International Journal of Analysis and Applications ISSN 2291-8639 Volume 4, Number 2 (2014), 201-215 http://www.etamaths.com COUPLED BEST PROXIMITY POINT THEOREM IN METRIC SPACES ANIMESH GUPTA1,∗, S.S. RAJPUT2 AND P.S. KAURAV2 Abstract. The purpose of this article is to generalized the result of W. S- intunavarat and P. Kumam [29]. We also give an example in support of our theorem for which result of W. Sintunavarat and P. Kumam [29] is not true. Moreover we establish the existence and convergence theorems of coupled best proximity points in metric spaces, we apply this results in a uniformly convex Banach space. Contents This article is organized in the following order. Section - 1 : In this section we give some basic concepts of the best proximity point theorems also we give some previous known results which are used to prove of our main result. Section - 2 : In this section we study the existence and convergence of coupled best proximity points for cyclic contraction pair. We also give an example in support of our Theorem. Section - 3 : In this section, we give the new coupled fixed point theorem for a cyclic contraction pair. We also give an example in support of our Theorem. Section - 4 : In this section authors would like to express their sincere thanks to the editorial board and referees. 1. Introduction and Preliminaries Fixed point theory is one of the most useful tools in analysis. The first result of fixed point theorem is given by Banach S. [4] by the general setting of complete metric space using which is known as Banach Contraction Principle. This principle has been generalized by many researchers in many ways like by [2], [9], [10], [24], [33], [34], [40] and so on. One of the important thing in [4] is Banach contraction principle is true for self mapping. In case of non self mapping (say T) the mapping does not has a fixed point. Then the researchers find an element x such that d(x,Tx) is minimum or near to zero for a given problem which implies that x and Tx are very closed says close proximity to each other. Due to this problem the theory of fixed point is converted into the theory of best proximity point. On the other words, proximity 2000 Mathematics Subject Classification. 47H10,54H25,46J10, 46J15. Key words and phrases. Coupled fixed point, Coupled Common Fixed Point, Coupled best proximity point, Mixed monotone. c©2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 201 202 GUPTA, RAJPUT AND KAURAV theory is a generalization of fixed point theory. Basically the proximity theory is useful tool to find proximity point when the given mapping is non self. Let A and B be two non empty subsets of X such that T : A → B then a point x ∈ A for which d(x,Tx) = d(A,B) is called a best proximity point of T . It should be noted that best proximity point reduced to fixed point when the mapping T is self mapping that is A = B. In 1969, Fan [12] presented a classical result for best approximation theorem which as follows, Theorem 1 ([12]). If A is a nonempty convex subset of a Hausdorff locally convex topological vector space B and T : A → B is continuous mapping, then there exists an element x ∈ A such that d(x,Tx) = d(Tx,A). Afterword a number of authors have derived extensions of Fan’s Theorem and the best approximation theorem in many directions such as Prolla [26], Sehgal and Singh [27, 28], Wlodarczyk and Plebaniak [43, 44, 45, 46], Vetrivel et al. [42], Eldred and Veramani [11], Mongkolkeha and Kumam [25] and Basha and Veeramani [5, 6, 7, 8] (see also [3, 15, 16, 17, 18, 19, 20, 21] and reference therein.) Beside this, Bhaskar and Lakshmikantham [13] introduced the notion of mixed monotone mapping and proved some coupled fixed point theorems for mapping satisfying mixed monotone property. After the result of [13] there are lots of work presented by many authors such as [1], [14], [30], [31] (see also reference therein.) The concept of coupled best proximity point theorem is introduced by W. Sintu- navarat and P. Kumam [29] and proved coupled best proximity theorem for cyclic contraction. Our purpose of this article is to generalized the result of [29] also we give an example in support of our main theorem. First we recall some basic definitions and examples that are related to the main results of this article. Throughout this article we denote by N the set of all positive integers and by R the set of all real numbers. For nonempty subsets A and B of a metric space (X,d), we let (1.1) d(A,B) = inf{d(x,y) : x ∈ A and y ∈ B} stands for the distance between A and B. A Banach spaces X is said to be (1) strictly convex if the following implication holds for all x,y ∈ X: ‖x‖ = ‖y‖ = 1 and x 6= y =⇒‖x+y 2 ‖ < 1. (2) uniformly convex if for each � with 0� ≤ 2, there exists δ > 0 such that thee following implication holds for all x,y ∈ X: ‖x‖≤ 1,‖y‖≤ 1 and ‖x−y‖≥ � =⇒‖x+y 2 ‖ < 1 − δ. It is easily to see that a uniformly convex Banach space X is strictly but the converges is not true. Definition 2 ([41]). Let A and B be nonempty subsets of a metric space (X,d). The ordered pair (A,B) satisfies the property UC if the following holds: COUPLED BEST PROXIMITY POINT THEOREM IN METRIC SPACES 203 If {xn} and {zn} are sequences in A and {yn} is a sequence in B such that d(xn,yn) → d(A,B) and d(zn,yn) → d(A,B), then d(xn,zn) → 0. Example 3. Let A and B be nonempty subsets of a metric space (X,d). The following are examples of a pair of nonempty subsets (A,B) satisfying the property UC. (1) Every pair of nonempty subsets A,B of a metric space (X,d) such that d(A,B) = 0. (2) Every pair of nonempty subsets A,B of a uniformly convex Banach space X such that A is convex. (3) Every pair of nonempty subsets A,B of a strictly convex Banach space which A is convex and relatively compact and the closure of B is weakly compact. Definition 4 ([39]). Let A and B be nonempty subsets of a metric space (X,d). The ordered pair (A,B) satisfies the property UC∗ if (A,B) has property UC and the following condition holds: If {xn} and {zn} are sequences in A and {yn} is a sequence in B satisfying: (1) d(zn,yn) → d(A,B) (2) For every � > 0 there exists N ∈ N such that d(xm,yn) ≤ d(A,B) + � for all m > n ≥ N. Then