Then 119879 has a unique fixed point 119911 isin 119883Moreover 119901(119911 119911) = 0
Corollary 14 extends and generalizes many existing fixedpoint theorems in the literature
By taking 120595(119904 119905) = (1 minus 119903)max119904 119905 where 119903 isin [0 1) inTheorem 13 we have the following result
Corollary 15 Let (119883 119901) be a 0-complete partial metric spacelet 119902 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119884 = ⋃119902
119894=1119860119894is a cyclic representation of 119884 with respect
to 119879(NZ3) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119909 119879119910) le 119903max 119901 (119909 119910) 119901 (119909 119879119909) 119901 (119910 119879119910)
1
2[119901 (119909 119879119910) + 119901 (119879119909 119910)]
(39)
where 119903 isin [0 1) Then 119879 has a unique fixed point 119911 belongingto⋂119902119894=1119860119894 moreover 119901(119911 119911) = 0
As a special case of Corollary 15 we obtain Matthewsrsquosversion of Banach contraction principle [10]
6 International Journal of Analysis
Corollary 16 Let (119883 119901) be a 0-complete partial metric spacelet 119901 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119883 = ⋃119902
119894=1119860119894is a cyclic representation of119883with respect
to 119879(NZ4) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119899
119909 119879119899
119910) lemax 119901 (119909 119910) 119901 (119909 119879119899119909) 119901 (119910 119879119899119910)
1
2[119901 (119909 119879
119899
119910) + 119901 (119910 119879119899
119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119899
119909))
(40)
where 119899 is a positive integer and 120595 [0infin)2
rarr [0infin) is alower semi-continuous mapping such that 120595(119905 119904) = 0 if andonly if 119905 = 119904 = 0 Then 119879 has a unique fixed point belonging to⋂119902
119894=1119860119894
4 Examples
The following example shows howTheorem 13 can be used Itis adapted from [38 Example 29]
Example 17 Consider the partial metric space (119883 119889) ofExample 8 (2) It is easy to see that it is 0-complete Considerthe following closed subsets of119883
1198601= [1 minus 2
minus119899
1] 119899 isin N cup 1
1198602= [1 1 + 2
minus119899
] 119899 isin N cup 1 (41)
119884 = 1198601cup 1198602 and define a mapping 119879 119884 rarr 119884 by
119879119909 =
[1 1 + 2minus(119899+1)
] if 119909 = [1 minus 2minus119899 1]
[1 minus 2minus(119899+1)
1] if 119909 = [1 1 + 2minus119899]
1 if 119909 = 1
(42)
Obviously 119884 = 1198601cup 1198602is a cyclic representation of 119884 with
respect to 119879 We will show that 119879 satisfies the contractivecondition (NZ2) of Definition 12 with the control function120595 [0 +infin)
2
rarr [0 +infin) given by 120595(119904 119905) = (12)max119904 119905Let (119909 119910) isin 119860
1times 1198602(the other possibility is treated
similarly) and consider the following cases
(1) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 lt 119896 that is119899 + 1 le 119896 Then 119901(119879119909 119879119910) = 119901([1 1 + 2
minus(119899+1)
] [1 minus
2minus(119896+1)
1]) = (12)(2minus119899
+ 2minus119896
) le (34) sdot 2minus119899
119872(119909 119910) =max 2minus119899 + 2minus119896 32sdot 2minus119899
3
2sdot 2minus119896
1
2(100381610038161003816100381610038162minus119899
minus 2minus(119896+1)
10038161003816100381610038161003816+100381610038161003816100381610038162minus119896
minus 2minus(119899+1)
10038161003816100381610038161003816)
=3
2sdot 2minus119899
(43)
and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot 2minus119899 = (34)sdot2minus119899 Hence the condition (NZ2) reduces to (34) sdot
2minus119899
le (32) sdot 2minus119899
minus (34) sdot 2minus119899 and holds true
(2) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 = 119896Then 119901(119879119909 119879119910) = 2
minus119899 119872(119909 119910) = 2 sdot 2minus119899 and
120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot 2 sdot 2minus119899
= 2minus119899 hence
(NZ2) reduces to 2minus119899 le 2 sdot 2minus119899 minus 2minus119899(3) 119909 = [1 minus 2
minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 gt 119896 that is119899 ge 119896 + 1 Then 119901(119879119909 119879119910) le (34) sdot 2
minus119896 119872(119909 119910) =
(32) sdot 2minus119896 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119896
= (34)sdot2minus119896 Hence (NZ2) reduces to (34)sdot2minus119896 le
(32) sdot 2minus119896
minus (34) sdot 2minus119896 and holds true
(4) 119909 = [1 minus 2minus119899
1] 119910 = 1 Then 119901(119879119909 119879119910) = 119901([1 1 +
