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Hindawi Publishing Corporation FixedPoint Theory and Applications Volume 2011, Article ID 363716, 14 pages doi:10.1155/2011/363716 ResearchArticleCommonCoupledFixedPointTheoremsforContractiveMappingsinFuzzyMetric Spaces Xin-Qi Hu School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China Correspondence should be addressed to Xin-Qi Hu, xqhu.math@whu.edu.cn Received 23 November 2010; Accepted 27 January 2011 Academic Editor: Ljubomir B. Ciric Copyright q 2011 Xin-Qi Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We prove a common fixed point theorem formappings under φ-contractive conditions infuzzymetric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of S.Sedghi et al. 2010 1. Introduction Since Zadeh 1 introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani 2, 3 gave the concept of fuzzymetric space and defined a Hausdorff topology on this fuzzymetric space which have very important applications in quantum particle physics particularly in connection with both string and E-infinity theory. Bhaskar and Lakshmikantham 4, Lakshmikantham and ´ Ciri ´ c 5 discussed the mixed monotone mappings and gave some coupled fixed pointtheorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al. 6 gave a coupled fixed point theorem for contractions infuzzymetric spaces, and Fang 7 gave some common fixed pointtheorems under φ-contractions for compatible and weakly compatible mappingsin Menger probabilistic metric spaces. Many authors 8– 23 have proved fixed pointtheoremsin intuitionistic fuzzymetric spaces or probabilistic metric spaces. In this paper, using similar proof as in 7, we give a new common fixed point theorem under weaker conditions than in 6 and give an example which shows that the result is a genuine generalization of the corresponding result in 6. 2 FixedPoint Theory and Applications 2. Preliminaries First we give some definitions. Definition 1 see 2.Abinaryoperation∗ : 0, 1 × 0, 1 → 0, 1 is continuous t-norm if ∗ is satisfying the following conditions: 1 ∗ is commutative and associative; 2 ∗ is continuous; 3 a ∗ 1 a for all a ∈ 0, 1; 4 a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ 0, 1. Definition 2 see 24.Letsup 0<t<1 Δt, t1. A t-norm Δ is said to be of H-type if the family of functions {Δ m t} ∞ m1 is equicontinuous at t 1, where Δ 1 t tΔt, Δ m1 t tΔ Δ m t ,m 1, 2, , t∈ 0, 1 . 2.1 The t-norm Δ M min is an example of t-norm of H-type, but there are some other t-norms Δ of H-type 24. Obviously, Δ is a H-type t norm if and only if for any λ ∈ 0, 1,thereexistsδλ ∈ 0, 1 such that Δ m t > 1 − λ for all m ∈ ,whent>1 − δ. Definition 3 see 2.A3-tupleX, M, ∗ is said to be a fuzzymetric space if X is an arbitrary nonempty set, ∗ is a continuous t-norm, and M is a fuzzy set on X 2 × 0, ∞ satisfying the following conditions, for each x, y, z ∈ X and t, s > 0: FM-1 Mx, y, t > 0; FM-2 Mx, y, t1 if and only if x y; FM-3 Mx, y, tMy, x, t; FM-4 Mx, y, t ∗ My, z, s ≤ Mx, z, t s; FM-5 Mx, y, · : 0, ∞ → 0, 1 is continuous. Let X, M, ∗ be a fuzzymetric space. For t>0, the open ball Bx, r, t with a center x ∈ X and a radius 0 <r<1isdefinedby B x, r, t y ∈ X : M x, y, t > 1 − r . 