for every � > 0 there exists N1 ∈ N such that d(xm,zn) ≤ d(A,B) + � for all m > n ≥ N1. Example 5 ([39]). Let A and B be nonempty subsets of a metric space (X,d). The following are examples of a pair of nonempty subsets (A,B) satisfying the property UC∗. (1) Every pair of nonempty subsets A,B of a metric space (X,d) such that d(A,B) = 0. (2) Every pair of nonempty closed subsets A,B of a uniformly convex Banach space X such that A is convex. Definition 6. Let A and B be nonempty subsets of a metric space (X,d) and T : A → B be a mapping. A point x ∈ A is said to be a best proximity point of T if it satisfies the condition that d(x,Tx) = d(A,B). It can be observed that a best proximity point reduces to a fixed point if the underlying mapping is a self mapping. Definition 7 ([13]). Let A be a nonempty subset of a metric space X and F : A×A → A. A point (x,y) ∈ A×A is called a coupled fixed point of F if x = F(x,y), y = F(y,x). 204 GUPTA, RAJPUT AND KAURAV 2. Coupled best proximity point theorems In this section we study the existence and convergence of coupled best proximity points for cyclic contraction pair. Definition 8. Let A and B be nonempty subsets of a metric space X and F : A×A → B. An ordered coupled (x,y) ∈ A×A is called a coupled best proximity point of F if, d(x,F(x,y)) = d(y,F(y,x)) = d(A,B). It is easy to see that if A = B in Definition 8, then a coupled best proximity point reduces to a coupled fixed point. Next,W. Sintunavarat and P. Kumam [29] introduce the notion of a cyclic con- traction for two mappings which as follows. Definition 9. Let A and B be nonempty subsets of a metric space X, F : A×A → B and G : B ×B → A. The ordered pair (F,G) is said to be a cyclic contraction if there exists a non-negative number α < 1 such that d(F(x,y),G(u,v)) ≤ α 2 [d(x,u) + d(y,v)] + (1 −α)d(A,B) for all (x,y) ∈ A×A and (u,v) ∈ B ×B. Now we introduced the following notion of cyclic contraction for two mappings which is generalization of [29] as follows. Definition 10. Let A and B be nonempty subsets of a metric space X, F : A×A → B and G : B ×B → A. The ordered pair (F,G) is said to be a cyclic contraction if there exists a non-negative number p + q < 1 such that d(F(x,y),G(u,v)) ≤ [pd(x,u) + qd(y,v)] + (1 − (p + q))d(A,B) for all (x,y) ∈ A×A and (u,v) ∈ B ×B. Note that if (F,G) is a cyclic contraction, then (G,F) is also a cyclic contraction. Also if we take p = q = α 2 in Definition 10 then we get Definition 9. Following example show that Definition 10 is generalization of Definition 9. Example 11. Let X = R with the usual metric d(x,y) =| x−y | also A = [6, 12] and B = [−12,−6]. It easy to see that d(A,B) = 12. Define F : A×A → B and G : B ×B → A by F(x,y) = −3x− 2y − 6 6 and G(x,y) = −3x− 2y + 6 6 . For arbitrary (x,y) ∈ A×A, (u,v) ∈ B ×B and fixed k = 1 2 , l = 1 3 , we get d(F(x,y),G(u,v)) = ∣∣∣∣−3x− 2y − 66 − −3u− 2v + 66 ∣∣∣∣ ≤ 3 | x−u | +2 | y −v | 6 + 2 = kd(x,u) + ld(y,v) + (1 − (k + l))d(A,B). This implies that (F,G) is a cyclic contraction with p = 1 2 and q = 1 3 . COUPLED BEST PROXIMITY POINT THEOREM IN METRIC SPACES 205 The