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Research ArticleApplication of Coupled Fixed Point Technique inSolving Integral Equations on Modified IntuitionisticFuzzy Metric Spaces
Bhavana Deshpande and Amrish Handa
Department of Mathematics Govt PG Arts and Science College Ratlam 457001 India
Correspondence should be addressed to Bhavana Deshpande bhavnadeshpandeyahoocom
Received 7 January 2014 Accepted 18 May 2014 Published 22 June 2014
Academic Editor RustomM Mamlook
Copyright copy 2014 B Deshpande and A HandaThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We establish a common coupled fixed point theorem for weakly compatible mappings on modified intuitionistic fuzzy metricspaces As an application of our result we study the existence and uniqueness of the solution to a nonlinear Fredholm integralequation We also give an example to demonstrate our result
1 Introduction
The concept of fuzzy metric space has been introduced inseveral ways In [1] Kramosil and Michalek introduced theconcept of fuzzy metric space Later on it is modified byGeorge and Veeramani [2] with the help of continuous t-norms and they defined the Hausdorff topology of fuzzymetric spaces
Atanassov [3] introduced and studied the concept ofintuitionistic fuzzy sets as a generalization of fuzzy setsAlaca et al [4] using the idea of intuitionistic fuzzy setsdefined the notion of intuitionistic fuzzy metric space withthe help of continuous t-norms and continuous t-conormsas a generalization of fuzzy metric space due to Kramosiland Michalek [1] In [5] Park generalized the notion offuzzy metric space given by George and Veeramani [2] andintroduced the notion of intuitionistic fuzzy metric space
Gregori et al [6] pointed out that topologies generated byfuzzy metric and intuitionistic fuzzy metric coincide In viewof this observation Saadati et al [7] modified the notion ofintuitionistic fuzzymetric and defined the notion ofmodifiedintuitionistic fuzzy metric spaces with the help of continuoust-representable
Bhaskar and Lakshmikantham [8] introduced the notionof coupled fixed point and mixed monotone mappings and
gave some coupled fixed point theorems As an applicationthey study the existence and uniqueness of solution for peri-odic boundary value problems Lakshmikantham and Ciric[9] introduced the concept of coupled coincidence point andproved some common coupled fixed point theorems Sedghiet al [10] gave a coupled fixed point theorem for contractionsin fuzzy metric space which was further generalized by Hu[11] In [12] Hu et al improved rectified and generalized theresult obtained in [11]
On the other hand many scientific and engineeringproblems can be described by integral equations Initial andboundary value problems can be transformed into Volterraor Fredholm integral equations Integral equations can alsobe created by many mathematical physics models suchas diffraction problems scattering in quantum mechanicsconformal mapping and water wave Integral equationsor integro-differential equations can be applied in scienceand engineering Many areas that are described by integralequations are Volterrarsquos population growth model biologicalspecies living together propagation of stocked fish in a newlake the heat radiation and so forth
Very recently Deshpande et al [13] proved a commonfixed point theorem formappings under 120601-contractive condi-tions on intuitionistic fuzzy metric spaces As an application
Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2014 Article ID 348069 11 pageshttpdxdoiorg1011552014348069
2 Advances in Fuzzy Systems
they study the existence and uniqueness of the solution to anonlinear Fredholm integral equation
In this paper we prove a common coupled fixed pointtheorem for weakly compatible mappings on modifiedintuitionistic fuzzy metric spaces As an application of ourresult we study the existence and uniqueness of the solutionto a nonlinear Fredholm integral equation which arisenaturally in the theory of signal processing linear forwardmodeling and inverse problems We also give an example tovalidate our result We extend and generalize the results ofHu [11] Hu et al [12] and Sedghi et al [10] in the settings ofmodified intuitionistic fuzzy metric spaces The result is thegenuine generalization of the result of Deshpande et al [13]
2 Preliminaries
Lemma 1 (Deschrijver and Kerre [14]) Consider the set 119871lowastand operation le
119871lowastdefined by
119871lowast
= (1199091 1199092) (1199091 1199092) isin [0 1]
2
1199091+ 1199092le 1 (1)
(1199091 1199092)le119871lowast(1199101 1199102) hArr 119909
1le 1199101 and 119909
2ge 1199102for every
(1199091 1199092) (1199101 1199102) isin 119871lowast Then (119871lowast le
119871lowast) is a complete lattice
Definition 2 (Atanassov [3]) An intuitionistic fuzzy setA120577120578
in a universe 119880 is an object A120577120578
= 120577A(119906) 120578A(119906) wherefor all 119906 isin 119880 120577A(119906) isin [0 1] and 120578A(119906) isin [0 1] arecalled the membership degree and nonmembership degreerespectively of 119906 in A
120577120578and further they satisfy 120577A(119906) +
120578A(119906) le 1 For every 119911119894= (119909119894 119910119894) isin 119871lowast if 119888
We denote its units by 0119871lowast = (0 1) and 1
119871lowast = (1 0)
Classically a triangular norm lowast = 119879 on [0 1] is defined as anincreasing commutative associative mapping 119879 [0 1]2 rarr[0 1] satisfying 119879(1 119909) = 1 lowast 119909 = 119909 for all 119909 isin [0 1]A triangular conorm 119878 = is defined as an increasingcommutative associative mapping 119878 [0 1]2 rarr [0 1]satisfying 119878(0 119909) = 0 119909 = 119909 for all 119909 isin [0 1] Usingthe lattice (119871lowast le
119871lowast) these definitions can be straightforwardly
extended
Definition 3 (Deschrijver et al [15]) A triangular norm (t-norm) on 119871lowast is a mapping T (119871lowast)2 rarr 119871lowast satisfying thefollowing conditions
Definition 4 (Deschrijver and Kerre [14] and Deschrijver etal [15]) A continuous t-normT on 119871lowast is called continuoust-representable if and only if there exist a continuous t-normlowast and a continuous t-conorm on [0 1] such that for all119909 = (119909
1 1199092) 119910 = (119910
1 1199102) isin 119871lowast
T (119909 119910) = (1199091lowast 1199101 1199092 1199102) (3)
Now define a sequenceT119899 recursively byT1 = T and