2.2 AsubsetA ⊂ X is called open if , for each x ∈ A,thereexistt>0and0<r<1suchthat Bx, r, t ⊂ A.Letτ denote the family of all open subsets of X.Thenτ is called the topology on X induced by the fuzzymetric M. This topology is Hausdorff and first countable. Example 1. Let X, d be a metric space. Define t-norm a ∗b ab and for all x, y ∈ X and t>0, Mx, y, tt/t dx, y.ThenX, M, ∗ is a fuzzymetric space. We call this fuzzymetric M induced by the metric d the standard fuzzy metric. FixedPoint Theory and Applications 3 Definition 4 see 2.LetX, M, ∗ be a fuzzymetric space, then 1 asequence{x n } in X is said to be convergent to x denoted by lim n →∞ x n x if lim n →∞ M x n ,x,t 1, 2.3 for all t>0; 2 asequence{x n } in X is said to be a Cauchy sequence if for any ε>0, there exists n 0 ∈ ,suchthat M x n ,x m ,t > 1 − ε, 2.4 for all t>0andn,m≥ n 0 ; 3 a fuzzymetric space X, M, ∗ is sa id to be complete if and only if every Cauchy sequence in X is convergent. Remark 1 see 25. 1 For all x, y ∈ X, Mx, y, · is nondecreasing. 2 It is easy to prove that if x n → x, y n → y, t n → t,then lim n →∞ M x n ,y n ,t n M x, y, t . 2.5 3 In a fuzzymetric space X, M, ∗, whenever Mx, y, t > 1 − r for x, y in X, t>0, 0 <r<1, we can find a t 0 ,0<t 0 <tsuch that Mx, y, t 0 > 1 − r. 4 For any r 1 >r 2 ,wecanfindanr 3 such that r 1 ∗ r 3 ≥ r 2 and for any r 4 we can find a r 5 such that r 5 ∗ r 5 ≥ r 4 r 1 ,r 2 ,r 3 ,r 4 ,r 5 ∈ 0, 1. Definition 5 see 6.LetX, M, ∗ be a fuzzymetric space. M is said to satisfy the n-property on X 2 × 0, ∞ if lim n →∞ M x, y, k n t n p 1, 2.6 whenever x, y ∈ X, k>1andp>0. Lemma 1. Let X, M, ∗ be a fuzzymetric space and M satisfies the n-property; then lim t →∞ M x, y, t 1, ∀x, y ∈ X. 2.7 Proof. If not, since Mx, y, · is nondecreasing and 0 ≤ Mx, y, · ≤ 1, there exists x 0 ,y 0 ∈ X such that lim t →∞ Mx 0 ,y 0 ,tλ<1, then for k>1, k n t → ∞ when n →∞as t>0and we get lim n →∞ Mx 0 ,y 0 ,k n t n p 0, which is a contraction. 4 FixedPoint Theory and Applications Remark 2. Condition 2.7 cannot guarantee the n-property. See the following example. Example 2. Let X, d be an ordinary metric space, a ∗ b ≤ ab for all a, b ∈ 0, 1,andψ be defined as following: ψ t ⎧ ⎪ ⎨ ⎪ ⎩ α √ t, 0 <t≤ 4, 1 − 1 ln t ,t>4, 2.8 where α 1/21 − 1/ ln 4.Thenψt is continuous and increasing in 0, ∞, ψt ∈ 0, 1 and lim t →∞ ψt1. Let M x, y, t ψ t dx,y , ∀x, y ∈ X, t > 0, 2.9 then X, M, ∗ is a fuzzymetric space and lim t →∞ M x, y, t lim t →∞ ψ t dx,y 1, ∀x, y ∈ X. 2.10 But for any x / y, p 1, k>1, t>0, lim n →∞ M x, y, k n t n p lim n →∞ ψk n t dx,y·n p lim n →∞ 1 − 1 ln k n t n·dx,y e −dx,y/ ln k / 1. 