following lemma plays an important role in our main results. Lemma 12. Let A and B be nonempty subsets of a metric space X, F : A×A → B and G : B ×B → A and (F,G) be a cyclic contraction. If (x0,y0) ∈ A×A and we define xn+1 = F(xn,yn), xn+2 = G(xn+1,yn+1) yn+1 = F(yn,xn), yn+2 = G(yn+1,xn+1) for all n ∈ N ∪ {0}, then d(xn,xn+1) → d(A,B), d(xn+1,xn+2) → d(A,B), d(yn,yn+1) → d(A,B) and d(yn+1,yn+2) → d(A,B). Proof. For each n ∈ N, we have d(xn,xn+1) = d(F(xn,yn),G(xn−1,yn−1)) ≤ pd(xn,xn−1) + qd(yn,yn−1) + (1 − (p + q))d(A,B) Similarly we have d(yn,yn+1) = d(F(yn,xn),G(yn−1,xn−1)) ≤ pd(yn,yn−1) + qd(xn,xn−1) + (1 − (p + q))d(A,B) Therefore, by letting dn = d(xn,xn+1) + d(yn,yn+1) by adding above inequality we have dn ≤ (p + q)dn−1 + 2(1 − (p + q))d(A,B) Similarly we can show that dn−1 ≤ (p + q)dn−2 + 2(1 − (p + q))d(A,B) Consequently we have d1 ≤ (p + q)d0 + 2(1 − (p + q))d(A,B) If d0 = 0 then (x0,y0) is a coupled best proximity point of F and G. Now let d0 > 0 for each n ≥ m we have d(xn,xm) ≤ d(xn,xn−1) + d(xn−1,xn−2) + ......... + d(xm+1,xm) d(yn,ym) ≤ d(yn,yn−1) + d(yn−1,yn−2) + ......... + d(ym+1,ym) which gives d(xn,xm) + d(yn,ym) ≤ dn−1 + dn−2 + dn−3....... + dm dn ≤ (p + q)nd0 + 2n(1 − (p + q)n)d(A,B) Taking n →∞ we have d(xn,xn+1) + d(yn,yn+1) → d(A,B) 206 GUPTA, RAJPUT AND KAURAV implies that d(xn,xn+1) → d(A,B) d(yn,yn+1) → d(A,B) for all n ∈ N. By similar argument, we also have d(xn+1,xn+2) → d(A,B), d(yn+1,yn+2) → d(A,B). � Lemma 13. Let A and B be nonempty subsets of a metric space X such that (A,B) and (B,A) have a property UC, F : A×A → B and G : B ×B → A and let the ordered pair (F,G) is a cyclic contraction. If (x0,y0) ∈ A×A and define xn+1 = F(xn,yn), xn+2 = G(xn+1,yn+1) yn+1 = F(yn,xn), yn+2 = G(yn+1,xn+1) for all n ∈ N ∪{0}, then for � > 0, there exists a positive integer N0 such that for all m > n ≥ N0 (2.1) pd(xm,xn+1) + qd(ym,yn+1) < d(A,B) + �. Proof. By Lemma 12, we have d(xn,xn+1) → d(A,B), d(xn+1,xn+2) → d(A,B), d(yn,yn+1) → d(A,B), d(yn+1,yn+2) → d(A,B). Since (A,B) has a property UC, we get d(xn,xn+2) → 0. A similar argument shows that d(yn,yn+2) → 0. As (B,A) has a property UC, we also have d(xn+1,xn+3) → 0, d(yn+1,yn+3) → 0. Suppose that (2.1) does not hold. Then there exists �′ > 0 such that for all k ∈ N, there is mk > nk ≥ k satisfying pd(xmk,xnk+1) + qd(ymk,ynk+1) ≥ d(A,B) + � ′. Further, corresponding to nk, we can choose mk in such a way that it is the smallest integer with mk > nk and satisfying above relation. Then pd(xmk−2,xnk+1) + qd(ymk−2,ynk+1) < d(A,B) + � ′. Therefore, we get d(A,B) + �′ ≤ pd(xmk,xnk+1) + qd(ymk,ynk+1) ≤ p[d(xmk,xmk−2) + d(xmk−2,xnk+1)] +q[d(ymk,ymk−2) + d(ymk−2,ynk+1)] < pd(xmk,xmk−2) + qd(ymk,ymk−2)] + d(A,B) + � ′. Letting k →∞, we obtain to see that pd(xmk,xnk+1) + qd(ymk,ynk+1) → d(A,B) + � ′. COUPLED BEST PROXIMITY POINT THEOREM IN METRIC SPACES 207 By using the triangle inequality, we get pd(xmk,xnk+1) + qd(ymk,ynk+1) ≤ p[d(xmk,xmk+2) + d(xmk+2,xnk+3) + d(xnk+3,xnk+1) +q[d(ymk,ymk+2) + d(ymk+2,ynk+3) + d(ynk+3,ynk+1)] = p[d(xmk,xmk+2) + d(G(xmk+1,ymk+1),F(xnk+2,ynk+2)) + d(xnk+3,xnk+1)] +q[d(ymk,ymk+2) + d(G(ymk+1,xmk+1),F(ynk+2,xnk+2)) + d(ynk+3,ynk+1)] ≤ p[d(xmk,xmk+2) + pd(xmk+1,xnk+2) + qd(ymk+1,ynk+2) +(1 − (p + q))d(A,B) + d(xnk+3,xnk+1)] +q[d(ymk,ymk+2) + pd(ymk+1,ynk+2) + qd(xmk+1,xnk+2) + (1 − (p + q))d(A,B) + d(ynk+3,ynk+1) ≤ (p + q)[d(xmk,xmk+2) + d(xnk+3,xnk+1) + d(ymk,ymk+2) + d(ynk+3,ynk+1)] +(p + q)2[d(xmk+1,xnk+2) + d(ymk+1,ynk+2)] + (1 − (p + q) 2)d(A,B). Taking k →∞, we get d(A,B) + �′ ≤ (p + q)2[d(A,B) + �′] + (1 − (p + q)2)d(A,B) = d(A,B) + (p + q)2�′ which contradicts. Therefore, we can conclude that (2.1) holds. � Lemma 14. Let A and B be nonempty subsets of a metric space X, (A,B) and (B,A) satisfy the property UC∗. Let F : A×A → B, G : B ×B → A and (F,G) be a cyclic contraction. If (x0,y0) ∈ A×A and define xn+1 = F(xn,yn) yn+1 = F(yn,xn) and xn+2 = G(xn+1,yn+1) yn+2 = G(yn+1,xn+1) for all n ∈ N∪{0}, then {xn}, {yn}, {xn+1} and {yn+1} are Cauchy sequences. Proof. By Lemma 12, we have d(xn,xn+1) → d(A,B) and d(xn+1,xn+2) → d(A,B). Since (A,B) satisfies property UC, we get d(xn,xn+2) → 0. Similarly, we also have d(xn+1,xn+3) → 0 because (B,A) satisfies property UC. We now show that for every � > 0 there exists N ∈ N such that (2.2) d(xm,xn+1) ≤ d(A,B) + � for all m > n ≥ N Suppose (2.2) not hold, then there exists � > 0 such that for all k ∈ N there exists mk > nk ≥ k such that (2.3) d(xmk,xnk+1) > d(A,B) + �. Further, corresponding to nk, we can choose mk in such a way that it is the smallest integer with mk > nk and satisfying above relation. Now we have d(A,B) + � < d(xmk,xnk+1) ≤ d(xmk,xmk−2) + d(xmk−2,xnk+1) ≤ d(x2mk,x2mk−2) + d(A,B) + �. 208 GUPTA, RAJPUT AND KAURAV Taking k →∞, we have d(x2mk,x2nk+1) → d(A,B) +�. By Lemma 13, there exists N ∈ N such that (2.4) pd(xmk,xnk+1) + qd(ymk,ynk+1) < d(A,B) + � for all m > n ≥ N. By using the triangle inequality, we get d(A,B) + � < d(xmk,xnk+1) ≤ d(xmk,xmk+2) + d(xmk+2,xnk+3) + d(xnk+3,xnk+1) = d(xmk,xmk+2) + d(G(xmk+1,ymk+1),F(xnk+2,ynk+2)) +d(xnk+3,xnk+1) ≤ d(xmk,xmk+2) + [pd(xmk+1,xnk+2) + qd(ymk+1,ynk+2)] +(1 − (p + q))d(A,B) + d(xnk+3,xnk+1) = p[d(F(xmk,ymk ),G(xnk+1,ynk+1))] + q[d(F(ymk,xmk ),G(ynk+1,xnk+1))] +(1 − (p + q))d(A,B) + d(xmk,xmk+2) + d(xnk+3,xnk+1) ≤ p [ p[d(xmk,xnk+1) + qd(ymk,ynk+1) + (1 − (p + q))d(A,B)] ] +q [ [pd(ymk,ynk+1) + qd(xmk,xnk+1) + (1 − (p + q))d(A,B)] ] +(1 − (p + q))d(A,B) + d(xmk,xmk+2) + d(xnk+3,xnk+1) = (p + q)2[d(xmk,xnk+1) + d(ymk,ynk+1)] +(1 − (p + q)2)d(A,B) + d(xmk,xmk+2) + d(xnk+3,xnk+1) < (p + q)2(d(A,B) + �) + (1 − (p + q)2)d(A,B) + d(xmk,xmk+2) + d(xnk+3,xnk+1) = (p + q)2� + d(A,B) + d(xmk,xmk+2) + d(xnk+3,xnk+1). Taking k →∞, we get d(A,B) + � ≤ d(A,B) + (p + q)2� which contradicts. Therefore, condition (2.2) holds. Since (2.2) holds and d(xn,xn+1) → d(A,B), by using property UC∗ of (A,B), we have {xn} is a Cauchy sequence. In similar way, we can prove that {yn},{xn+1} and {yn+1} are Cauchy sequences. � Here we state the main results of this article in the existence and convergence of coupled best proximity points for cyclic contraction pairs on nonempty subsets of metric spaces satisfying the property UC∗. Theorem 