2minus(119899+1)
] 1) = (12) sdot 2minus119899 119872(119909 119910) = 119901(119909 119879119909) =
(32) sdot 2minus119899 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119899
= (34) sdot 2minus119899 (NZ2) reduces to (12) sdot 2
minus119899
le
(32) sdot 2minus119899
minus (34) sdot 2minus119899
(5) The case 119909 = 1 119910 = [1 1 + 2minus119896
] is treated symmet-rically
(6) The case 119909 = 119910 = 1 is trivial
We conclude that all conditions of Theorem 13 aresatisfied The mapping 119879 has a unique fixed point 1 isin 119860
1cap
1198602
Here is another example showing the use of Theorem 13
Example 18 Let 119883 = [0 1] be equipped with the partialmetric given as
119901 (119909 119910) =
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 if 119909 119910 isin [0 1)
1 if 119909 = 1 or 119910 = 1(44)
Then (119883 119901) is a 0-complete partial metric space Let 1198601=
[0 12] 1198602= [12 1] and 119879 119883 rarr 119883 be given as
119879119909 =
1
2 if 119909 isin [0 1)
1
6 if 119909 = 1
(45)
Obviously 119883 = 1198601cup 1198602is a cyclic representation of 119883 with
respect to 119879 We will check the contractive condition (NZ2)of Definition 12 with the control function 120595 [0 +infin)
2
rarr
[0 +infin) given by120595(119904 119905) = (119904+119905)(2+119904+119905) Let (119909 119910) isin 1198601times1198602
(the other possibility is treated symmetrically) Consider thefollowing possible cases
(1) 119909 isin [0 12] 119910 isin [12 1) Then 119901(119879119909 119879119910) =
119901(12 12) = 0 119872(119909 119910) = max119910 minus 119909 12 minus
119909 119910 minus 12 (12)(12 minus 119909 + 119910 minus 12) = 119910 minus 119909120595(119901(119909 119910) 119901(119909 119879119909)) = (119910 + 12 minus 2119909)(52 + 119910 minus 2119909)It is easy to check that
119901 (119879119909 119879119910) = 0 le 119910 minus 119909 minus119910 + 12 minus 2119909
52 + 119910 minus 2119909
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(46)
holds for the given values of 119909 and 119910
International Journal of Analysis 7
(2) 119909 isin [0 12] 119910 = 1 Then 119901(119879119909 119879119910) = 119901(12 16) =
13 119872(119909 119910) = 1 and 120595(119901(119909 119910) 119901(119909 119879119909)) =
120595(1 12 minus 119909) = (32 minus 119909)(72 minus 119909) The condition(NZ2) reduces to
1
3le 1 minus
3 minus 2119909
7 minus 2119909(47)
and can be checked directlyThus all the conditions of Theorem 13 are fulfilled and
we conclude that 119879 has a unique fixed point 12 isin 1198601cap 1198602
We state amore involved example that is inspiredwith theone from [48]
Example 19 Let119883 sub ℓ1119883 ni 119909 = (119909
119899)infin
119899=1if and only if 119909
119899ge 0
for each 119899 isin N Define a partial metric 119901 on119883 by
119901 ((119909119899) (119910119899)) =
infin
sum
119899=1
max 119909119899 119910119899 (48)
(it is easy to check that axioms (1199011)ndash(1199014) hold true) Let 120572 isin
(0 1) be fixed denote 0 = (0)infin119899=1
and consider the subsets1198601
and1198602of119883 defined by119860
1= 1198601015840
cup 0 1198602= 11986010158401015840
cup 0 where
1198601015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 or 119899 = 2119896 minus 1 119896 isin N
120572119899
119899 = 2119896 ge 2119897119897 = 1 2
11986010158401015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 minus 1 or 119899 = 2119896 119896 isin N
120572119899
119899 = 2119896 minus 1 ge 2119897 minus 1119897 = 1 2
(49)
Denote 119884 = 1198601cup 1198602(obviously 119860
1cap 1198602= 0)
Consider the mapping 119879 119884 rarr 119884 given by
119879 (0) = 0
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 )
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897+1
1205722119897+2
0 1205722119897+4
0 )
(50)
Obviously 119879(1198601) sub 119860
2and 119879(119860
2) sub 119860
1 hence 119884 = 119860
1cup
1198602is a cyclic representation of 119884 with respect to 119879 Take 120595
[0 +infin)2
rarr [0 +infin) defined by 120595(119904 119905) = (1 minus 120572)max119904 119905Let us check the contractive condition (NZ2) of Theo-
rem 13 Take 119909 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ) isin 1198601and
119910 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119898
1205722119898+1
0 1205722119898+3
0 ) isin 1198602and assume for
example that 119897 le 119898 (the case 119897 gt 119898 is treated similarly aswell as the case when 119909 or 119910 is equal to 0) Then
119901 (119909 119910) = 1205722119897