2.11 Define Φ{φ : R → R },whereR 0, ∞ and each φ ∈ Φ satisfies the following conditions: φ-1 φ is nondecreasing; φ-2 φ is upper semicontinuous from the right; φ-3 ∞ n0 φ n t < ∞ for all t>0, where φ n1 tφφ n t, n ∈ . It is easy to prove that, if φ ∈ Φ,thenφt <tfor all t>0. Lemma 2 see 7. Let X, M, ∗ be a fuzzymetric space, where ∗ is a continuous t-norm of H-type. If there exists φ ∈ Φ such that if M x, y, φ t ≥ M x, y, t , 2.12 for all t>0,thenx y. Definition 6 see 5.Anelementx, y ∈ X×X is called a coupled fixed point of the mapping F : X × X → X if F x, y x, F y, x y. 2.13 FixedPoint Theory and Applications 5 Definition 7 see 5.Anelementx, y ∈ X × X is called a coupled coincidence point of the mappings F : X × X → X and g : X → X if F x, y g x ,F y, x g y . 2.14 Definition 8 see 7.Anelementx, y ∈ X × X is called a commoncoupled fixed point of the mappings F : X ×X → X and g : X → X if x F x, y g x ,y F y, x g y . 2.15 Definition 9 see 7.Anelementx ∈ X is called a common fixed point of the mappings F : X × X → X and g : X → X if x g x F x, x . 2.16 Definition 10 see 7. The mappings F : X×X → X and g : X → X are said to be compatible if lim n →∞ M gF x n ,y n ,F g x n ,g y n ,t 1, lim n →∞ M gF y n ,x n ,F g y n ,g x n ,t 1, 2.17 for all t>0 whenever {x n } and {y n } are sequences in X,suchthat lim n →∞ F x n ,y n lim n →∞ g x n x, lim n →∞ F y n ,x n lim n →∞ g y n y, 2.18 for all x, y ∈ X are satisfied. Definition 11 see 7. The mappings F : X ×X → X and g : X → X are called commutative if g F x, y F gx, gy , 2.19 for all x, y ∈ X. Remark 3. It is easy to prove that, if F and g are commutative, then they are compatible. 3. Main Results For convenience, we denote M x, y, t n Mx, y, t ∗ Mx, y, t ∗···∗Mx, y, t n , 3.1 for all n ∈ . 6 FixedPoint Theory and Applications Theorem 1. Let X, M, ∗ be a complete FM-space, where ∗ is a continuous t-norm of H-type satisfying 2.7.LetF : X × X → X and g : X → X be two mappings and there exists φ ∈ Φ such that M F x, y ,F u, v ,φ t ≥ M g x ,g u ,t ∗ M g y ,g v ,t , 3.2 for all x, y, u, v ∈ X, t>0. Suppose that FX ×X ⊆ gX,and g is continuous, F and g are compatible. Then there exist x, y ∈ X such that x gxFx, x,thatis,F and g have a unique common fixed pointin X. Proof. Let x 0 ,y 0 ∈ X be two arbitrary points in X.SinceFX × X ⊆ gX, we can choose x 1 ,y 1 ∈ X such that gx 1 Fx 0 ,y 0 and gy 1 Fy 0 ,x 0 . Continuing in this way we can construct two sequences {x n } and {y n } in X such that g x n1 F x n ,y n ,g y n1 F y n ,x n , ∀n ≥ 0. 3.3 The proof is divided into 4 steps. Step 1. Prove that {gx n } and {gy n } are Cauchy sequences. Since ∗ is a t-norm of H-type, for any λ>0, there exists a μ>0suchthat 1 − μ ∗ 1 − μ ∗···∗1 − μ k ≥ 1 − λ, 3.4 for all k ∈ . Since Mx, y, · is continuous and lim t →∞ Mx, y, t1forallx, y ∈ X,thereexists t 0 > 0suchthat M gx 0 ,gx 1 ,t 0 ≥ 1 − μ, M gy 0 ,gy 1 ,t 0 ≥ 1 − μ. 3.5 On the other hand, since φ ∈ Φ, by condition φ-3 we have ∞ n1 φ n t 0 < ∞.Thenfor any t>0, there exists n 0 ∈ such that t> ∞ kn 0 φ k t 0 . 3.6 From condition 3.2,wehave M gx 1 ,gx 2 ,φ t 0 M F x 0 ,y 0 ,F x 1 ,y 1 ,φ t 0 ≥ M gx 0 ,gx 1 ,t 0 ∗ M gy 0 ,gy 1 ,t 0 , M gy 1 ,gy 2 ,φ t 0 M F y 0 ,x 0 ,F y 1 ,x 1 ,φ t 0 ≥ M gy 0 ,gy 1 ,t 0 ∗ M gx 0 ,gx 1 ,t 0 . 