15. Let A and B be nonempty closed subsets of a metric space X such that (A,B) and (B,A) have a property UC∗, F : A×A → B and G : B ×B → A and let the ordered pair (F,G) is a cyclic contraction. If (x0,y0) ∈ A×A and define xn+1 = F(xn,yn) yn+1 = F(yn,xn) and xn+2 = G(xn+1,yn+1) yn+2 = G(yn+1,xn+1) COUPLED BEST PROXIMITY POINT THEOREM IN METRIC SPACES 209 for all n ∈ N ∪{0}. Then F has a coupled best proximity point (r,s) ∈ A3 and G has a coupled best proximity point (p′,q′,r′) ∈ B3. Moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. Furthermore, if r = s and r′ = s′, then d(r,r′) + d(s,s′) = 2d(A,B). Proof. By Lemma 12, we get d(xn,xn+1) → d(A,B). Using Lemma 14, we have {xn} and {yn} are Cauchy sequences. Thus, there exists r,s ∈ A such that xn → r, yn → s. We obtain that (2.5) d(A,B) ≤ d(r,xn−1) ≤ d(r,xn) + d(xn,xn−1). Letting n →∞ in (2.5), we have d(r,xn−1) → d(A,B). By a similar argument we also have d(s,yn−1) → d(A,B). It follows that d(xn,F(r,s)) = d(G(xn−1,yn−1),F(r,s)) ≤ pd(xn−1,p) + qd(yn−1,q) + (1 − (p + q))d(A,B). Taking n →∞, we get d(p,F(p,q,r)) = d(A,B). Similarly, we can prove that d(s,F(s,r)) = d(A,B) Therefore, we have (r,s) is a coupled best proximity point of F. In similar way, we can prove that there exists r′,s′ ∈ B such that xn+1 → r′ and yn+1 → s′. Moreover, we have d(r′,G(r′,s′)) = d(A,B), and d(s′,F(s′,r′)) = d(A,B) and so (r′,s′) is a coupled best proximity point of G. Finally, we assume that r = s and r′ = s′ and then we show that d(r,r′) + d(s,s′) = 2d(A,B). For all n ∈ N, we obtain that d(xn,xn+1) = d(G(xn−1,yn−1),F(xn,yn)) ≤ pd(xn−1,xn) + qd(yn−1,yn) + (1 − (p + q))d(A,B). Letting n →∞, we have (2.6) d(r,r′) ≤ pd(r,r′) + d(s,s′) + (1 − (p + q))d(A,B). For all n ∈ N, we have d(yn,yn+1) = d(G(yn−1,xn−1),F(yn,xn)) ≤ pd(yn−1,yn) + qd(xn−1,xn) + (1 − (p + q))d(A,B). Letting n →∞, we have d(s,s′) ≤ pd(s,s′) + d(r,r′) + (1 − (p + q))d(A,B). Similarly we can write, It follows from (2.6)and (2.7) that d(r,r′) + d(s,s′) ≤ pd(r,r′) + qd(s,s′) + 2(1 − (p + q))d(A,B) 210 GUPTA, RAJPUT AND KAURAV which implies that (2.7) d(r,r′) + d(s,s′) ≤ 2d(A,B). Since d(A,B) ≤ d(r,r′) and d(A,B) ≤ d(s,s′), we have (2.8) d(r,r′) + d(s,s′) ≥ 2d(A,B). From (2.7) and (2.8), we get (2.9) d(r,r′) + d(s,s′) = 2d(A,B). This complete the proof. � Note that every pair of nonempty closed subsets A,B of a uniformly convex Banach space X such that A is convex satisfies the property UC. Therefore, we obtain the following corollary. Corollary 16. Let A and B be nonempty closed convex subsets of a uniformly convex Banach space X, F : A×A → B and G : B ×B → A and let the ordered pair (F,G) be a cyclic contraction. Let (x0,y0) ∈ A×A and define xn+1 = F(xn,yn), xn+2 = G(xn+1,yn+1) yn+1 = F(yn,xn), yn+2 = G(yn+1,xn+1) for all n ∈ N ∪{0}. Then F has a coupled best proximity point (r,s) ∈ A × A and G has a coupled best proximity point (r′,s′) ∈ B × B. Moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. Furthermore, if r = s and r′ = s′, then d(r,r′) + d(s,s′) = 2d(A,B). Next, we give some illustrative example of Corollary 16. Example 17. Consider