+ sdot sdot sdot + 1205722119898minus2
+1205722119898
1 minus 120572
119901 (119879119909 119879119910) = 1205722119897+1
+ sdot sdot sdot + 1205722119898minus1
+1205722119898+1
1 minus 120572
119901 (119909 119879119909) =1205722119897
1 minus 120572 119901 (119910 119879119910) =
1205722119898+1
1 minus 120572
(51)
Hence
119901 (119879119909 119879119910) = 120572119901 (119909 119910) le 120572 sdot1205722119897
1 minus 120572
=1205722119897
1 minus 120572minus (1 minus 120572)
1205722119897
1 minus 120572
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(52)
Obviously 119879 has a unique fixed point 0 isin 1198601cap 1198602
Finally we present an example showing that in certainsituations the existence of a fixed point can be concludedunder partial metric conditions while the same cannot beobtained using the standard metric
Example 20 Let 119883 = R+ be equipped with the usual partialmetric 119901(119909 119910) = max119909 119910 Suppose 119860
1= [0 1] 119860
2=
[0 12] 1198603= [0 16] 119860
4= [0 142] and 119884 = ⋃
4
119894=1119860119894
Consider the mapping 119879 119884 rarr 119884 defined by
119879119909 =1199092
1 + 119909 (53)
It is clear that ⋃4119894=1119860119894is a cyclic representation of 119884 with
respect to 119879 Further consider the function 120595 [0 +infin)2
rarr
[0 +infin) given by
120595 (119904 119905) =119904 + 119905
2 + 119904 + 119905 (54)
Take an arbitrary pair (119909 119910) isin 119860119894times119860119894+1
with say 119910 le 119909 (theother possibility can be treated in a similar way) Then
119901 (119879119909 119879119910) = max 1199092
1 + 1199091199102
1 + 119910 =
1199092
1 + 119909 (55)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
![Page 6: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/6.jpg)
6 International Journal of Analysis
Corollary 16 Let (119883 119901) be a 0-complete partial metric spacelet 119901 isin N 119860
1 1198602 119860
119902be nonempty closed subsets of 119883
119884 = ⋃119902
119894=1119860119894 and 119879 119884 rarr 119884 such that
(NZ1) 119883 = ⋃119902
119894=1119860119894is a cyclic representation of119883with respect
to 119879(NZ4) for any (119909 119910) isin 119860
119894times119860119894+1 119894 = 1 2 119902 (with119860
119902+1=
1198601)
119901 (119879119899
119909 119879119899
119910) lemax 119901 (119909 119910) 119901 (119909 119879119899119909) 119901 (119910 119879119899119910)
1
2[119901 (119909 119879
119899
119910) + 119901 (119910 119879119899
119909)]
minus 120595 (119901 (119909 119910) 119901 (119909 119879119899
119909))
(40)
where 119899 is a positive integer and 120595 [0infin)2
rarr [0infin) is alower semi-continuous mapping such that 120595(119905 119904) = 0 if andonly if 119905 = 119904 = 0 Then 119879 has a unique fixed point belonging to⋂119902
119894=1119860119894
4 Examples
The following example shows howTheorem 13 can be used Itis adapted from [38 Example 29]
Example 17 Consider the partial metric space (119883 119889) ofExample 8 (2) It is easy to see that it is 0-complete Considerthe following closed subsets of119883
1198601= [1 minus 2
minus119899
1] 119899 isin N cup 1
1198602= [1 1 + 2
minus119899
] 119899 isin N cup 1 (41)
119884 = 1198601cup 1198602 and define a mapping 119879 119884 rarr 119884 by
119879119909 =
[1 1 + 2minus(119899+1)
] if 119909 = [1 minus 2minus119899 1]
[1 minus 2minus(119899+1)
1] if 119909 = [1 1 + 2minus119899]
1 if 119909 = 1
(42)
Obviously 119884 = 1198601cup 1198602is a cyclic representation of 119884 with
respect to 119879 We will show that 119879 satisfies the contractivecondition (NZ2) of Definition 12 with the control function120595 [0 +infin)
2
rarr [0 +infin) given by 120595(119904 119905) = (12)max119904 119905Let (119909 119910) isin 119860
1times 1198602(the other possibility is treated
similarly) and consider the following cases
(1) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 lt 119896 that is119899 + 1 le 119896 Then 119901(119879119909 119879119910) = 119901([1 1 + 2
minus(119899+1)
] [1 minus
2minus(119896+1)
1]) = (12)(2minus119899
+ 2minus119896
) le (34) sdot 2minus119899
119872(119909 119910) =max 2minus119899 + 2minus119896 32sdot 2minus119899
3
2sdot 2minus119896
1
2(100381610038161003816100381610038162minus119899
minus 2minus(119896+1)
10038161003816100381610038161003816+100381610038161003816100381610038162minus119896
minus 2minus(119899+1)
10038161003816100381610038161003816)
=3
2sdot 2minus119899
(43)
and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot 2minus119899 = (34)sdot2minus119899 Hence the condition (NZ2) reduces to (34) sdot
