International Journal of Analysis and Applications ISSN 2291-8639 Volume 5, Number 1 (2014), 33-44 http://www.etamaths.com EXISTENCE AND CONVERGENCE OF BEST PROXIMITY POINTS FOR SEMI CYCLIC CONTRACTION PAIRS BALWANT SINGH THAKUR∗, AJAY SHARMA Abstract. In this article, we introduce the notion of a semi cyclic ϕ-contraction pair of mappings, which contains semi cyclic contraction pairs as a subclass. Existence and convergence results of best proximity points for semi cyclic ϕ- contraction pair of mappings are obtained. 1. Introduction As it is well known, fixed point theory is an indispensable tool for solving various equations involving self-mappings, defined on subsets of a metric space or a normed linear space. Nevertheless, when the mapping T (say) is a non-self one, then it is possible that the equation Tx = x has no solution and, in this case, we have to focus the study on the problem of finding an element x which is in the closest proximity to Tx in some sense; in such circumstances, it may be speculated to determine an element x for which the ”distance error” d(x,Tx) is minimum. Let A and B be nonempty subsets of a metric space (X,d) and T a mapping from A to B. Since d(x,Tx) is greater than or equals to the distance between A and B for all x in A, a best proximity theorem offers sufficient conditions for the existence of an element x, called a best proximity point of the mapping T, satisfying the condition that d(x,Tx) = dist(A,B). Also, it is interesting to see that best proximity point theorems emerge as a natural generalization of fixed point theorems, because a best proximity point reduces to a fixed point if the mapping under consideration turns out to be a self-mapping. Now, let us consider S and T be given non-self mappings from A to B, where A and B are nonempty subsets of a metric space. As S and T are non-self mappings, the equations Sx = x and Tx = x do not necessarily have a common solution, called a common fixed point of the mappings S and T. Therefore, in such cases of non-existence of a common fixed points, it is attempted to find a point x that is closest to both Sx and Tx in some sense. Common best proximity theorems, explore the existence of such optimal solutions, known as best proximity point. In view of the fact that, for any element x in A, the distance between x and Sx, and the distance between x and Tx are at least the distance between the sets A and B, a common best proximity theorem states that, under certain conditions, there exists a point x satisfying d(x,Sx) = d(x,Tx) = dist(A,B). 2010 Mathematics Subject Classification. 47H10, 54H25. Key words and phrases. Best proximity point, fixed point, semi cyclic ϕ-contraction, metric space, Banach space. 