uniformly convex Banach space X = R with the usual norm. Let A = [1, 2] and B = [−1,−2].Thus d(A,B) = 2. Define F : A×A → B and G : B ×B → A by F(x,y) = −2x− 3y − 1 6 and G(x,y) = −2x− 3y + 1 6 . For arbitrary (x,y) ∈ A×A and (u,v) ∈ B ×B and fixed p = 1 3 and q = 1 2 we get d(F(x,y),G(u,v)) = ∣∣∣∣−x−y − 16 − −u−v + 16 ∣∣∣∣ ≤ 2|x−u| + 3|y −v| 6 + 1 3 = 1 3 d(x,u) + 1 2 d(y,v) + (1 − (p + q))d(A,B) This implies that (F,G) is a cyclic contraction with α = 1 2 . Since A and B are closed convex, we have (A,B) and (B,A) satisfy the property UC∗. Therefore, all hypothesis of Corollary 16 hold. So F has a coupled best proximity point and G has a coupled best proximity point. We note that a point (1, 1) ∈ A × A is a unique COUPLED BEST PROXIMITY POINT THEOREM IN METRIC SPACES 211 coupled best proximity point of F and a point (−1,−1, ) ∈ B×B is a unique coupled best proximity point of G. Furthermore, we get d(1,−1) + d(1,−1) = 4 = 2d(A,B). Next, we give the coupled best proximity point result in compact subsets of metric spaces. Theorem 18. Let A and B be nonempty compact subsets of a metric space X, F : A × A → B and G : B × B → A and let the ordered pair (F,G) be a cyclic contraction. Let (x0,y0) ∈ A×A and define xn+1 = F(xn,yn), xn+2 = G(xn+1,yn+1) yn+1 = F(yn,xn), yn+2 = G(yn+1,xn+1) for all n ∈ N ∪{0}. Then F has a coupled best proximity point (r,s) ∈ A × A and G has a coupled best proximity point (r′,s′) ∈ B × B. Moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. Furthermore, if r = s and r′ = s′, then d(p,p′) + d(q,q′) + d(r,r′) = 2d(A,B). Proof. Since x0,y0 ∈ A and xn+1 = F(xn,yn), xn+2 = G(xn+1,yn+1) yn+1 = F(yn,xn), yn+2 = G(yn+1,xn+1) for all n ∈ N∪{0}, we have xn,yn ∈ A and xn+1,yn+1 ∈ A for all n ∈ N∪{0}. As A is compact, the sequences {xn} and {yn} have convergent subsequences {xnk} and {ynk} respectively, such that xnk → r ∈ A, ynk → s ∈ A. Now, we have (2.10) d(A,B) ≤ d(r,xnk−1) ≤ d(r,xnk ) + d(xnk,xnk−1) By Lemma 12, we have d(xnk,xnk−1) → d(A,B). Taking k → ∞ in (2.10), we get d(r,xnk−1) → d(A,B). By a similar argument we observe that d(s,xnk−1) → d(A,B). Note that d(A,B) ≤ d(xnk,F(r,s)) = d(G(xnk−1,ynk−1),F(r,s)) ≤ pd(xnk−1,r) + qd(ynk−1,s) + (1 − (p + q))d(A,B). Taking k →∞, we get d(r,F(r,s)) = d(A,B). Similarly, we can prove that d(s,F(s,r)) = d(A,B). Thus F has a coupled best proximity (r,s) ∈ A × A. In similar way, since B is compact, we can also prove that G has a coupled best proximity point (r′,s′) ∈ B ×B. For d(r,r′) + d(s,s′) = 2d(A,B) similar to the final step of the proof of Theorem 15. This complete the proof. � 212 GUPTA, RAJPUT AND KAURAV 3. Coupled Fixed Point Theorems In this section, we give the new coupled fixed point theorem for a cyclic contrac- tion pair. Theorem 19. Let A and B be nonempty closed subsets of a metric space X, F : A × A → B and G : B × B → A and let the ordered pair (F,G) be a cyclic contraction. Let (x0,y0) ∈ A×A and define xn+1 = F(xn,yn), xn+2 = G(xn+1,yn+1) yn+1 = F(yn,xn), yn+2 = G(yn+1,xn+1) for all n ∈ N∪{0}. If d(A,B) = 0, then F has a coupled fixed point (r,s) ∈ A×A and G has a coupled