2minus119899
le (32) sdot 2minus119899
minus (34) sdot 2minus119899 and holds true
(2) 119909 = [1 minus 2minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 = 119896Then 119901(119879119909 119879119910) = 2
minus119899 119872(119909 119910) = 2 sdot 2minus119899 and
120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot 2 sdot 2minus119899
= 2minus119899 hence
(NZ2) reduces to 2minus119899 le 2 sdot 2minus119899 minus 2minus119899(3) 119909 = [1 minus 2
minus119899
1] 119910 = [1 1 + 2minus119896
] and 119899 gt 119896 that is119899 ge 119896 + 1 Then 119901(119879119909 119879119910) le (34) sdot 2
minus119896 119872(119909 119910) =
(32) sdot 2minus119896 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119896
= (34)sdot2minus119896 Hence (NZ2) reduces to (34)sdot2minus119896 le
(32) sdot 2minus119896
minus (34) sdot 2minus119896 and holds true
(4) 119909 = [1 minus 2minus119899
1] 119910 = 1 Then 119901(119879119909 119879119910) = 119901([1 1 +
2minus(119899+1)
] 1) = (12) sdot 2minus119899 119872(119909 119910) = 119901(119909 119879119909) =
(32) sdot 2minus119899 and 120595(119901(119909 119910) 119901(119909 119879119909)) = (12) sdot (32) sdot
2minus119899
= (34) sdot 2minus119899 (NZ2) reduces to (12) sdot 2
minus119899
le
(32) sdot 2minus119899
minus (34) sdot 2minus119899
(5) The case 119909 = 1 119910 = [1 1 + 2minus119896
] is treated symmet-rically
(6) The case 119909 = 119910 = 1 is trivial
We conclude that all conditions of Theorem 13 aresatisfied The mapping 119879 has a unique fixed point 1 isin 119860
1cap
1198602
Here is another example showing the use of Theorem 13
Example 18 Let 119883 = [0 1] be equipped with the partialmetric given as
119901 (119909 119910) =
1003816100381610038161003816119909 minus 1199101003816100381610038161003816 if 119909 119910 isin [0 1)
1 if 119909 = 1 or 119910 = 1(44)
Then (119883 119901) is a 0-complete partial metric space Let 1198601=
[0 12] 1198602= [12 1] and 119879 119883 rarr 119883 be given as
119879119909 =
1
2 if 119909 isin [0 1)
1
6 if 119909 = 1
(45)
Obviously 119883 = 1198601cup 1198602is a cyclic representation of 119883 with
respect to 119879 We will check the contractive condition (NZ2)of Definition 12 with the control function 120595 [0 +infin)
2
rarr
[0 +infin) given by120595(119904 119905) = (119904+119905)(2+119904+119905) Let (119909 119910) isin 1198601times1198602
(the other possibility is treated symmetrically) Consider thefollowing possible cases
(1) 119909 isin [0 12] 119910 isin [12 1) Then 119901(119879119909 119879119910) =
119901(12 12) = 0 119872(119909 119910) = max119910 minus 119909 12 minus
119909 119910 minus 12 (12)(12 minus 119909 + 119910 minus 12) = 119910 minus 119909120595(119901(119909 119910) 119901(119909 119879119909)) = (119910 + 12 minus 2119909)(52 + 119910 minus 2119909)It is easy to check that
119901 (119879119909 119879119910) = 0 le 119910 minus 119909 minus119910 + 12 minus 2119909
52 + 119910 minus 2119909
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(46)
holds for the given values of 119909 and 119910
International Journal of Analysis 7
(2) 119909 isin [0 12] 119910 = 1 Then 119901(119879119909 119879119910) = 119901(12 16) =
13 119872(119909 119910) = 1 and 120595(119901(119909 119910) 119901(119909 119879119909)) =
120595(1 12 minus 119909) = (32 minus 119909)(72 minus 119909) The condition(NZ2) reduces to
1
3le 1 minus
3 minus 2119909
7 minus 2119909(47)
and can be checked directlyThus all the conditions of Theorem 13 are fulfilled and
we conclude that 119879 has a unique fixed point 12 isin 1198601cap 1198602
We state amore involved example that is inspiredwith theone from [48]
Example 19 Let119883 sub ℓ1119883 ni 119909 = (119909
119899)infin
119899=1if and only if 119909
119899ge 0
for each 119899 isin N Define a partial metric 119901 on119883 by
119901 ((119909119899) (119910119899)) =
infin
sum
119899=1
max 119909119899 119910119899 (48)
(it is easy to check that axioms (1199011)ndash(1199014) hold true) Let 120572 isin
(0 1) be fixed denote 0 = (0)infin119899=1
and consider the subsets1198601
and1198602of119883 defined by119860
1= 1198601015840
cup 0 1198602= 11986010158401015840
cup 0 where
1198601015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 or 119899 = 2119896 minus 1 119896 isin N
120572119899
119899 = 2119896 ge 2119897119897 = 1 2
11986010158401015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 minus 1 or 119899 = 2119896 119896 isin N
120572119899
119899 = 2119896 minus 1 ge 2119897 minus 1119897 = 1 2
(49)