T119899
(119909(1)
119909(119899+1)
) = T (T119899minus1
(119909(1)
119909(119899)
) 119909(119899+1)
)
(4)
for 119899 ge 2 and 119909(119894) isin 119871lowast
Definition 5 (Deschrijver and Kerre [14] andDeschrijver et al[15]) A negator on 119871lowast is any decreasing mappingN 119871lowast rarr119871lowast satisfyingN(0
119871lowast) = 1
119871lowast andN(1
119871lowast) = 0
119871lowast IfN(N(119909)) =
119909 for all 119909 isin 119871lowast then N is called an involutive negator Anegator on [0 1] is a decreasing mapping119873 [0 1] rarr [0 1]satisfying 119873(0) = 1 and 119873(1) = 0 119873
119904denotes the standard
negator on [0 1] defined as for all 119909 isin [0 1]119873119904(119909) = 1 minus 119909
Definition 6 (Saadati et al [7]) Let119872119873 be fuzzy sets from1198832 times (0 + infin) to [0 1] such that 119872(119909 119910 119905) + 119873(119909 119910 119905) le 1for all 119909 119910 isin 119883 and 119905 gt 0 The 3-tuple (119883 M
119872119873T) is said
to be a modified intuitionistic fuzzy metric space if 119883 is anarbitrary (nonempty) setT is a continuous t-representableandM
119872119873is a mapping 1198832 times (0 + infin) rarr 119871lowast satisfying the
following conditions for every 119909 119910 119911 isin 119883 and 119905 119904 gt 0
(a) M119872119873
(119909 119910 119905)gt119871lowast0119871lowast
(b) M119872119873
(119909 119910 119905) = 1119871lowast if and only if 119909 = 119910
(c) M119872119873
(119909 119910 119905) = M119872119873
(119910 119909 119905)(d) M
119872119873(119909 119910 119905+119904)ge
119871lowastT(M
119872119873(119909 119911 119905)M
119872119873(119911 119910 119904))
(e) M119872119873
(119909 119910 sdot) (0 infin) rarr 119871lowast is continuous
In this caseM119872119873
is called amodified intuitionistic fuzzymetric Here
The sequence 119909119899 is said to be convergent to 119909 isin 119883 in the
modified intuitionistic fuzzy metric space (119883M119872119873
T) and
denoted by 119909119899
M119872119873
997888997888997888997888rarr 119909 if M119872119873
(119909119899 119909 119905) rarr 1
119871lowast whenever
119899 rarr infin for every 119905 gt 0 A modified intuitionistic fuzzymetric space is said to be complete if and only if every Cauchysequence is convergent
Lemma 12 (Saadati and Park [17]) Let M119872119873
be a modifiedintuitionistic fuzzy metric Then for any 119905 gt 0M
119872119873(119909 119910 119905)
is nondecreasing with respect to 119905 in (119871lowast le119871lowast) for all 119909 119910 in119883
Definition 13 (Saadati et al [7]) Let (119883M119872119873
T) be amodified intuitionistic fuzzy metric space For 119905 gt 0 definethe open ball119861(119909 119903 119905)with center119909 isin 119883 and radius 0 lt 119903 lt 1as
A subset 119860 sub 119883 is called open if for each 119909 isin 119860 thereexist 119905 gt 0 and 0 lt 119903 lt 1 such that 119861(119909 119903 119905) sub 119860 Let120591119872119873
denote the family of all open subsets of119883 120591119872119873
is calledthe topology induced by modified intuitionistic fuzzy metricThis topology is Hausdorff
Definition 14 (Saadati et al [7]) Let (119883M119872119873
T) be amodified intuitionistic fuzzy metric space M is said to becontinuous on119883 times 119883 times (0infin) if
lim119899rarrinfin
M119872119873
(119909119899 119910119899 119905119899) = M
119872119873(119909 119910 119905) (9)
whenever a sequence (119909119899 119910119899 119905119899) in119883times119883times(0infin) converges
to a point (119909 119910 119905) isin 119883 times 119883 times (0infin) that is
lim119899rarrinfin
M119872119873
(119909119899 119910 119905) = M
119872119873(119909 119910 119905)
lim119899rarrinfin
M119872119873
(119909 119910119899 119905) = M
119872119873(119909 119910 119905)
lim119899rarrinfin
M119872119873
(119909 119910 119905119899) = M
119872119873(119909 119910 119905)
(10)
Lemma 15 (Saadati et al [7]) Let (119883M119872119873
T) be a mod-ified intuitionistic fuzzy metric space Then M is continuousfunction on 119883 times 119883 times (0infin)
Definition 16 (Bhaskar and Lakshmikantham [8]) An ele-ment (119909 119910) isin 119883 times 119883 is called a coupled fixed point of themapping 119865 119883 times 119883 rarr 119883 if
Definition 17 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883 times 119883 is called a coupledcoincidence point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if
Definition 18 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883times119883 is called a commoncoupled fixed point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if
Definition 19 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element 119909 isin 119883 is called a common fixedpoint of the mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 if
Definition 20 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty setThemappings119865 119883times119883 rarr 119883 and119892 119883 rarr 119883are said to be commutative if
Definition 21 (Fang [18]) Let (119883M119872119873
T) be a modifiedintuitionistic fuzzy metric spaceThemappings 119865 119883times119883 rarr119883 and 119892 119883 rarr 119883 are said to be compatible if
Definition 22 (Abbas et al [19]) Let 119883 be a nonempty setThe mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 are calledweakly compatible mappings if 119865(119909 119910) = 119892(119909) 119865(119910 119909) =119892(119910) implies that 119892119865(119909 119910) = 119865(119892119909 119892119910) and 119892119865(119910 119909) =119865(119892119910 119892119909) for all 119909 119910 isin 119883
4 Advances in Fuzzy Systems
3 Main Results
Definition 23 Let sup0lt119905lt1
T(119905 119905) = 1119871lowast A continuous t-
representableT is said to be continuous t-representable ofH-type if the family of functions T119898(119905)infin
119898=1is equicontinuous
at 119905 = 1119871lowast where
T1
(119905) = T (119905 119905) T119898+1
(119905) = T (119905T119898
(119905))
119898 = 1 2 119905 isin 119871lowast
(18)
ObviouslyT is aH-type t-representable if and only if for any0 lt 120582 lt 1 there exists 0 lt 120583 lt 1 such that
T119898
(119905) gt119871lowast (119873119904(120582) 120582) forall119898 isin 119873 when 119905gt
119871lowast (119873119904(120583) 120583)
(19)
Remark 24 In a modified intuitionistic fuzzy metric space(119883M