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. 33 34 BALWANT SINGH THAKUR∗, AJAY SHARMA In their elegant paper [1], Kirk et al. introduced the notion of cyclical contrac- tive mapping, and proved fixed point results for this class of mappings, while in [2], Eldred and Veeramani studied existence and convergence results of best proximity points in a more general case. In [3], Thagafi and Shahzad introduced a new class of mappings known as cyclic ϕ-contraction, and proved their convergence and exis- tence results for best proximity points. Recently, Gabeleh and Abkar [4], introduced results on best proximity points for semi-cylic contractive pairs in Banach spaces. For other recent results on this topic, see Chandok and Postolache [5], Shatanawi and Postolache [6]. This paper aims to develop further these results. In this respect, we introduce the new notion of a semi cyclic ϕ-contraction pair of mappings, which contains semi cyclic contraction pairs as a subclass. Existence and convergence results of best proximity points for semi cyclic ϕ-contraction pair of mappings, in the framework of a uniformly convex Banach space [7], are obtained. Our results are extensions of several results as in relevant items from the reference section of this paper, as well as in the literature in general. In particular, our results reduce to those of Gabeleh and Abkar [4]. 2. Preliminaries Let A and B be nonempty subsets of a metric space (X,d) and let T : A∪B → A ∪ B such that T(A) ⊆ B and T(B) ⊆ A. The mapping T is said to be cyclic contraction [1] if for some k ∈ (0, 1) we have d(Tx,Ty) ≤ k d(x,y), x ∈ A, y ∈ B. Kirk et al. [1] proved that if A∩B 6= Ø then the cyclic mapping T has a unique fixed point in A∩B. But what happens when A∩B is not necessarily nonempty. In this situation, a mapping T is said to be cyclic contraction [2] if for some k ∈ (0, 1) we have d(Tx,Ty) ≤ k d(x,y) + (1 −k)dist(A,B), for all x ∈ A and y ∈ B. Eldred and Veeramani [2] proved existence, uniqueness and convergence for best proximity points of cyclic contraction mapping T . Thagafi and Shahzad [3] introduced the following new class of mapping known as cyclic ϕ- contraction: The mapping T is said to be cyclic ϕ-contraction if ϕ: [0, +∞) → [0, +∞) is strictly increasing mapping and d(Tx,Ty) ≤ d(x,y) −ϕ(d(x,y)) + ϕ(dist(A,B)), for all x ∈ A and y ∈ B. We can see from the following example [3] that a cyclic ϕ-contraction mapping need not be a cyclic contraction. Example 2.1. Let X = R with usual metric. For A = B = [0, 1], define T : A∪B → A ∪ B by the formula Tx = x 1+x . If ϕ(t) = t 2 1+t for t ≥ 0, then T is cyclic ϕ- contraction mapping which is not a cyclic contraction. Recently, Gabeleh and Akbar [4] introduced the concept of a semi-cyclic