Denote 119884 = 1198601cup 1198602(obviously 119860
1cap 1198602= 0)
Consider the mapping 119879 119884 rarr 119884 given by
119879 (0) = 0
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 )
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897+1
1205722119897+2
0 1205722119897+4
0 )
(50)
Obviously 119879(1198601) sub 119860
2and 119879(119860
2) sub 119860
1 hence 119884 = 119860
1cup
1198602is a cyclic representation of 119884 with respect to 119879 Take 120595
[0 +infin)2
rarr [0 +infin) defined by 120595(119904 119905) = (1 minus 120572)max119904 119905Let us check the contractive condition (NZ2) of Theo-
rem 13 Take 119909 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ) isin 1198601and
119910 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119898
1205722119898+1
0 1205722119898+3
0 ) isin 1198602and assume for
example that 119897 le 119898 (the case 119897 gt 119898 is treated similarly aswell as the case when 119909 or 119910 is equal to 0) Then
119901 (119909 119910) = 1205722119897
+ sdot sdot sdot + 1205722119898minus2
+1205722119898
1 minus 120572
119901 (119879119909 119879119910) = 1205722119897+1
+ sdot sdot sdot + 1205722119898minus1
+1205722119898+1
1 minus 120572
119901 (119909 119879119909) =1205722119897
1 minus 120572 119901 (119910 119879119910) =
1205722119898+1
1 minus 120572
(51)
Hence
119901 (119879119909 119879119910) = 120572119901 (119909 119910) le 120572 sdot1205722119897
1 minus 120572
=1205722119897
1 minus 120572minus (1 minus 120572)
1205722119897
1 minus 120572
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(52)
Obviously 119879 has a unique fixed point 0 isin 1198601cap 1198602
Finally we present an example showing that in certainsituations the existence of a fixed point can be concludedunder partial metric conditions while the same cannot beobtained using the standard metric
Example 20 Let 119883 = R+ be equipped with the usual partialmetric 119901(119909 119910) = max119909 119910 Suppose 119860
1= [0 1] 119860
2=
[0 12] 1198603= [0 16] 119860
4= [0 142] and 119884 = ⋃
4
119894=1119860119894
Consider the mapping 119879 119884 rarr 119884 defined by
119879119909 =1199092
1 + 119909 (53)
It is clear that ⋃4119894=1119860119894is a cyclic representation of 119884 with
respect to 119879 Further consider the function 120595 [0 +infin)2
rarr
[0 +infin) given by
120595 (119904 119905) =119904 + 119905
2 + 119904 + 119905 (54)
Take an arbitrary pair (119909 119910) isin 119860119894times119860119894+1
with say 119910 le 119909 (theother possibility can be treated in a similar way) Then
119901 (119879119909 119879119910) = max 1199092
1 + 1199091199102
1 + 119910 =
1199092
1 + 119909 (55)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Page 7: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/7.jpg)
International Journal of Analysis 7
(2) 119909 isin [0 12] 119910 = 1 Then 119901(119879119909 119879119910) = 119901(12 16) =
13 119872(119909 119910) = 1 and 120595(119901(119909 119910) 119901(119909 119879119909)) =
120595(1 12 minus 119909) = (32 minus 119909)(72 minus 119909) The condition(NZ2) reduces to
1
3le 1 minus
3 minus 2119909
7 minus 2119909(47)
and can be checked directlyThus all the conditions of Theorem 13 are fulfilled and
we conclude that 119879 has a unique fixed point 12 isin 1198601cap 1198602
We state amore involved example that is inspiredwith theone from [48]
Example 19 Let119883 sub ℓ1119883 ni 119909 = (119909
119899)infin
119899=1if and only if 119909
119899ge 0
for each 119899 isin N Define a partial metric 119901 on119883 by
119901 ((119909119899) (119910119899)) =
infin
sum
119899=1
max 119909119899 119910119899 (48)
(it is easy to check that axioms (1199011)ndash(1199014) hold true) Let 120572 isin
(0 1) be fixed denote 0 = (0)infin119899=1
and consider the subsets1198601
and1198602of119883 defined by119860
1= 1198601015840
cup 0 1198602= 11986010158401015840
cup 0 where
1198601015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 or 119899 = 2119896 minus 1 119896 isin N
120572119899
119899 = 2119896 ge 2119897119897 = 1 2
11986010158401015840
ni 119909119897
= (119909119897
119899)infin
119899=1
iff 119909119897
119899=
0 119899 lt 2119897 minus 1 or 119899 = 2119896 119896 isin N
120572119899
119899 = 2119896 minus 1 ge 2119897 minus 1119897 = 1 2
(49)
Denote 119884 = 1198601cup 1198602(obviously 119860
1cap 1198602= 0)
Consider the mapping 119879 119884 rarr 119884 given by
119879 (0) = 0
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 )
119879((0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897