119872119873T) whenever M
119872119873(119909 119910 119905)gt
119871lowast(119873119904(119903) 119903) for
119909 119910 isin 119883 119905 gt 0 and 0 lt 119903 lt 1 we can find a 1199050 0 lt 119905
0lt 119905
such that M119872119873
(119909 119910 1199050) gt (119873
119904(119903) 119903)
Remark 25 For convenience we denote
[M119872119873
(119909 119910 119905)]119899
= T119899minus1
(M119872119873
(119909 119910 119905)) (20)
for all 119899 isin 119873
Definition 26 Let (119883M119872119873
T) be a modified intuitionisticfuzzy metric spaceM
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 Suppose that 119865(119883 times119883) sube 119892(119883)and 119865(119883 times 119883) or 119892(119883) is complete Then there exists a unique119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
Proof Let 1199090 1199100isin 119883 be two arbitrary points in 119883 Since
119865(119883 times 119883) sube 119892(119883) we can choose 1199091 1199101isin 119883 such that
1198921199091= 119865 (119909
0 1199100) 119892119910
1= 119865 (119910
0 1199090) (31)
Continuing in this way we can construct two sequences 119909119899
and 119910119899 in119883 such that
119892119909119899+1
= 119865 (119909119899 119910119899) 119892119910
119899+1= 119865 (119910
119899 119909119899) forall119899 ge 0 (32)
The proof is divided into 4 steps
Step 1 Prove that 119892119909119899 and 119892119910
119899 are Cauchy sequences
SinceT is a continuous t-representable of H-type there-fore for any 120582 gt 0 there exists a 120583 gt 0 such that
Letting 119899 rarr infin in the above inequality we get
M119872119873
(119909 119910 120601 (1199050))
ge119871lowastT (M
119872119873(119909 119910 119905
0) M119872119873
(119910 119909 1199050))
(65)
Thus by (60) (61) (62) and (65) we have
M119872119873
(119909 119910 119905)
ge119871lowastM119872119873
(119909 119910infin
sum119896=1198990
120601119896
(1199050))
ge119871lowast M119872119873
(119909 119910 1206011198990 (1199050))
ge119871lowastT ([M
119872119873(119909 119910 119905
0)]21198990minus1
[M119872119873
(119910 119909 1199050)]21198990minus1
)
ge119871lowastT21198990+1
minus3
(119873119904(120583) 120583)
ge119871lowast (119873119904(120582) 120582)
(66)
which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31
Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence
Corollary 32 Let (119883M119872119873
T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
8 Advances in Fuzzy Systems
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
they study the existence and uniqueness of the solution to anonlinear Fredholm integral equation
In this paper we prove a common coupled fixed pointtheorem for weakly compatible mappings on modifiedintuitionistic fuzzy metric spaces As an application of ourresult we study the existence and uniqueness of the solutionto a nonlinear Fredholm integral equation which arisenaturally in the theory of signal processing linear forwardmodeling and inverse problems We also give an example tovalidate our result We extend and generalize the results ofHu [11] Hu et al [12] and Sedghi et al [10] in the settings ofmodified intuitionistic fuzzy metric spaces The result is thegenuine generalization of the result of Deshpande et al [13]
2 Preliminaries
Lemma 1 (Deschrijver and Kerre [14]) Consider the set 119871lowastand operation le
119871lowastdefined by
119871lowast
= (1199091 1199092) (1199091 1199092) isin [0 1]
2
1199091+ 1199092le 1 (1)
(1199091 1199092)le119871lowast(1199101 1199102) hArr 119909
1le 1199101 and 119909
2ge 1199102for every
(1199091 1199092) (1199101 1199102) isin 119871lowast Then (119871lowast le
119871lowast) is a complete lattice
Definition 2 (Atanassov [3]) An intuitionistic fuzzy setA120577120578
in a universe 119880 is an object A120577120578
= 120577A(119906) 120578A(119906) wherefor all 119906 isin 119880 120577A(119906) isin [0 1] and 120578A(119906) isin [0 1] arecalled the membership degree and nonmembership degreerespectively of 119906 in A
120577120578and further they satisfy 120577A(119906) +
120578A(119906) le 1 For every 119911119894= (119909119894 119910119894) isin 119871lowast if 119888
We denote its units by 0119871lowast = (0 1) and 1
119871lowast = (1 0)
Classically a triangular norm lowast = 119879 on [0 1] is defined as anincreasing commutative associative mapping 119879 [0 1]2 rarr[0 1] satisfying 119879(1 119909) = 1 lowast 119909 = 119909 for all 119909 isin [0 1]A triangular conorm 119878 = is defined as an increasingcommutative associative mapping 119878 [0 1]2 rarr [0 1]satisfying 119878(0 119909) = 0 119909 = 119909 for all 119909 isin [0 1] Usingthe lattice (119871lowast le
119871lowast) these definitions can be straightforwardly
extended
Definition 3 (Deschrijver et al [15]) A triangular norm (t-norm) on 119871lowast is a mapping T (119871lowast)2 rarr 119871lowast satisfying thefollowing conditions
Definition 4 (Deschrijver and Kerre [14] and Deschrijver etal [15]) A continuous t-normT on 119871lowast is called continuoust-representable if and only if there exist a continuous t-normlowast and a continuous t-conorm on [0 1] such that for all119909 = (119909
1 1199092) 119910 = (119910
1 1199102) isin 119871lowast
T (119909 119910) = (1199091lowast 1199101 1199092 1199102) (3)
Now define a sequenceT119899 recursively byT1 = T and
T119899
(119909(1)
119909(119899+1)
) = T (T119899minus1
(119909(1)
119909(119899)
) 119909(119899+1)
)
(4)
for 119899 ge 2 and 119909(119894) isin 119871lowast
Definition 5 (Deschrijver and Kerre [14] andDeschrijver et al[15]) A negator on 119871lowast is any decreasing mappingN 119871lowast rarr119871lowast satisfyingN(0
119871lowast) = 1
119871lowast andN(1
119871lowast) = 0
119871lowast IfN(N(119909)) =
119909 for all 119909 isin 119871lowast then N is called an involutive negator Anegator on [0 1] is a decreasing mapping119873 [0 1] rarr [0 1]satisfying 119873(0) = 1 and 119873(1) = 0 119873
119904denotes the standard
negator on [0 1] defined as for all 119909 isin [0 1]119873119904(119909) = 1 minus 119909
Definition 6 (Saadati et al [7]) Let119872119873 be fuzzy sets from1198832 times (0 + infin) to [0 1] such that 119872(119909 119910 119905) + 119873(119909 119910 119905) le 1for all 119909 119910 isin 119883 and 119905 gt 0 The 3-tuple (119883 M
119872119873T) is said