con- traction pair: Definition 2.1. Let S,T be two self mappings on A∪B. The pair (S,T) is called a semi-cyclic contraction if the following conditions hold: EXISTENCE AND CONVERGENCE OF BEST PROXIMITY POINTS 35 (i) S(A) ⊆ B, T(B) ⊆ A ; (ii) ∃α ∈ (0, 1), such that d(Sx,Ty) ≤ αd(x,y) + (1−α)dist(A,B) , x ∈ A, y ∈ B. Clearly when S = T, a semi-cyclic contraction pair reduces to a cyclic contraction mapping, already studied by Eldred and Veeramani [2]. But there exist a semi-cyclic contraction pair which is not cyclic, the following example [4] illustrates it. Example 2.2. Let X = R2 and for all (x,y) ∈ R2 define ‖(x,y)‖ = max{|x|, |y|}. Let A = { (x,y) ∈ R2 : 1 2 ≤ x ≤ 1, y = 0 } , B = { (x,y) ∈ R2 : x = 0 , 1 ≤ y ≤ 2 } . Then A and B are closed and dist(A,B) = 1. Define S,T : A∪B → A∪B by S(x,y) = { (0, 1), (x,y) ∈ A (x,y), (x,y) ∈ B, T(x,y) = { (x,y), (x,y) ∈ A (y/2, 0), (x,y) ∈ B. Here S(A) ⊆ B and T(B) ⊆ A but S(B) * A and T(A) * B, hence neither S nor T is cyclic. On the other hand, if b = (0,y) ∈ B and a = (x, 0) ∈ A, then ‖Tb−Sa‖ = ‖T(0,y) −S(x, 0)‖ = ‖(y/2,−1)‖ = 1. Similarly ‖a− b‖ = ‖(x,−y)‖ = max{x, |−y|} = y. Therefore ‖Tb−Sa‖ = 1 ≤ 1 2 |y| + 1 2 < 1 2 ‖b−a‖ + 1 2 dist(A,B). Hence (S,T) is a semi-cyclic contraction pair. We now introduce the following new class of semi-cyclic contraction pair. Definition 2.2. Let A and B be nonempty subsets of a metric space (X,d) and let S,T : A∪B → A∪B such that S(A) ⊆ B and T(B) ⊆ A. Then (S,T) is said to be a semi-cyclic ϕ-contraction if ϕ: [0, +∞) → [0, +∞) is strictly increasing mapping and (1) d(Sx,Ty) ≤ d(x,y) −ϕ (d(x,y)) + ϕ (dist(A,B)) , for all x ∈ A and y ∈ B. A semi-cyclic contraction pair is semi-cyclic ϕ-contraction pair with ϕ(t) = (1− α)t for t ≥ 0 and 0 < α < 1. We now give an example to illustrate that a semi-cyclic ϕ-contraction pair need not necessary a semi-cyclic contraction pair Example 2.3. Let X = R with the usual metric. Let A = [1, 2], B = [−2,−1], then dist(A,B) = 2. Define T,S : A∪B → A∪B by S(x) =   −1 −x 2 , x ∈ A −1 + x 2 , x ∈ B, T(x) =   1 + x 2 , x ∈ A 1 −x 2 , x ∈ B. Clearly S(A) ⊆ B and T(B) ⊆ A. Take a ∈ A, b ∈ B and ϕ(t) = t 2 1+8t for t ≥ 0, then (S,T) is a semi-cyclic ϕ-contraction pair. On the other hand, if a = 2 ∈ A, b = −2 ∈ B and α ∈ ( 0, 1 2 ) , then d(Tb,Sa) = 3 > α · 4 + (1 −α) · 2 = α dist(A,B) + (1 −α)dist(A,B). Hence (S,T) is not a semi-cyclic contraction pair. 36 BALWANT SINGH THAKUR∗, AJAY SHARMA 3. Main results Consider x0 ∈ A, then Sx0 ∈ B, so there exists y0 ∈ B such that y0 = Sx0. Now Ty0 ∈ A, so there exists x1 ∈ A such that x1 = Ty0. Inductively, we define sequences {xn} and {yn} in A and B, respectively by (2) xn+1 = Tyn, yn = Sxn. For all x ∈ A and y ∈ B, we have dist(A,B) ≤ d(x,y). Since ϕ is a