1205722119897+1
0 1205722119897+3
0 ))
= (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897+1
1205722119897+2
0 1205722119897+4
0 )
(50)
Obviously 119879(1198601) sub 119860
2and 119879(119860
2) sub 119860
1 hence 119884 = 119860
1cup
1198602is a cyclic representation of 119884 with respect to 119879 Take 120595
[0 +infin)2
rarr [0 +infin) defined by 120595(119904 119905) = (1 minus 120572)max119904 119905Let us check the contractive condition (NZ2) of Theo-
rem 13 Take 119909 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119897minus1
1205722119897
0 1205722119897+2
0 ) isin 1198601and
119910 = (0 0⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
2119898
1205722119898+1
0 1205722119898+3
0 ) isin 1198602and assume for
example that 119897 le 119898 (the case 119897 gt 119898 is treated similarly aswell as the case when 119909 or 119910 is equal to 0) Then
119901 (119909 119910) = 1205722119897
+ sdot sdot sdot + 1205722119898minus2
+1205722119898
1 minus 120572
119901 (119879119909 119879119910) = 1205722119897+1
+ sdot sdot sdot + 1205722119898minus1
+1205722119898+1
1 minus 120572
119901 (119909 119879119909) =1205722119897
1 minus 120572 119901 (119910 119879119910) =
1205722119898+1
1 minus 120572
(51)
Hence
119901 (119879119909 119879119910) = 120572119901 (119909 119910) le 120572 sdot1205722119897
1 minus 120572
=1205722119897
1 minus 120572minus (1 minus 120572)
1205722119897
1 minus 120572
= 119872(119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909))
(52)
Obviously 119879 has a unique fixed point 0 isin 1198601cap 1198602
Finally we present an example showing that in certainsituations the existence of a fixed point can be concludedunder partial metric conditions while the same cannot beobtained using the standard metric
Example 20 Let 119883 = R+ be equipped with the usual partialmetric 119901(119909 119910) = max119909 119910 Suppose 119860
1= [0 1] 119860
2=
[0 12] 1198603= [0 16] 119860
4= [0 142] and 119884 = ⋃
4
119894=1119860119894
Consider the mapping 119879 119884 rarr 119884 defined by
119879119909 =1199092
1 + 119909 (53)
It is clear that ⋃4119894=1119860119894is a cyclic representation of 119884 with
respect to 119879 Further consider the function 120595 [0 +infin)2
rarr
[0 +infin) given by
120595 (119904 119905) =119904 + 119905
2 + 119904 + 119905 (54)
Take an arbitrary pair (119909 119910) isin 119860119894times119860119894+1
with say 119910 le 119909 (theother possibility can be treated in a similar way) Then
119901 (119879119909 119879119910) = max 1199092
1 + 1199091199102
1 + 119910 =
1199092
1 + 119909 (55)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
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
![Page 8: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/8.jpg)
8 International Journal of Analysis
On the other hand
119872(119909 119910) =max119901 (119909 119910) 119901(119909 1199092
1 + 119909) 119901(119910
1199102
1 + 119910)
1
2(119901(119909
1199102
1 + 119910) + 119901(119910
1199092
1 + 119909))
=max119909 119909 119910 12(119909 +max119910 119909
2
1 + 119909) = 119909
119872 (119909 119910) minus 120595 (119901 (119909 119910) 119901 (119909 119879119909)) = 119909 minus2119909
2 + 2119909=
1199092
1 + 119909
(56)
Hence condition (NZ2) is satisfied as well as other condi-tions of Theorem 13 (with 119902 = 4) We deduce that 119879 has aunique fixed point 119911 = 0 isin 119860
1cap 1198602cap 1198603cap 1198604
On the other hand consider the same problem in thestandard metric 119889(119909 119910) = |119909minus119910| and take 119909 = 1 and 119910 = 12Then
119889 (119879119909 119879119910) =
1003816100381610038161003816100381610038161003816
1
2minus1
6
1003816100381610038161003816100381610038161003816=1
3
119872 (119909 119910) = max 121
21
31
2(5
6+ 0) =
1
2
(57)
and hence
119872(119909 119910) minus 120595 (119889 (119909 119910) 119889 (119909 119879119909)) =1
2minus1
3=1
6 (58)
Thus condition (NZ2) for 119901 = 119889 does not hold and theexistence of a fixed point of 119879 cannot be derived using thestandard metric
Remark 21 The results of this paper are obtained under theassumption that the given partial metric space is 0-completeTaking into account Lemma 10 and Example 11 it followsthat they also hold if the space is complete but that ourassumption is weaker
Acknowledgments
The authors thank the referees for valuable suggestionsthat helped them to improve the text The second authoris thankful to the Ministry of Science and TechnologicalDevelopment of Serbia
References