3.7 FixedPoint Theory and Applications 7 Similarly, we can also get M gx 2 ,gx 3 ,φ 2 t 0 M F x 1 ,y 1 ,F x 2 ,y 2 ,φ 2 t 0 ≥ M gx 1 ,gx 2 ,φ t 0 ∗ M gy 1 ,gy 2 ,φ t 0 ≥ M gx 0 ,gx 1 ,t 0 2 ∗ M gy 0 ,gy 1 ,t 0 2 , M gy 2 ,gy 3 ,φ 2 t 0 M F y 1 ,x 1 ,F y 2 ,x 2 ,φ 2 t 0 ≥ M gy 0 ,gy 1 ,t 0 2 ∗ M gx 0 ,gx 1 ,t 0 2 . 3.8 Continuing in the same way we can get M gx n ,gx n1 ,φ n t 0 ≥ M gx 0 ,gx 1 ,t 0 2 n−1 ∗ M gy 0 ,gy 1 ,t 0 2 n−1 , M gy n ,gy n1 ,φ n t 0 ≥ M gy 0 ,gy 1 ,t 0 2 n−1 ∗ M gx 0 ,gx 1 ,t 0 2 n−1 . 3.9 So, from 3.5 and 3.6,form>n≥ n 0 ,wehave M gx n ,gx m ,t ≥ M gx n ,gx m , ∞ kn 0 φ k t 0 ≥ M gx n ,gx m , m−1 kn φ k t 0 ≥ M gx n ,gx n1 ,φ n t 0 ∗ M gx n1 ,gx n2 ,φ n1 t 0 ∗···∗M gx m−1 ,gx m ,φ m−1 t 0 ≥ M gy 0 ,gy 1 ,t 0 2 n−1 ∗ M gx 0 ,gx 1 ,t 0 2 n−1 ∗ M gy 0 ,gy 1 ,t 0 2 n ∗ M gx 0 ,gx 1 ,t 0 2 n ∗···∗ M gy 0 ,gy 1 ,t 0 2 m−2 ∗ M gx 0 ,gx 1 ,t 0 2 m−2 M gy 0 ,gy 1 ,t 0 2 m−nmn−3 ∗ M gx 0 ,gx 1 ,t 0 2 m−nmn−3 ≥ 1 − μ ∗ 1 − μ ∗···∗1 − μ 2 2m−nmn−3 ≥ 1 − λ, 3.10 which implies that M gx n ,gx m ,t > 1 −λ, 3.11 for all m, n ∈ with m>n≥ n 0 and t>0. So {gx n } is a Cauchy sequence. Similarly, we can get that {gy n } is also a Cauchy sequence. 8 FixedPoint Theory and Applications Step 2. Prove that g and F have a coupled coincidence point. Since X complete, there exist x, y ∈ X such that lim n →∞ F x n ,y n lim n →∞ g x n x, lim n →∞ F y n ,x n lim n →∞ g y n y. 3.12 Since F and g are compatible, we have by 3.12, lim n →∞ M gF x n ,y n ,F g x n ,g y n ,t 1, lim n →∞ M gF y n ,x n ,F g y n ,g x n ,t 1. 3.13 for all t>0. Next we prove that gxFx, y and gyFy, x. For all t>0, by condition 3.2,wehave M gx, F x, y ,φ t ≥ M ggx n1 ,F x, y ,φ k 1 t ∗ M gx, ggx n1 ,φ t − φ k 1 t M gF x n ,y n ,F x, y ,φ k 1 t ∗ M gx, ggx n1 ,φ t − φ k 1 t ≥ M gF x n ,y n ,F gx n ,gy n ,φ k 1 t − φ k 2 t ∗ M F gx n ,gy n ,F x, y ,φ k 2 t ∗ M gx, ggx n1 ,φ t − φ k 1 t ≥ M gF x n ,y n ,F gx n ,gy n ,φ k 1 t − φ k 2 t ∗ M ggx n ,gx,k 2 t ∗ M ggy n ,gy,k 2 t ∗ M gx, ggx n1 ,φ t − φ k 1 t , 3.14 for all 0 <k 2 <k 1 < 1. Let n →∞,sinceg and F are compatible, with the continuity of g,we get M gx, F x, y ,φ t ≥ 1, 3.15 which implies that gx Fx, y. Similarly, we can get gy Fy, x. Step 3. Prove that gx y and gy x. Since ∗ is a t-norm of H-type, for any λ>0, there exists an μ>0suchthat 1 − μ ∗ 1 − μ ∗···∗1 − μ k ≥ 1 − λ, 3.16 for all k ∈ . Since Mx, y, · is continuous and lim t →∞ Mx, y, t1forallx, y ∈ X,thereexists t 0 > 0suchthatMgx,y,t 0 ≥ 1 − μ and Mgy,x,t 0 ≥ 1 − μ. FixedPoint Theory and Applications 9 On the other hand, since φ ∈ Φ, by condition φ-3 we have ∞ n1 φ n t 0 < ∞.Thenfor any t>0, there exists n 0 ∈ such that t> ∞ kn 0 φ k t 0 .Since M gx, gy n1 ,φ t 0 M F x, y ,F y n ,x n ,φ t 0 ≥ M gx, gy n ,t 0 ∗ M gy,gx n ,t 0 , 3.17 letting n →∞,weget M gx,y,φ t 0 ≥ M gx,y,t 0 ∗ M gy,x,t 0 . 3.18 Similarly, we can get M gy,x,φ t 0 ≥ M gx,y,t 0 ∗ M gy,x,t 0 . 3.19 From 3.18 and 3.19 we have M gx,y,φ t 0 ∗ M gy,x,φ t 0 ≥ M gx,y,t 0 2 ∗ M gy,x,t 0 2 . 3.20 By this way, we can get for all n ∈ , M gx,y,φ n t 0 ∗ M gy,x,φ n t 0 ≥ M gx,y,φ n−1 t 0 2 ∗ M gy,x,φ n−1 t 0 2 ≥ M gx,y,t 0 2 n ∗ M gy,x,t 0 2 n . 