to be a modified intuitionistic fuzzy metric space if 119883 is anarbitrary (nonempty) setT is a continuous t-representableandM
119872119873is a mapping 1198832 times (0 + infin) rarr 119871lowast satisfying the
following conditions for every 119909 119910 119911 isin 119883 and 119905 119904 gt 0
(a) M119872119873
(119909 119910 119905)gt119871lowast0119871lowast
(b) M119872119873
(119909 119910 119905) = 1119871lowast if and only if 119909 = 119910
(c) M119872119873
(119909 119910 119905) = M119872119873
(119910 119909 119905)(d) M
119872119873(119909 119910 119905+119904)ge
119871lowastT(M
119872119873(119909 119911 119905)M
119872119873(119911 119910 119904))
(e) M119872119873
(119909 119910 sdot) (0 infin) rarr 119871lowast is continuous
In this caseM119872119873
is called amodified intuitionistic fuzzymetric Here
The sequence 119909119899 is said to be convergent to 119909 isin 119883 in the
modified intuitionistic fuzzy metric space (119883M119872119873
T) and
denoted by 119909119899
M119872119873
997888997888997888997888rarr 119909 if M119872119873
(119909119899 119909 119905) rarr 1
119871lowast whenever
119899 rarr infin for every 119905 gt 0 A modified intuitionistic fuzzymetric space is said to be complete if and only if every Cauchysequence is convergent
Lemma 12 (Saadati and Park [17]) Let M119872119873
be a modifiedintuitionistic fuzzy metric Then for any 119905 gt 0M
119872119873(119909 119910 119905)
is nondecreasing with respect to 119905 in (119871lowast le119871lowast) for all 119909 119910 in119883
Definition 13 (Saadati et al [7]) Let (119883M119872119873
T) be amodified intuitionistic fuzzy metric space For 119905 gt 0 definethe open ball119861(119909 119903 119905)with center119909 isin 119883 and radius 0 lt 119903 lt 1as
A subset 119860 sub 119883 is called open if for each 119909 isin 119860 thereexist 119905 gt 0 and 0 lt 119903 lt 1 such that 119861(119909 119903 119905) sub 119860 Let120591119872119873
denote the family of all open subsets of119883 120591119872119873
is calledthe topology induced by modified intuitionistic fuzzy metricThis topology is Hausdorff
Definition 14 (Saadati et al [7]) Let (119883M119872119873
T) be amodified intuitionistic fuzzy metric space M is said to becontinuous on119883 times 119883 times (0infin) if
lim119899rarrinfin
M119872119873
(119909119899 119910119899 119905119899) = M
119872119873(119909 119910 119905) (9)
whenever a sequence (119909119899 119910119899 119905119899) in119883times119883times(0infin) converges
to a point (119909 119910 119905) isin 119883 times 119883 times (0infin) that is
lim119899rarrinfin
M119872119873
(119909119899 119910 119905) = M
119872119873(119909 119910 119905)
lim119899rarrinfin
M119872119873
(119909 119910119899 119905) = M
119872119873(119909 119910 119905)
lim119899rarrinfin
M119872119873
(119909 119910 119905119899) = M
119872119873(119909 119910 119905)
(10)
Lemma 15 (Saadati et al [7]) Let (119883M119872119873
T) be a mod-ified intuitionistic fuzzy metric space Then M is continuousfunction on 119883 times 119883 times (0infin)
Definition 16 (Bhaskar and Lakshmikantham [8]) An ele-ment (119909 119910) isin 119883 times 119883 is called a coupled fixed point of themapping 119865 119883 times 119883 rarr 119883 if
Definition 17 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883 times 119883 is called a coupledcoincidence point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if
Definition 18 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883times119883 is called a commoncoupled fixed point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if
Definition 19 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element 119909 isin 119883 is called a common fixedpoint of the mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 if
Definition 20 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty setThemappings119865 119883times119883 rarr 119883 and119892 119883 rarr 119883are said to be commutative if
Definition 21 (Fang [18]) Let (119883M119872119873
T) be a modifiedintuitionistic fuzzy metric spaceThemappings 119865 119883times119883 rarr119883 and 119892 119883 rarr 119883 are said to be compatible if
Definition 22 (Abbas et al [19]) Let 119883 be a nonempty setThe mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 are calledweakly compatible mappings if 119865(119909 119910) = 119892(119909) 119865(119910 119909) =119892(119910) implies that 119892119865(119909 119910) = 119865(119892119909 119892119910) and 119892119865(119910 119909) =119865(119892119910 119892119909) for all 119909 119910 isin 119883
4 Advances in Fuzzy Systems
3 Main Results
Definition 23 Let sup0lt119905lt1
T(119905 119905) = 1119871lowast A continuous t-
representableT is said to be continuous t-representable ofH-type if the family of functions T119898(119905)infin
119898=1is equicontinuous
at 119905 = 1119871lowast where
T1
(119905) = T (119905 119905) T119898+1
(119905) = T (119905T119898
(119905))
119898 = 1 2 119905 isin 119871lowast
(18)
ObviouslyT is aH-type t-representable if and only if for any0 lt 120582 lt 1 there exists 0 lt 120583 lt 1 such that
T119898
(119905) gt119871lowast (119873119904(120582) 120582) forall119898 isin 119873 when 119905gt
119871lowast (119873119904(120583) 120583)
(19)
Remark 24 In a modified intuitionistic fuzzy metric space(119883M
119872119873T) whenever M
119872119873(119909 119910 119905)gt
119871lowast(119873119904(119903) 119903) for
119909 119910 isin 119883 119905 gt 0 and 0 lt 119903 lt 1 we can find a 1199050 0 lt 119905
0lt 119905
such that M119872119873
(119909 119910 1199050) gt (119873
119904(119903) 119903)
Remark 25 For convenience we denote
[M119872119873
(119909 119910 119905)]119899
= T119899minus1
(M119872119873
(119909 119910 119905)) (20)
for all 119899 isin 119873
Definition 26 Let (119883M119872119873
T) be a modified intuitionisticfuzzy metric spaceM
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 Suppose that 119865(119883 times119883) sube 119892(119883)and 119865(119883 times 119883) or 119892(119883) is complete Then there exists a unique119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
Proof Let 1199090 1199100isin 119883 be two arbitrary points in 119883 Since
119865(119883 times 119883) sube 119892(119883) we can choose 1199091 1199101isin 119883 such that
1198921199091= 119865 (119909
0 1199100) 119892119910
1= 119865 (119910