strictly increasing function, we deduce that ϕ(dist(A,B)) ≤ ϕ(d(x,y)). Also (S,T) is semi cyclic ϕ-contraction pair, hence d(Sx,Ty) ≤ d(x,y) −ϕ (d(x,y)) + ϕ (dist(A,B)) ≤ d(x,y). By (2), we have d(xn,Sxn) = d(Tyn−1,Sxn) ≤ d(yn−1,xn) = d(xn−1,Sxn−1), and d(xn+1,yn) = d(Tyn,Sxn) ≤ d(yn,xn) = d(yn,Tyn−1). Also, d(yn+1,Tyn) = d(Sxn+1,Tyn) ≤ d(xn+1,yn) = d(Tyn,Sxn) ≤ d(yn,xn) = d(yn,Tyn−1) . We summarize these results in: Lemma 3.1. Let (X,d) be a metric space and let A,B be nonempty subsets of X. Let S,T : A ∪ B → A ∪ B such that the pair (S,T) is semi-cyclic ϕ−contraction. For x0 ∈ A ∪ B the sequences {xn} and {yn} are generated by (2). Then for all x ∈ A, y ∈ B, and n ≥ 1, we have (i) −ϕ (d(x,y)) + ϕ (dist(A,B)) ≤ 0, (ii) d(Sx,Ty) ≤ d(x,y), (iii) d (xn,Sxn) ≤ d (xn−1,Sxn−1), (iv) d (xn+1,yn) ≤ d (yn,Tyn−1), (v) d (yn+1,Tyn) ≤ d (yn,Tyn−1) . We now state and prove the following result which will be needed in what follows. Theorem 3.1. Let (X,d) be a metric space and let A,B be nonempty subsets of X. Let S,T : A∪B → A∪B such that the pair (S,T) is semi-cyclic ϕ-contraction. For x0 ∈ A∪B the sequences {xn} and {yn} are generated by (2). Then d(xn,Sxn) → dist(A,B) and d(yn,Tyn−1) → dist(A,B). Proof. Let dn = d(xn,Sxn). It follows from Lemma 3.1(iii), that {dn} is decreasing and bounded, so limn→∞dn = t0, for some t0 ≥ dist(A,B). If t0 = dist(A,B) there is nothing to prove, so assume t0 > dist(A,B). Since d(Sx,Ty) ≤ d(x,y) −ϕ(d(x,y)) + ϕ(dist(A,B)), for all x ∈ A, y ∈ B, EXISTENCE AND CONVERGENCE OF BEST PROXIMITY POINTS 37 we have dn+1 = d (xn+1,Sxn+1) = d (Tyn,Sxn+1) ≤ d (yn,xn+1) = d (Sxn,Tyn) ≤ d (xn,yn) −ϕ (d (xn,yn)) + ϕ (dist(A,B)) = d (xn,Sxn) −ϕ (d (xn,Sxn)) + ϕ (dist(A,B)) = dn −ϕ (dn) + ϕ (dist(A,B)) . Hence, ϕ (dist(A,B)) ≤ ϕ(dn) = dn −dn+1 + ϕ (dist(A,B)) . Thus ϕ (dist(A,B)) ≤ lim n→∞ ϕ (dn) ≤ ϕ (dist(A,B)) , which shows that (3) lim n→∞ ϕ (dn) = ϕ(t0) = ϕ (dist(A,B)) . On the other hand, since ϕ is strictly increasing and dn ≥ t0 > dist(A,B), we have lim n→∞ ϕ (dn) ≥ ϕ(t0) > ϕ (dist(A,B)) , which contradicts to (3). Consequently, t0 = dist(A,B). Similarly, using Lemma 3.1, it can be shown that d(yn,Tyn−1) → dist(A,B). � Remark 3.1. Proposition 3.1 of [4] is a special case of Theorem 3.1. Theorem 3.2. Let (X,d) be a metric space and let A,B be nonempty subsets of X. Let S,T : A∪B → A∪B such that the pair (S,T) is semi-cyclic ϕ-contraction. For x0 ∈ A ∪ B the sequences {xn} and {yn} are generated by (2). If both {xn} and {yn} have a convergent subsequence in A and B respectively, then there exists x ∈ A and y ∈ B such that d(x,Sx) = dist(A,B) = d(y,Ty). Proof. Let {ynk} be a subsequence of {yn} such that ynk → y. Since dist(A,B) ≤ d (Tynk,y) ≤ d (ynk,y) + d (ynk,Tynk ) , letting k →∞, by