[1] W A Kirk P S Srinivasan and P Veeramani ldquoFixed points formappings satisfying cyclical contractive conditionsrdquo Fixed PointTheory vol 4 no 1 pp 79ndash89 2003
[2] M Pacurar and I A Rus ldquoFixed point theory for cyclic 120593-contractionsrdquo Nonlinear Analysis vol 72 no 3-4 pp 1181ndash11872010
[3] E Karapınar ldquoFixed point theory for cyclic weak 120601-contrac-tionrdquo Applied Mathematics Letters vol 24 no 6 pp 822ndash8252011
[4] E Karapinar M Jleli and B Samet ldquoFixed point resultsfor almost generalized cyclic (Ψ 120601)-weak contractive typemappings with applicationsrdquoAbstract and Applied Analysis vol2012 Article ID 917831 17 pages 2012
[5] E Karapinar and K Sadarangani ldquoFixed point theory for cyclic(120593 minus 120595)-contractionsrdquo Fixed Point Theory and Applications vol2011 article 69 2011
[6] E Karapinar and SMoradi ldquoFixed point theory for cyclic gene-talized (120601 120601)-contraction mappingsrdquo Annali DellrsquoUniversitarsquo diFerrara In press
[7] E Karapinar and H K Nashine ldquoFixed point theorem forcyclic weakly Chatterjea type contractionsrdquo Journal of AppliedMathematics vol 2012 Article ID 165698 15 pages 2012
[8] E Karapinar and H K Nashine ldquoFixed point theorems forKanaan type cyclicweakly contractionsrdquoNonlinearAnalysis andOptimization In press
[9] H K Nashine ldquoCyclic generalized 120595-weakly contractive map-pings and fixed point results with applications to integralequationsrdquo Nonlinear Analysis vol 75 no 16 pp 6160ndash61692012
[10] S G Matthews ldquoPartial metric topologyrdquo Annals of the NewYork Academy of Sciences vol 728 pp 183ndash197 1994 Proceed-ings of the 8th Summer Conference on General Topology andApplications
[11] R Heckmann ldquoApproximation of metric spaces by partialmetric spacesrdquo Applied Categorical Structures vol 7 no 1-2 pp71ndash83 1999
[12] S Oltra and O Valero ldquoBanachrsquos fixed point theorem forpartial metric spacesrdquo Rendiconti dellrsquoIstituto di MatematicadellrsquoUniversita di Trieste vol 36 no 1-2 pp 17ndash26 2004
[13] O Valero ldquoOn Banach fixed point theorems for partial metricspacesrdquo Applied General Topology vol 6 no 2 pp 229ndash2402005
[14] H-P A Kunzi H Pajoohesh and M P Schellekens ldquoPartialquasi-metricsrdquoTheoretical Computer Science vol 365 no 3 pp237ndash246 2006
[15] M Bukatin R Kopperman S Matthews and H PajooheshldquoPartial metric spacesrdquo American Mathematical Monthly vol116 no 8 pp 708ndash718 2009
[16] S Romaguera ldquoAKirk type characterization of completeness forpartial metric spacesrdquo Fixed Point Theory and Applications vol2010 Article ID 493298 6 pages 2010
[17] D Ilic V Pavlovic and V Rakocevic ldquoSome new extensions ofBanachrsquos contraction principle to partial metric spacerdquo AppliedMathematics Letters vol 24 no 8 pp 1326ndash1330 2011
[18] D Ilic V Pavlovic and V Rakocevic ldquoExtensions of theZamfirescu theorem to partialmetric spacesrdquoMathematical andComputer Modelling vol 55 no 3-4 pp 801ndash809 2012
[19] E Karapinar ldquoGeneralizations of Caristi Kirkrsquos theorem onpartial metric spacesrdquo Fixed Point Theory and Applications vol2011 article 4 2011
[20] C Di Bari Z Kadelburg H K Nashine and S RadenovicldquoCommon fixed points of g-quasicontractions and relatedmappings in 0-complete partial metric spacesrdquo Fixed PointTheory and Applications vol 2010 article 113 2012
[21] K P Chi E Karapınar and T D Thanh ldquoA generalizedcontraction principle in partial metric spacesrdquo Mathematicaland Computer Modelling vol 55 no 5-6 pp 1673ndash1681 2012
[22] L Ciric B Samet H Aydi and C Vetro ldquoCommon fixed pointsof generalized contractions on partial metric spaces and anapplicationrdquo Applied Mathematics and Computation vol 218no 6 pp 2398ndash2406 2011
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
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
![Page 9: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/9.jpg)
International Journal of Analysis 9
[23] E Karapinar ldquoWeak 120601-contraction on partial metric spacesrdquoJournal of Computational Analysis and Applications vol 14 no2 pp 206ndash210 2012
[24] E Karapınar and U Yuksel ldquoSome common fixed point theo-rems in partial metric spacesrdquo Journal of Applied Mathematicsvol 2011 Article ID 263621 16 pages 2011
[25] E Karapinar ldquoA note on common fixed point theorems inpartial metric spacesrdquo Miskolc Mathematical Notes vol 12 no2 pp 185ndash191 2011
[26] S Romaguera ldquoFixed point theorems for generalized contrac-tions on partial metric spacesrdquo Topology and Its Applicationsvol 159 no 1 pp 194ndash199 2012