0 1199090) (31)
Continuing in this way we can construct two sequences 119909119899
and 119910119899 in119883 such that
119892119909119899+1
= 119865 (119909119899 119910119899) 119892119910
119899+1= 119865 (119910
119899 119909119899) forall119899 ge 0 (32)
The proof is divided into 4 steps
Step 1 Prove that 119892119909119899 and 119892119910
119899 are Cauchy sequences
SinceT is a continuous t-representable of H-type there-fore for any 120582 gt 0 there exists a 120583 gt 0 such that
Letting 119899 rarr infin in the above inequality we get
M119872119873
(119909 119910 120601 (1199050))
ge119871lowastT (M
119872119873(119909 119910 119905
0) M119872119873
(119910 119909 1199050))
(65)
Thus by (60) (61) (62) and (65) we have
M119872119873
(119909 119910 119905)
ge119871lowastM119872119873
(119909 119910infin
sum119896=1198990
120601119896
(1199050))
ge119871lowast M119872119873
(119909 119910 1206011198990 (1199050))
ge119871lowastT ([M
119872119873(119909 119910 119905
0)]21198990minus1
[M119872119873
(119910 119909 1199050)]21198990minus1
)
ge119871lowastT21198990+1
minus3
(119873119904(120583) 120583)
ge119871lowast (119873119904(120582) 120582)
(66)
which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31
Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence
Corollary 32 Let (119883M119872119873
T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
8 Advances in Fuzzy Systems
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
The sequence 119909119899 is said to be convergent to 119909 isin 119883 in the
modified intuitionistic fuzzy metric space (119883M119872119873
T) and
denoted by 119909119899
M119872119873
997888997888997888997888rarr 119909 if M119872119873
(119909119899 119909 119905) rarr 1
119871lowast whenever
119899 rarr infin for every 119905 gt 0 A modified intuitionistic fuzzymetric space is said to be complete if and only if every Cauchysequence is convergent
Lemma 12 (Saadati and Park [17]) Let M119872119873
be a modifiedintuitionistic fuzzy metric Then for any 119905 gt 0M
119872119873(119909 119910 119905)
is nondecreasing with respect to 119905 in (119871lowast le119871lowast) for all 119909 119910 in119883
Definition 13 (Saadati et al [7]) Let (119883M119872119873
T) be amodified intuitionistic fuzzy metric space For 119905 gt 0 definethe open ball119861(119909 119903 119905)with center119909 isin 119883 and radius 0 lt 119903 lt 1as
A subset 119860 sub 119883 is called open if for each 119909 isin 119860 thereexist 119905 gt 0 and 0 lt 119903 lt 1 such that 119861(119909 119903 119905) sub 119860 Let120591119872119873
denote the family of all open subsets of119883 120591119872119873
is calledthe topology induced by modified intuitionistic fuzzy metricThis topology is Hausdorff
Definition 14 (Saadati et al [7]) Let (119883M119872119873
T) be amodified intuitionistic fuzzy metric space M is said to becontinuous on119883 times 119883 times (0infin) if
lim119899rarrinfin
M119872119873
(119909119899 119910119899 119905119899) = M
119872119873(119909 119910 119905) (9)
whenever a sequence (119909119899 119910119899 119905119899) in119883times119883times(0infin) converges
to a point (119909 119910 119905) isin 119883 times 119883 times (0infin) that is
lim119899rarrinfin
M119872119873
(119909119899 119910 119905) = M
119872119873(119909 119910 119905)
lim119899rarrinfin
M119872119873
(119909 119910119899 119905) = M
119872119873(119909 119910 119905)
lim119899rarrinfin
M119872119873
(119909 119910 119905119899) = M
119872119873(119909 119910 119905)
(10)
Lemma 15 (Saadati et al [7]) Let (119883M119872119873
T) be a mod-ified intuitionistic fuzzy metric space Then M is continuousfunction on 119883 times 119883 times (0infin)
Definition 16 (Bhaskar and Lakshmikantham [8]) An ele-ment (119909 119910) isin 119883 times 119883 is called a coupled fixed point of themapping 119865 119883 times 119883 rarr 119883 if
Definition 17 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883 times 119883 is called a coupledcoincidence point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if
Definition 18 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element (119909 119910) isin 119883times119883 is called a commoncoupled fixed point of the mappings 119865 119883 times 119883 rarr 119883 and119892 119883 rarr 119883 if
Definition 19 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty set An element 119909 isin 119883 is called a common fixedpoint of the mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 if
Definition 20 (Lakshmikantham and Ciric [9]) Let 119883 be anonempty setThemappings119865 119883times119883 rarr 119883 and119892 119883 rarr 119883are said to be commutative if
Definition 21 (Fang [18]) Let (119883M119872119873
T) be a modifiedintuitionistic fuzzy metric spaceThemappings 119865 119883times119883 rarr119883 and 119892 119883 rarr 119883 are said to be compatible if
Definition 22 (Abbas et al [19]) Let 119883 be a nonempty setThe mappings 119865 119883 times 119883 rarr 119883 and 119892 119883 rarr 119883 are calledweakly compatible mappings if 119865(119909 119910) = 119892(119909) 119865(119910 119909) =119892(119910) implies that 119892119865(119909 119910) = 119865(119892119909 119892119910) and 119892119865(119910 119909) =119865(119892119910 119892119909) for all 119909 119910 isin 119883
4 Advances in Fuzzy Systems
3 Main Results
Definition 23 Let sup0lt119905lt1
T(119905 119905) = 1119871lowast A continuous t-
representableT is said to be continuous t-representable ofH-type if the family of functions T119898(119905)infin
119898=1is equicontinuous
at 119905 = 1119871lowast where
T1
(119905) = T (119905 119905) T119898+1
(119905) = T (119905T119898
(119905))
119898 = 1 2 119905 isin 119871lowast
(18)
ObviouslyT is aH-type t-representable if and only if for any0 lt 120582 lt 1 there exists 0 lt 120583 lt 1 such that
T119898
(119905) gt119871lowast (119873119904(120582) 120582) forall119898 isin 119873 when 119905gt
119871lowast (119873119904(120583) 120583)
(19)
Remark 24 In a modified intuitionistic fuzzy metric space(119883M
119872119873T) whenever M
119872119873(119909 119910 119905)gt
119871lowast(119873119904(119903) 119903) for
119909 119910 isin 119883 119905 gt 0 and 0 lt 119903 lt 1 we can find a 1199050 0 lt 119905
0lt 119905
such that M119872119873
(119909 119910 1199050) gt (119873