Theorem 3.1, we have d (y,Tynk ) → dist(A,B). Now, for each k ≥ 1 dist(A,B) ≤ d (Ty,ynk ) = d (Ty,Sxnk ) ≤ d (y,xnk ) ≤ d (y,ynk ) + d (ynk,xnk ) = d (y,ynk ) + d (Sxnk,xnk ) , 38 BALWANT SINGH THAKUR∗, AJAY SHARMA i.e., dist(A,B) ≤ d (y,ynk ) + d (Sxnk,xnk ) , letting k →∞ we conclude that d(Ty,y) = dist(A,B). Similarly, it can be proved that d(x,Sx) = dist(A,B). � Remark 3.2. Proposition 3.2 of [4] is a special case of Theorem 3.2. Theorem 3.3. Let (X,d) be a metric space and let A,B be nonempty subsets of X. Let S,T : A∪B → A∪B such that the pair (S,T) is semi-cyclic ϕ-contraction. Then the sequences {xn} and {yn} generated by (2) are bounded. Proof. By Theorem 3.1 we have that d (xn,Sxn) → dist(A,B) as n → ∞, so it is enough to prove that the sequence {Sxn} is bounded. If not, then for each M > 0 there exists n ∈ N such that d(x1,SxN ) > M and d (x1,SxN−1) ≤ M. We obtain M < d (x1,SxN ) = d (Ty0,SxN ) , Furthermore, we have M < d (x1,SxN ) = d (Ty0,SxN ) ≤ d (y0,xN ) = d (Sx0,TyN−1) ≤ d (x0,yN−1) −ϕ (d (x0,yN−1)) + ϕ (dist (A,B)) ≤ d (x0,x1) + d (x1,SxN−1) −ϕ (d (x0,SxN−1)) + ϕ (dist (A,B)) , i.e. M < d (x0,x1) + M −ϕ (d (x0,SxN−1)) + ϕ (dist (A,B)) , so, ϕ (d (x0,SxN−1)) < d (x0,x1) + ϕ (dist (A,B)) . Since, ϕ is unbounded function, we can choose M such that ϕ(M) > d(x0,x1) + ϕ (dist(A,B)) . Now, M < d (x1,SxN ) ≤ d (y0,xN ) = d (Sx0,TyN−1) ≤ d (x0,yN−1) = d (x0,SxN−1) . We deduce that ϕ(M) < ϕ (d (x1,SxN )) ≤ ϕ (d (x0,SxN−1)) < d (x0,x1) + ϕ (dist (A,B)) , a contradiction. Similarly, we can prove boundedness of {yn} in B. � Remark 3.3. Proposition 3.3 of [4] is a special case of Theorem 3.3. In 1936, Clarkson [7] introduced the notion of uniform convexity of norm in a Banach space. EXISTENCE AND CONVERGENCE OF BEST PROXIMITY POINTS 39 Definition 3.1. A Banach space X is said to be uniformly convex if and only if given ε > 0, there exists δ(ε) > 0 such that (4) ‖x‖≤ 1 ‖y‖≤ 1 ‖x−y‖≥ ε   ⇒ ∥∥∥∥x + y2 ∥∥∥∥ ≤ 1 − δ(ε), where δ : [0, 2] → [0, 1] given by δ(ε) = inf { 1 − ∥∥∥∥x + y2 ∥∥∥∥ : ‖x‖≤ 1,‖y‖≤ 1,‖x−y‖≥ ε } . The function δ is known as modulus of convexity of a Banach space X. The implication (4) has following more general form. For x,y,p ∈ X, R > 0 and r ∈ [0, 2R] ‖x−p‖≤ R ‖y −p‖≤ R ‖x−y‖≥ r   ⇒ ∥∥∥∥p− x + y2 ∥∥∥∥ ≤ (1 −δ( rR )) R. Now, define a sequence {zn} in A∪B in the following manner: (5) zn = { Tyk , n = 2k Sxk , n = 2k − 1. Lemma 3.2. Let A and B be nonempty convex subsets of a uniformly convex Banach space X and let S,T : A∪B → A∪B are semi-cyclic ϕ-contraction mappings such that T(A) ⊆ B and S(B) ⊆ A. For x0 ∈ A∪B, the sequences {xn} and {yn} generated by (2). The sequences {zn} is generated by (5), then ‖z2n+2 −z2n‖→ 0 and ‖z2n+3 −z2n+1‖→ 0 as n →∞. Proof. To show ‖z2n+2 −z2n‖ → 0 