fixed point (r′,s′) ∈ B×B. Moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. Furthermore, if r = r′ and s = s′, then F and G have a common coupled fixed point in (A∩B)2. Proof. Since d(A,B) = 0, we get (A,B) and (B,A) satisfy the property UC. There- fore, by Theorem 15, claim that F has a coupled best proximity point (r,s) ∈ A×A that is (3.1) d(r,F(r,s)) = d(s,F(s,r)) = d(A,B) and G has a coupled best proximity point (r′,s′) ∈ B ×B that is (3.2) d(r′,G(r′,s′)) = d(s′,G(s′,r′)) = d(A,B). From (3.1) and d(A,B) = 0 , we conclude that r = F(r,s), s = F(s,r). that is (r,s) is a coupled fixed point of F . It follows from (3.2) and d(A,B) = 0, we get r′ = G(r′,s′), and s′ = G(s′,r′) that is (r′,s′) is a coupled fixed point of G. Next, we assume that r = r′ and s = s′ and then we show that F and G have a unique common coupled fixed point in (A∩B)2. From Theorem 15, we get (3.3) d(r,r′) + d(s,s′) = 2d(A,B). Since d(A,B) = 0, we get d(r,r′) + d(s,s′) = 0 which implies that r = r′ and s = s′. Therefore, we conclude that (r,s) ∈ (A∩B)2 is common coupled fixed point of F and G. � Example 20. Consider X = R with the usual metric, A = [−2, 0] and B = [0, 2]. Define F : A×A → B and G : B ×B → A by F(x,y) = − 2x + 3y 6 and G(u,v) = − 2u + 3v 6 . COUPLED BEST PROXIMITY POINT THEOREM IN METRIC SPACES 213 Then d(A,B) = 0 and (F,G) is a cyclic contraction with p = 1 3 and q = 1 2 . Indeed, for arbitrary (x,y) ∈ A×A and (u,v) ∈ B ×B, we have d(F(x,y),G(u,v)) = ∣∣∣∣−2x + 3y6 + 2u + 3v6 ∣∣∣∣ ≤ 1 6 (2 | x−u | +3 | y −v |) ≤ pd(x,u) + qd(y,v) + (1 − (p + q))d(A,B). Therefore, all hypothesis of Theorem 19 hold. So F and G have a common coupled fixed point and this point is (0, 0) ∈ (A∩B)2. If we take A = B in Theorem 19, then we get the following results. Corollary 21. Let A be a nonempty closed subset of a complete metric space X, F : A × A → A and G : A × A → A and let the ordered pair (F,G) be a cyclic contraction. Let (x0,y0) ∈ A×A and define xn+1 = F(xn,yn) yn+1 = F(yn,xn) and xn+2 = G(xn+1,yn+1) yn+2 = G(yn+1,xn+1) for all n ∈ N ∪{0}. Then F has a coupled fixed point (r,s) ∈ A×A and G has a coupled fixed point (r′,s′) ∈ B ×B. Moreover, we have xn → r, yn → s, xn+1 → r′, yn+1 → s′. Furthermore, if r = r′ and s = s′, then F and G have a common coupled fixed point in A×A. We take F = G in Corollary 21, then we get the following results Corollary 22. Let A be nonempty closed subsets of a complete metric space X, F : A×A → A and d(F(x,y),F(u,v)) ≤ pd(x,u) + qd(y,v) for all (x,y), (u,v) ∈ A×A. Then F has a coupled fixed point (r,s) ∈ A×A. 4. Acknowledgements The authors thank the editor and the referees for their useful comments and suggestions for improving the quality of this research. References [1] M. Abbas, W. Sintunavarat, P. 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Obczynski, Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 72, 794–805 (2010) 1Department of Mathematics, Vidhyapeeth Institute of Science & Technology, B- hopal - INDIA 2Department of Mathematics, Govt. P.G. College, Gadarwara, Dist- Narsingpur (M.P.) - INDIA ∗Corresponding author