119904(119903) 119903)
Remark 25 For convenience we denote
[M119872119873
(119909 119910 119905)]119899
= T119899minus1
(M119872119873
(119909 119910 119905)) (20)
for all 119899 isin 119873
Definition 26 Let (119883M119872119873
T) be a modified intuitionisticfuzzy metric spaceM
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 Suppose that 119865(119883 times119883) sube 119892(119883)and 119865(119883 times 119883) or 119892(119883) is complete Then there exists a unique119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
Proof Let 1199090 1199100isin 119883 be two arbitrary points in 119883 Since
119865(119883 times 119883) sube 119892(119883) we can choose 1199091 1199101isin 119883 such that
1198921199091= 119865 (119909
0 1199100) 119892119910
1= 119865 (119910
0 1199090) (31)
Continuing in this way we can construct two sequences 119909119899
and 119910119899 in119883 such that
119892119909119899+1
= 119865 (119909119899 119910119899) 119892119910
119899+1= 119865 (119910
119899 119909119899) forall119899 ge 0 (32)
The proof is divided into 4 steps
Step 1 Prove that 119892119909119899 and 119892119910
119899 are Cauchy sequences
SinceT is a continuous t-representable of H-type there-fore for any 120582 gt 0 there exists a 120583 gt 0 such that
Letting 119899 rarr infin in the above inequality we get
M119872119873
(119909 119910 120601 (1199050))
ge119871lowastT (M
119872119873(119909 119910 119905
0) M119872119873
(119910 119909 1199050))
(65)
Thus by (60) (61) (62) and (65) we have
M119872119873
(119909 119910 119905)
ge119871lowastM119872119873
(119909 119910infin
sum119896=1198990
120601119896
(1199050))
ge119871lowast M119872119873
(119909 119910 1206011198990 (1199050))
ge119871lowastT ([M
119872119873(119909 119910 119905
0)]21198990minus1
[M119872119873
(119910 119909 1199050)]21198990minus1
)
ge119871lowastT21198990+1
minus3
(119873119904(120583) 120583)
ge119871lowast (119873119904(120582) 120582)
(66)
which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31
Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence
Corollary 32 Let (119883M119872119873
T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
8 Advances in Fuzzy Systems
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 Suppose that 119865(119883 times119883) sube 119892(119883)and 119865(119883 times 119883) or 119892(119883) is complete Then there exists a unique119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
Proof Let 1199090 1199100isin 119883 be two arbitrary points in 119883 Since
119865(119883 times 119883) sube 119892(119883) we can choose 1199091 1199101isin 119883 such that
1198921199091= 119865 (119909
0 1199100) 119892119910
1= 119865 (119910
0 1199090) (31)
Continuing in this way we can construct two sequences 119909119899
and 119910119899 in119883 such that
119892119909119899+1
= 119865 (119909119899 119910119899) 119892119910
119899+1= 119865 (119910
119899 119909119899) forall119899 ge 0 (32)
The proof is divided into 4 steps
Step 1 Prove that 119892119909119899 and 119892119910
119899 are Cauchy sequences
SinceT is a continuous t-representable of H-type there-fore for any 120582 gt 0 there exists a 120583 gt 0 such that
Letting 119899 rarr infin in the above inequality we get
M119872119873
(119909 119910 120601 (1199050))
ge119871lowastT (M
119872119873(119909 119910 119905
0) M119872119873
(119910 119909 1199050))
(65)
Thus by (60) (61) (62) and (65) we have
M119872119873
(119909 119910 119905)
ge119871lowastM119872119873
(119909 119910infin
sum119896=1198990
120601119896
(1199050))
ge119871lowast M119872119873
(119909 119910 1206011198990 (1199050))
ge119871lowastT ([M
119872119873(119909 119910 119905
0)]21198990minus1
[M119872119873
(119910 119909 1199050)]21198990minus1
)
ge119871lowastT21198990+1
minus3
(119873119904(120583) 120583)
ge119871lowast (119873119904(120582) 120582)
(66)
which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31
Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence
Corollary 32 Let (119883M119872119873
T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
8 Advances in Fuzzy Systems
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 Suppose that 119865(119883 times119883) sube 119892(119883)and 119865(119883 times 119883) or 119892(119883) is complete Then there exists a unique119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
Proof Let 1199090 1199100isin 119883 be two arbitrary points in 119883 Since
119865(119883 times 119883) sube 119892(119883) we can choose 1199091 1199101isin 119883 such that
1198921199091= 119865 (119909
0 1199100) 119892119910
1= 119865 (119910
0 1199090) (31)
Continuing in this way we can construct two sequences 119909119899
and 119910119899 in119883 such that
119892119909119899+1
= 119865 (119909119899 119910119899) 119892119910
119899+1= 119865 (119910
119899 119909119899) forall119899 ge 0 (32)
The proof is divided into 4 steps
Step 1 Prove that 119892119909119899 and 119892119910
119899 are Cauchy sequences
SinceT is a continuous t-representable of H-type there-fore for any 120582 gt 0 there exists a 120583 gt 0 such that
Letting 119899 rarr infin in the above inequality we get
M119872119873
(119909 119910 120601 (1199050))
ge119871lowastT (M
119872119873(119909 119910 119905
0) M119872119873
(119910 119909 1199050))
(65)
Thus by (60) (61) (62) and (65) we have
M119872119873
(119909 119910 119905)
ge119871lowastM119872119873
(119909 119910infin
sum119896=1198990
120601119896
(1199050))
ge119871lowast M119872119873
(119909 119910 1206011198990 (1199050))
ge119871lowastT ([M
119872119873(119909 119910 119905
0)]21198990minus1
[M119872119873
(119910 119909 1199050)]21198990minus1
)
ge119871lowastT21198990+1
minus3
(119873119904(120583) 120583)
ge119871lowast (119873119904(120582) 120582)
(66)
which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31
Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence
Corollary 32 Let (119883M119872119873
T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
8 Advances in Fuzzy Systems
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
Letting 119899 rarr infin in the above inequality we get
M119872119873
(119909 119910 120601 (1199050))
ge119871lowastT (M
119872119873(119909 119910 119905
0) M119872119873
(119910 119909 1199050))
(65)
Thus by (60) (61) (62) and (65) we have
M119872119873
(119909 119910 119905)