as n → ∞, assume the contrary. Then there exists ε0 > 0 such that for each k ≥ 1, there exists nk ≥ k so that (6) ‖z2nk+2 −z2nk‖≥ ε0 . Choose ε > 0 so that ( 1 −δ ( ε0 dist(A,B)+ε )) (dist(A,B) + ε) < dist(A,B). By Theorem 3.1, we have ‖z2nk+2 −z2nk+1‖ = ‖Tynk+1 −Sxnk+1‖ ≤‖ynk+1 −xnk+1‖ = ‖Sxnk+1 −xnk+1‖→ dist(A,B), hence, there exists N1 such that (7) ‖z2nk+2 −z2nk+1‖≤ dist(A,B) + ε, ∀nk ≥ N1 . Also, ‖z2nk −z2nk+1‖ = ‖Tynk −Sxnk+1‖ ≤‖ynk −xnk+1‖ = ‖ynk −Tynk‖ = ‖Sxnk −Tynk‖ ≤‖xnk −ynk‖ = ‖Tynk−1 −ynk‖→ dist(A,B), 40 BALWANT SINGH THAKUR∗, AJAY SHARMA so, there exists N2 such that (8) ‖z2nk −z2nk+1‖≤ dist(A,B) + ε, ∀nk ≥ N2 . Let N = max{N1,N2}. It follows from the uniform convexity of X and (6)-(8) that∥∥∥∥z2nk+2 + z2nk2 −z2nk+1 ∥∥∥∥ ≤ ( 1 − δ ( ε0 dist(A,B) + ε )) (dist(A,B) + ε) , ∀nk ≥ N. Since A is convex z2nk+2+z2nk 2 ∈ A, the choice of ε and the fact that δ is strictly increasing imply that∥∥∥∥z2nk+2 + z2nk2 −z2nk+1 ∥∥∥∥ < dist(A,B) , ∀nk ≥ N, a contradiction. By a similar argument, we can show that ‖z2n+3 −z2n+1‖→ 0, as n →∞. � Theorem 3.4. Let A and B be nonempty convex subsets of a uniformly convex Banach space X and let S,T : A ∪ B → A ∪ B are semi-cyclic ϕ−contraction mappings such that T(A) ⊆ B and S(B) ⊆ A. For x0 ∈ A the sequences {zn} is generated by (5). Then, for each ε > 0, there exists a positive integer N0 such that for all m ≥ n ≥ N0, ‖z2m −z2n+1‖ < dist(A,B) + ε. Proof. Suppose the contrary. Then there exists ε0 > 0 such that for each k ≥ 1, there is mk > nk ≥ k satisfying (9) ‖z2mk −z2nk+1‖≥ dist(A,B) + ε0 and (10) ∥∥z2(mk−1) −z2nk+1∥∥ < dist(A,B) + ε0. It follows from (9) and (10), that dist(A,B) + ε0 ≤‖z2mk −z2nk+1‖ ≤ ∥∥z2mk −z2(mk−1)∥∥ + ∥∥z2(mk−1) −z2nk+1∥∥ < ∥∥z2mk −z2(mk−1)∥∥ + dist(A,B) + ε0, i.e. dist(A,B) + ε0 ≤‖z2mk −z2nk+1‖ < ∥∥z2mk −z2(mk−1)∥∥ + dist(A,B) + ε0, letting k →∞, Lemma 3.2 implies that ∥∥z2mk −z2(mk−1)∥∥ → 0, hence (11) lim k→∞ ‖z2mk −z2nk+1‖ = dist(A,B) + ε0. EXISTENCE AND CONVERGENCE OF BEST PROXIMITY POINTS 41 Since (S,T) is semi cyclic ϕ−contraction pair, by Lemma 3.1(i),(ii) we obtain ‖z2mk −z2nk+1‖≤‖z2mk −z2mk+2‖ + ‖z2mk+2 −z2nk+3‖ + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖Tymk+1 −Sxnk+2‖ + ‖z2nk+3 −z2nk+1‖ ≤‖z2mk −z2mk+2‖ + ‖ymk+1 −xnk+2‖ + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖Sxmk+1 −Tynk+1‖ + ‖z2nk+3 −z2nk+1‖ ≤‖z2mk −z2mk+2‖ + ‖xmk+1 −ynk+1‖−ϕ (‖xmk+1 −ynk+1‖) + ϕ (dist(A,B)) + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖Tymk −Sxnk+1‖−ϕ (‖Tymk −Sxnk+1‖) + ϕ (dist(A,B)) + ‖z2nk+3 −z2nk+1‖ = ‖z2mk −z2mk+2‖ + ‖z2mk −z2nk+1‖−ϕ (‖z2mk −z2nk+1‖) + ϕ (dist(A,B)) + ‖z2nk+3 −z2nk+1‖ . Letting k →∞, using Lemma 3.2 and (11), we get dist(A,B) + ε0 ≤ dist(A,B) + ε0 − lim k→∞ ϕ (‖z2mk −z2nk+1‖) + ϕ (dist(A,B)) ≤ dist(A,B) + ε0. Hence, we obtain (12) lim k→∞ ϕ (‖z2mk −z2nk+1‖) = ϕ (dist(A,B)) . Since ϕ is strictly increasing, from (9) and (12), it follows that ϕ (dist(A,B) + ε0) ≤ lim k→∞ ϕ (‖z2mk −z2nk+1‖) = ϕ (dist(A,B)) < ϕ (dist(A,B) + ε0) , a contradiction. � Theorem 3.5. Let A and B be nonempty closed convex subsets of a uniformly convex Banach space X and let S,T : A∪B → A∪B are semi-cyclic ϕ-contraction mappings such that T(A) ⊆ B and S(B) ⊆ A. For x0 ∈ A the sequences {zn} is generated by (5). If dist(A,B) = 0 then (S,T) have a unique common fixed point in A∩B. Proof. Let ε > 0 be given. By Theorem 3.1, we have ‖z2n −z2n+1‖ = ‖Tyn −Sxn+1‖ ≤‖yn −xn+1‖ ≤‖xn −yn‖ = ‖xn −Sxn‖ → dist(A,B) = 0. Hence, for given ε > 0, there exists a positive integer N1 such that ‖z2n −z2n+1‖ < ε, for all n ≥ N1. By Theorem 3.4, there exists a positive integer N2 such that ‖z2m −z2n+1‖ < ε 42 BALWANT SINGH THAKUR∗, AJAY SHARMA for all m > n ≥ N2. Let N = max{N1,N2}. Then, for all m > n ≥ N2, we have ‖z2m −z2n‖≤‖z2m −z2n+1‖ + ‖z2n+1 −z2n‖ < 2ε. Thus {z2n} is a Cauchy sequence in A, since A is closed subset of a complete space X then there exists z ∈ A such that z2n → z as n →∞. Since {z2n−1}⊆ B, and B is closed, it follows that z ∈ B, and finally z ∈ A∩B. It follows from Theorem 3.2 that ‖z −Tz‖ = dist(A,B) = 0 = ‖z −Sz‖ . So, z is a common fixed point of S and T and hence z ∈ F(T ∩S) ⊆ A∩B. We claim that the fixed point z is unique. In fact, if Tw = w = Sw for some w ∈ A∩B, z 6= w, then ‖z −w‖ = ‖Tz −Sw‖≤‖z −w‖−ϕ (‖z −w‖) + ϕ(0), it follows that ϕ(0) < ϕ (‖z −w‖) ≤ ϕ(0), a contradiction. Let rn = ‖zn −z‖ for each n ≥ 0. As the sequence {rn} is bounded and decreas- ing and r2n → 0 as n → ∞, we conclude that rn → 0 as n → ∞, Thus zn → z as n →∞. � Theorem 3.6. Let A and B be nonempty closed convex subsets of a uniformly convex Banach space X and let S,T : A∪B → A∪B are semi-cyclic ϕ-contraction mappings such that T(A) ⊆ B and S(B) ⊆ A. For x0 ∈ A the sequences {zn} is generated by (5). Then {z2n} and {z2n+1} are Cauchy sequences. Proof. If dist(A,B) = 0 then the result follows from Theorem 3.5. So assume that dist(A,B) > 0. To show that {z2n} is a Cauchy sequence in A we assume the contrary. Then there exists ε0 > 0 such that for each k ≥ 1, there exists mk > nk ≥ k so that (13) ‖z2mk −z2nk‖≥ ε0. Choose ε > 0 so that ( 1 −δ ( ε0 dist(A,B)+ε )) (dist(A,B) + ε) < dist(A,B). By Theorem 3.1, we have ‖z2nk −z2nk+1‖ = ‖Tynk −Sxnk+1‖ ≤‖ynk −xnk+1‖ = ‖Sxnk −Tynk‖ ≤‖xnk −ynk‖ = ‖xnk −Sxnk‖→ dist(A,B), hence, there exists a positive integer N1 such that (14) ‖z2nk −z2nk+1‖≤ dist(A,B) + ε, ∀nk ≥ N1. Also, by Theorem 3.4, there exists a positive integer N2 such that (15) ‖z2mk −z2nk+1‖≤ dist(A,B) + ε, ∀mk > nk ≥ N2 . EXISTENCE AND CONVERGENCE