3.21 Then, we have M gx,y,t ∗ M gy,x,t ≥ M gx, y, ∞ kn 0 φ k t 0 ∗ M gy,x, ∞ kn 0 φ k t 0 ≥ M gx,y,φ n 0 t 0 ∗ M gy,x,φ n 0 t 0 ≥ M gx,y,t 0 2 n 0 ∗ M gy,x,t 0 2 n 0 ≥ 1 − μ ∗ 1 − μ ∗···∗1 − μ 2 2n 0 ≥ 1 −λ. 3.22 So for any λ>0wehave M gx,y,t ∗ M gy,x,t ≥ 1 − λ, 3.23 for all t>0. We can get that gx y and gy x. 10 FixedPoint Theory and Applications Step 4. Prove that x y. Since ∗ is a t-norm of H-type, for any λ>0, there exists an μ>0suchthat 1 − μ ∗ 1 − μ ∗···∗1 − μ k ≥ 1 − λ, 3.24 for all k ∈ . Since Mx, y, · is continuous and lim t →∞ Mx, y, t1, there exists t 0 > 0suchthat Mx, y, t 0 ≥ 1 − μ. On the other hand, since φ ∈ Φ, by condition φ-3 we have ∞ n1 φ n t 0 < ∞.Thenfor any t>0, there exists n 0 ∈ such that t> ∞ kn 0 φ k t 0 . Since for t 0 > 0, M gx n1 ,gy n1 ,φ t 0 M F x n ,y n ,F y n ,x n ,φ t 0 ≥ M gx n ,gy n ,t 0 ∗ M gy n ,gx n ,t 0 . 3.25 Letting n →∞yields M x, y, φ t 0 ≥ M x, y, t 0 ∗ M y, x, t 0 . 3.26 Thus we have M x, y, t ≥ M x, y, ∞ kn 0 φ k t 0 ≥ M x, y, φ n 0 t 0 ≥ M x, y, t 0 2 n 0 ∗ M y, x, t 0 2 n 0 ≥ 1 − μ ∗ 1 − μ ∗···∗1 − μ 2 2n 0 ≥ 1 − λ, 3.27 which implies that x y. Thus we have proved that F and g ha ve a unique common fixed pointin X. This completes the proof of the Theorem 1. Taking g I the identity mapping in Theorem 1, we get the following consequence. Corollary 1. Let X, M, ∗ be a complete FM-space, where ∗ is a continuous t-norm of H-type satisfying 2.7.LetF : X × X → X and there exists φ ∈ Φ such that M F x, y ,F u, v ,φ t ≥ M x, u, t ∗ M y, v, t , 3.28 for all x, y, u, v ∈ X, t>0. [...]... nonlinear contractions in c c partially ordered metric spaces,” FixedPoint Theory and Applications, vol 2008, Article ID 131294, 11 pages, 2008 14 A Aliouche, F Merghadi, and A Djoudi, “A related fixed point theorem in two fuzzymetric spaces,” Journal of Nonlinear Science and its Applications, vol 2, no 1, pp 19–24, 2009 ´ c 15 L Ciri´ , Common fixed pointtheoremsfor a family of non-self mappings in. .. results for Banach contractions and Edelstein contractivemappings on fuzzymetric spaces,” Chaos, Solitons and Fractals, vol 42, no 1, pp 146–154, 2009 ´ c 19 S Shakeri, L J B Ciri´ , and R Saadati, Common fixed point theorem in partially ordered L -fuzzy metric spaces,” FixedPoint Theory and Applications, vol 2010, Article ID 125082, 13 pages, 2010 ´ c 20 L Ciri´ , B Samet, and C Vetro, Common fixed point. .. Abbas, B Damjanovi´ , and R Saadati, Commonfuzzy fixed pointtheoremsin ordered c metric spaces,” Mathematical and Computer Modelling, vol 53, no 9-10, pp 1737–1741, 2011 23 T Kamran and N Caki´ , “Hybrid tangential property and coincidence point theorems, ” FixedPoint c Theory, vol 9, no 2, pp 487–496, 2008 24 O Hadˇ i´ and E Pap, FixedPoint Theory in Probabilistic Metric Spaces, vol 536 of Mathematics... 