[27] H K Nashine and E Karapinar ldquoFixed point results in orbitallycomplete partial metric spacesrdquo Bulletin of the MalaysianMathematical Sciences Society In press
[28] H Aydi ldquoFixed point theorems for generalized weakly con-tractive condition in ordered partial metric spacesrdquo Journal ofNonlinear Analysis and Optimization vol 2 no 2 pp 269ndash2842011
[29] H Aydi ldquoCommon fixed point results for mappings satusfying(Ψ 120601)-weak contractions in ordered partial metric spacesrdquoInternational Journal of Mathematics and Statistics vol 12 pp53ndash64 2012
[30] H Aydi E Karapınar and W Shatanawi ldquoCoupled fixed pointresults for (120595 120593)-weakly contractive condition in ordered par-tialmetric spacesrdquoComputers ampMathematics with Applicationsvol 62 no 12 pp 4449ndash4460 2011
[31] S Romaguera ldquoMatkowskirsquos type theorems for generalized con-tractions on (ordered) partial metric spacesrdquo Applied GeneralTopology vol 12 no 2 pp 213ndash220 2011
[32] B Samet ldquoCoupled fixed point theorems for a generalizedMeir-Keeler contraction in partially ordered metric spacesrdquoNonlinear Analysis vol 72 no 12 pp 4508ndash4517 2010
[33] B Samet M Rajovic R Lazovic and R Stoiljkovic ldquoCommonfixed point results for nonlinear contractions in ordered partialmetric spacesrdquo Fixed Point Theory and Applications vol 2011article 71 2011
[34] H K Nashine Z Kadelburg and S Radenovic ldquoCommonfixed point theorems for weakly isotone increasing mappingsin ordered partial metric spacesrdquo Mathematical and ComputerModelling In press
[35] H K Nashine Z Kadelburg S Radenovic and J K KimldquoFixed point theorems under Hardy-Rogers weak contractiveconditions on 0-complete ordered partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 180 2012
[36] D Paesano and P Vetro ldquoSuzukirsquos type characterizations ofcompleteness for partial metric spaces and fixed points forpartially ordered metric spacesrdquo Topology and Its Applicationsvol 159 no 3 pp 911ndash920 2012
[37] C Di Bari and P Vetro ldquoFixed points for weak 120601-contractionson partial metric spacesrdquo International Journal of ContemporaryMathematical Sciences vol 1 pp 5ndash12 2011
[38] M Abbas T Nazir and S Romaguera ldquoFixed point resultsfor generalized cyclic contraction mappings in partial metricspacesrdquo Revista de la Real Academia de Ciencias Exactas Fısicasy Naturales A vol 106 pp 287ndash297 2012
[39] R P Agarwal M A Alghamdi and N Shahzad ldquoFixed pointtheory for cyclic generalized contractions in partial metricspacesrdquo Fixed Point Theory and Applications vol 2012 article40 2012
[40] E Karapınar and I S Yuce ldquoFixed point theory for cyclic gen-eralized weak 120593-contraction on partial metric spacesrdquo AbstractandAppliedAnalysis vol 2012 Article ID491542 12 pages 2012
[41] E Karapinar N Shobkolaei S Sedghi and S M VaezpourldquoA common fixed point theorem for cyclic operators in partialmetric spacesrdquo Filomat vol 26 pp 407ndash414 2012
[42] M S 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
[43] B S Choudhury ldquoUnique fixed point theorem for weak C-contractive mappingsrdquo Kathmandu University Journal of Sci-ence Engineering and Technology vol 5 pp 6ndash13 2009
[44] A G B Ahmad Z M Fadail H K Nashine Z Kadelburg andS Radenovic ldquoSome new common fixed point results throughgeneralized altering distances on partial metric spacesrdquo FixedPoint Theory and Applications vol 2012 article 120 2012
[45] H Aydi ldquoA common fixed point result by altering distancesinvolving a contractive condition of integral type in partialmetric spacesrdquo Demonstratio Mathematica In press
[46] T Abdeljawad E Karapınar andK Tas ldquoExistence and unique-ness of a common fixed point on partial metric spacesrdquo AppliedMathematics Letters vol 24 no 11 pp 1900ndash1904 2011
[47] E Karapınar and I M Erhan ldquoFixed point theorems for opera-tors on partial metric spacesrdquo Applied Mathematics Letters vol24 no 11 pp 1894ndash1899 2011
[48] M A Petric ldquoBest proximity point theorems for weak cyclicKannan contractionsrdquo Filomat vol 25 no 1 pp 145ndash154 2011
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
![Page 10: Research Article Cyclic Contractions and Fixed Point ...downloads.hindawi.com/journals/ijanal/2013/726387.pdf · InternationalJournalofAnalysis eorem (see [ ]). Let (, ) be a complete](https://reader033.fdocuments.in/reader033/viewer/2022050103/5f4217caffab12582f5c33fa/html5/thumbnails/10.jpg)
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