OF BEST PROXIMITY POINTS 43 Let N = max{N1,N2}. It follows from the uniform convexity of X and (13)-(15) that∥∥∥∥z2mk + z2nk2 −z2nk+1 ∥∥∥∥ ≤ ( 1 −δ ( ε0 dist(A,B) + ε )) (dist(A,B) + ε) , ∀mk > nk ≥ N. Since A is convex z2mk +z2nk 2 ∈ A, the choice of ε and the fact that δ is strictly increasing imply that∥∥∥∥z2mk + z2nk2 −z2nk+1 ∥∥∥∥ < dist(A,B), ∀mk > nk ≥ N, a contradiction. Thus {z2n} is a Cauchy sequence in A. By a similar argument we can show that {z2n+1} is a Cauchy sequence in B. � Theorem 3.7. Let A and B be nonempty closed convex subsets of a uniformly convex Banach space X and let S,T : A∪B → A∪B are semi-cyclic ϕ-contraction mappings such that T(A) ⊆ B and S(B) ⊆ A. For x0 ∈ A, the sequences {zn} is generated by (5). Then there exists unique x in A and y in B such that z2n → y, z2n+1 → x and ‖x−Sx‖ = dist(A,B) = ‖y −Ty‖ . Proof. Since {z2n} is a Cauchy sequence, we can find a y ∈ B such that {z2n} converges to y. It follows from Theorem 3.2 that ‖y −Ty‖ = dist(A,B). Similarly we can show that the sequence {z2n+1} is convergent to some x ∈ A and ‖x−Sx‖ = dist(A,B). To prove uniqueness, assume that there is another w ∈ A such that ‖w −Sw‖ = dist(A,B). Since dist(A,B) ≤‖TSx−Sx‖≤‖Sx−x‖−ϕ (‖Sx−x‖) + ϕ (dist(A,B)) = dist(A,B) −ϕ (dist(A,B)) + ϕ (dist(A,B)) = dist(A,B), it follows that ‖TSx−Sx‖ = ‖x−Sx‖, this in turn gives TSx = x. Similarly, we can see that TSw = w. Now if w 6= x, then ‖x−Sw‖ > dist(A,B), from which we get ‖Sx−w‖ = ‖Sx−TSw‖≤‖x−Sw‖−ϕ (‖x−Sw‖) + ϕ (dist(A,B)) < ‖x−Sw‖−ϕ (dist(A,B)) + ϕ (dist(A,B)) = ‖x−Sw‖ = ‖TSx−Sw‖ ≤‖Sx−w‖−ϕ(‖Sx−w‖) + ϕ (dist(A,B)) ≤‖Sx−w‖ , which is a contraction. Thus x = w. Similarly, we can see the uniqueness of y ∈ B. This completes the proof. � 4. Conclusion In this paper, the new notion of a semi cyclic ϕ-contraction pair of mappings, which contains semi cyclic contraction pairs as a subclass, is introduced. Existence and convergence results of best proximity points for semi cyclic ϕ-contraction pair 44 BALWANT SINGH THAKUR∗, AJAY SHARMA of mappings are obtained. Our results are extensions of several results as in relevant items from the reference section of this paper, as well as in the literature in general. In particular, our results reduce to those of Gabeleh and Abkar [4]. References [1] Kirk, W.A., Srinivasan, P.S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 4, 79-89 (2003) [2] Eldred, A.A., Veeramani, P.: Existence and convergence of best proximinity points. J. Math. Anal. Appl. 323, 1001-1006 (2006) [3] Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665–3671 (2009) [4] Gabeleh, M., Abkar, A.: Best proximity points for semi-cylic contractive pairs in Banach spaces, Int. Math. 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