11 L B Ciri´ , S N Jeˇ i´ , and J S Ume, “The existence theoremsfor fixed and periodic points of sc nonexpansive mappingsin intuitionistic fuzzymetric spaces,” Chaos, Solitons and Fractals, vol 37, no 3, pp 781–791, 2008 ´ c 12 L Ciri´ and V Lakshmikantham, Coupled random fixed pointtheoremsfor nonlinear contractions in partially ordered metric spaces,” Stochastic Analysis and Applications, vol... mappingsin convex metric spaces,” Nonlinear Analysis Theory, Methods & Applications, vol 71, no 5-6, pp 1662–1669, 2009 16 K P R Rao, A Aliouche, and G R Babu, “Related fixed pointtheoremsinfuzzymetric spaces,” Journal of Nonlinear Science and its Applications, vol 1, no 3, pp 194–202, 2008 ´ c 17 L Ciri´ and N Caki´ , “On common fixed pointtheoremsfor non-self hybrid mappingsin convex c metric spaces,”... Ciri´ , B Samet, and C Vetro, Common fixed pointtheoremsfor families of occasionally weakly compatible mappings, ” Mathematical and Computer Modelling, vol 53, no 5-6, pp 631–636, 2011 14 FixedPoint Theory and Applications ´ c 21 L Ciri´ , M Abbas, R Saadati, and N Hussain, Common fixed points of almost generalized contractivemappingsin ordered metric spaces,” Applied Mathematics and Computation,... spaces,” Fuzzy Sets and Systems, vol 90, no 3, pp 365–368, 1997 4 T G Bhaskar and V Lakshmikantham, Fixedpointtheoremsin partially ordered metric spaces and applications,” Nonlinear Analysis Theory, Methods & Applications, vol 65, no 7, pp 1379–1393, 2006 ´ c 5 V Lakshmikantham and L Ciri´ , Coupled fixed pointtheoremsfor nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis... Applications, vol 70, no 12, pp 4341–4349, 2009 6 S Sedghi, I Altun, and N Shobe, Coupled fixed pointtheoremsfor contractions infuzzymetric spaces,” Nonlinear Analysis Theory, Methods & Applications, vol 72, no 3-4, pp 1298–1304, 2010 7 J.-X Fang, Common fixed pointtheorems of compatible and weakly compatible maps in Menger spaces,” Nonlinear Analysis Theory, Methods & Applications, vol 71, no 5-6, pp 1833–1843,... contractions in partially ordered ¸ probabilistic metric spaces,” Topology and its Applications, vol 156, no 17, pp 2838–2844, 2009 9 D O’Regan and R Saadati, “Nonlinear contraction theoremsin probabilistic spaces,” Applied Mathematics and Computation, vol 195, no 1, pp 86–93, 2008 10 S Jain, S Jain, and L Bahadur Jain, “Compatibility of type P in modified intuitionistic fuzzymetric space,” Journal of Nonlinear... F and g in X Acknowledgment The author is grateful to the referees for their valuable comments and suggestions References 1 L A Zadeh, Fuzzy sets,” Information and Computation, vol 8, pp 338–353, 1965 2 A George and P Veeramani, “On some results infuzzymetric spaces,” Fuzzy Sets and Systems, vol 64, no 3, pp 395–399, 1994 3 A George and P Veeramani, “On some results of analysis forfuzzymetric spaces,” . Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 363716, 14 pages doi:10.1155/2011/363716 Research Article Common Coupled Fixed Point Theorems for Contractive. coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang 7 gave some common fixed point theorems under φ-contractions for compatible and weakly compatible mappings in Menger. probabilistic metric spaces. Many authors 8– 23 have proved fixed point theorems in intuitionistic fuzzy metric spaces or probabilistic metric spaces. In this paper, using similar proof as in 7,