ge119871lowastM119872119873
(119909 119910infin
sum119896=1198990
120601119896
(1199050))
ge119871lowast M119872119873
(119909 119910 1206011198990 (1199050))
ge119871lowastT ([M
119872119873(119909 119910 119905
0)]21198990minus1
[M119872119873
(119910 119909 1199050)]21198990minus1
)
ge119871lowastT21198990+1
minus3
(119873119904(120583) 120583)
ge119871lowast (119873119904(120582) 120582)
(66)
which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31
Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence
Corollary 32 Let (119883M119872119873
T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
8 Advances in Fuzzy Systems
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
Letting 119899 rarr infin in the above inequality we get
M119872119873
(119909 119910 120601 (1199050))
ge119871lowastT (M
119872119873(119909 119910 119905
0) M119872119873
(119910 119909 1199050))
(65)
Thus by (60) (61) (62) and (65) we have
M119872119873
(119909 119910 119905)
ge119871lowastM119872119873
(119909 119910infin
sum119896=1198990
120601119896
(1199050))
ge119871lowast M119872119873
(119909 119910 1206011198990 (1199050))
ge119871lowastT ([M
119872119873(119909 119910 119905
0)]21198990minus1
[M119872119873
(119910 119909 1199050)]21198990minus1
)
ge119871lowastT21198990+1
minus3
(119873119904(120583) 120583)
ge119871lowast (119873119904(120582) 120582)
(66)
which implies that 119909 = 119910 Thus we have proved that 119865 and 119892have a unique common fixed point in 119883 This completes theproof of Theorem 31
Taking 119892 = 119868 (the identity mapping) in Theorem 31 weget the following consequence
Corollary 32 Let (119883M119872119873
T) be a complete modifiedintuitionistic fuzzy metric space where T is continuous t-representable of H-type satisfying (22) Let 119865 119883 times 119883 rarr 119883and there exist 120601 isin Φ such that
for all 119909 119910 119906 V isin 119883 and 119905 gt 0 where 0 lt 119896 lt 1 Suppose that119865(119883 times 119883) sube 119892(119883) and 119865(119883 times 119883) or 119892(119883) is complete Thenthere exists a unique 119909 isin 119883 such that 119909 = 119892(119909) = 119865(119909 119909)
8 Advances in Fuzzy Systems
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
Remark 34 ComparingTheorem 31 in the present paperwithTheorem 31 in [13] we can see that Theorem 31 is a genuinegeneralization of Theorem 31 in the sense that
(1) we only use the completeness of 119892(119883) or 119865(119883 times 119883)(2) we drop off the continuity of 119892(3) the concept of compatible mappings has been
replaced by weakly compatible mappings
Next we give an example to demonstrate Theorem 31
Example 35 Let 119883 = 0 1 12 13 1119899 andT(119886 119887) = (min119886
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(69)
Then (119883M119872119873
T) is a modified intuitionistic fuzzy metricspace Let 120601(119905) = 1199052 Let 119892 119883 rarr 119883 and 119865 119883 times 119883 rarr 119883be defined as
119892 (119909) =
0 119909 = 0
1 119909 =1
2119899 + 1
1
2119899 + 1 119909 =
1
2119899
119865 (119909 119910) =
1
(2119899 + 1)4 (119909 119910) = (
1
21198991
2119899)
0 otherwise
(70)
Let 119909119899= 119910119899= 12119899 We have
119892119909119899=
1
2119899 + 1997888rarr 0
119865 (119909119899 119910119899) =
1
(2119899 + 1)4997888rarr 0 as 119899 997888rarr infin
so 119892 and 119865 are not compatible From 119865(119909 119910) = 119892(119909) and119865(119910 119909) = 119892(119910) we can get (119909 119910) = (0 0) and we have119892119865(0 0) = 119865(1198920 1198920) which implies that 119865 and 119892 are weaklycompatible The following result is easy to be verified
119905
119883 + 119905ge min 119905
119884 + 119905
119905
119885 + 119905
119883
119883 + 119905le max 119884
119884 + 119905
119885
119885 + 119905
lArrrArr 119883 le max 119884 119885 forall119883 119884 119885 ge 0 119905 gt 0
(73)
By the definition ofM119872119873
120601 and the result above we can getinequality (30)
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 le max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 (75)
Now we verify inequality (75) Let 119860 = 12119899 119899 isin 119873 119861 =119883 minus 119860 By the symmetry and without loss of generality(119909 119910) (119906 V) have six possibilities
Case 1 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119861 It is obvious that (75)holds
Case 2 (119909 119910) isin 119861 times 119861 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 3 (119909 119910) isin 119861 times 119861 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816 =2119899
2119899 + 1
(76)
which implies that (75) holds
Case 4 (119909 119910) isin 119861 times 119860 (119906 V) isin 119861 times 119860 It is obvious that (75)holds
Case 5 (119909 119910) isin 119861 times 119860 (119906 V) isin 119860 times 119860 If 119906 = V (75) holds If119906 = V let 119909 isin 119861 119910 = 12119895 119906 = V = 12119899 and then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
Case 6 (119909 119910) isin 119860 times 119860 and (119906 V) isin 119860 times 119860 If 119909 = 119910 and 119906 = V(75) holds If 119909 = 119910 119906 = V let 119909 = 12119894 119910 = 12119895 119894 = 119895 and119906 = V = 12119899 Then
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816 =2
(2119899 + 1)4
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
max 1003816100381610038161003816119892119909 minus 1198921199061003816100381610038161003816 1003816100381610038161003816119892119910 minus 119892V
1003816100381610038161003816
=1
2119894 + 1minus
1
2119899 + 1
(80)
So (75) holds Then all the conditions in Theorem 31 aresatisfied and 0 is the unique common fixed point of 119892 and119865
4 Application to Integral Equations
As an application of the coupled fixed point theorems estab-lished in Section 3 of our paper we study the existence anduniqueness of the solution to a Fredholm nonlinear integralequation We will consider the following integral equation
for all 119901 isin 119868 = [119886 119887]Let Θ denote the set of all functions 120579 [0 +infin) rarr
[0 +infin) satisfying the following
(i120579) 120579 is nondecreasing
(ii120579) 120579(119901) le 119901
We assume that the functions 1198701 1198702 119891 119892 fulfill the
following conditions
Assumption 36 (i) Consider
1198701(119901 119902) ge 0 119870
2(119901 119902) le 0 forall119901 119902 isin 119868 (82)
(ii) There exists positive numbers 120582 120583 and 120579 isin Θ suchthat for all 119909 119910 isin 119877 with 119909 ge 119910 the following conditionshold
0 le 119891 (119902 119909) minus 119891 (119902 119910) le 120582120579 (119909 minus 119910)
minus120583120579 (119909 minus 119910) le 119892 (119902 119909) minus 119892 (119902 119910) le 0(83)
(iii) Consider
max 120582 120583 sup119901isin119868
int119887
119886
[1198701(119901 119902) minus 119870
2(119901 119902)] 119889119902 le
1
8 (84)
Theorem 37 Consider the integral equation (81) with1198701 1198702isin 119862(119868 times 119868R) 119891 119892 isin 119862(119868 times RR) and ℎ isin 119862(119868 119877)
Suppose that Assumption 36 is satisfied Then the integralequation (81) has a unique solution in 119862(119868 119877)
Proof Consider 119883 = 119862(119868 119877) It is easy to check that(119883M
119872119873T) is a complete modified intuitionistic fuzzy
metric space with respect to the modified intuitionistic fuzzymetric
M119872119873
(119909 119910 119905) = (119905
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816
1003816100381610038161003816119909 minus 1199101003816100381610038161003816
119905 +1003816100381610038161003816119909 minus 119910
1003816100381610038161003816)
forall119909 119910 isin 119883 119905 gt 0
(85)
for all 119909 119910 isin 119883 and 119905 gt 0 with T(119886 119887) =(min119886
1 1198871max119886
2 1198872) for all 119886 = (119886
1 1198862) and 119887 = (119887
1 1198872) isin
119871lowast Define now the mapping 119865 119883 times 119883 rarr 119883 by
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816
1199052 +1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816)
= (119905
119905 + 21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816
21003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)
1003816100381610038161003816119905 + 2
1003816100381610038161003816119865 (119909 119910) minus 119865 (119906 V)1003816100381610038161003816)
ge119871lowast (
119905
119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +max |119909 minus 119906| 1003816100381610038161003816119910 minus V1003816100381610038161003816
)
Advances in Fuzzy Systems 11
ge119871lowast (min 119905
119905 + |119909 minus 119906|
119905
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010
119905 +1003816100381610038161003816119910 minus V1003816100381610038161003816
max |119909 minus 119906|
119905 + |119909 minus 119906|
1003816100381610038161003816119910 minus V1003816100381610038161003816119905 +
1003816100381610038161003816119910 minus V1003816100381610038161003816)
ge119871lowastT (M
119872119873(119909 119906 119905) M
119872119873(119910 V 119905))
(97)
which is the condition in (67) shows that all hypotheses ofCorollary 32 are satisfied This proves that 119865 has a uniquefixed point 119909 isin 119883 that is 119909 = 119865(119909 119909) and therefore119909 isin 119862(119868 119877) is the unique solution of the integral equation(81)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] I Kramosil and JMichalek ldquoFuzzymetric and statistical metricspacesrdquo Kybernetika vol 11 no 5 pp 336ndash344 1975
[2] A George and P Veeramani ldquoOn some results in fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 64 no 3 pp 395ndash399 1994
[3] K T Atanassov ldquoIntuitionistic fuzzy setsrdquo Fuzzy Sets andSystems vol 20 no 1 pp 87ndash96 1986
[4] C Alaca D Turkoglu and C Yildiz ldquoFixed points in intuition-istic fuzzy metric spacesrdquo Chaos Solitons amp Fractals vol 29 no5 pp 1073ndash1078 2006
[5] J H Park ldquoIntuitionistic fuzzy metric spacesrdquo Chaos Solitonsamp Fractals vol 22 no 5 pp 1039ndash1046 2004
[6] V Gregori S Romaguera and P Veeramani ldquoA note onintuitionistic fuzzy metric spacesrdquo Chaos Solitons amp Fractalsvol 28 no 4 pp 902ndash905 2006
[7] R Saadati S Sedghi and N Shobe ldquoModified intuitionisticfuzzy metric spaces and some fixed point theoremsrdquo ChaosSolitons amp Fractals vol 38 no 1 pp 36ndash47 2008
[8] T G Bhaskar and V Lakshmikantham ldquoFixed point theoremsin partially ordered metric spaces and applicationsrdquo NonlinearAnalysis Theory Methods and Applications vol 65 no 7 pp1379ndash1393 2006
[9] V Lakshmikantham and L Ciric ldquoCoupled fixed point the-orems for nonlinear contractions in partially ordered metricspacesrdquo Nonlinear Analysis Theory Methods and Applicationsvol 70 no 12 pp 4341ndash4349 2009
[10] S Sedghi I Altun andN Shobe ldquoCoupled fixedpoint theoremsfor contractions in fuzzy metric spacesrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1298ndash1304 2010
[11] X Q Hu ldquoCommon coupled fixed point theorems for contrac-tive mappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2011 Article ID 363716 14 pages 2011
[12] X Q Hu M X Zheng B Damjanovic and X F ShaoldquoCommon coupled fixed point theorems for weakly compatiblemappings in fuzzy metric spacesrdquo Fixed Point Theory andApplications vol 2013 article 220 2013
[13] B Deshpande S Sharma and A Handa ldquoCommon coupledfixed point theorems for nonlinear contractive condition onintuitionistic fuzzy metric space with application to integral
equationsrdquo Journal of the Korean Society of MathematicalEducation B The Pure and Applied Mathematics vol 20 no 3pp 159ndash180 2013
[14] G Deschrijver and E E Kerre ldquoOn the relationship betweensome extensions of fuzzy set theoryrdquo Fuzzy Sets and Systemsvol 133 no 2 pp 227ndash235 2003
[15] G Deschrijver C Cornelis and E E Kerre ldquoOn the repre-sentation of intuitionistic fuzzy t-norms and t-conormsrdquo IEEETransactions on Fuzzy Systems vol 12 no 1 pp 45ndash61 2004
[16] J X Fang ldquoOn fixed point theorems in fuzzy metric spacesrdquoFuzzy Sets and Systems vol 46 no 1 pp 107ndash113 1992
[17] R Saadati and J H Park ldquoOn the modified intuitionistic fuzzytopological spacesrdquo Chaos Solitons amp Fractals vol 27 no 2 pp331ndash334 2006
[18] J X Fang ldquoCommon fixed point theorems of compatible andweakly compatible maps inMenger spacesrdquoNonlinear AnalysisTheoryMethods andApplications vol 71 no 5-6 pp 1833ndash18432009
[19] M Abbas M A Khan and S Radenovic ldquoCommon coupledfixed point theorems in cone metric spaces for w-compatiblemappingsrdquo Applied Mathematics and Computation vol 217 no1 pp 195ndash202 2010