Cyclic pairs and common best proximity points in uniformly convex Banach spaces

Abstract In this article, we survey the existence, uniqueness and convergence of a common best proximity point for a cyclic pair of mappings, which is equivalent to study of a solution for a nonlinear programming problem in the setting of uniformly convex Banach spaces. Finally, we provide an extension of Edelstein’s fixed point theorem in strictly convex Banach spaces. Examples are given to illustrate our main conclusions.

Notice that best proximity point results have been studied to find necessary conditions such that the minimization problem min x2A[B d.x; T x/; (1) has at least one solution, where T is a cyclic mapping defined on A [ B.
In 2006, a class of cyclic mappings was introduced in [3] as follows. After that in 2009, a generalized class of cyclic contractions was introduced as below. It is remarkable to note that the class of cyclic '-contraction mappings contains the class of cyclic contractions as a subclass by considering '.t / D .1 ˛/t for t 0 and for some˛2 OE0; 1OE. Next theorem guarantees the existence, uniqueness and convergence of a best proximity point for cyclic 'contractions in uniformly convex Banach spaces.
Theorem 0.6 (Theorem 8 of [4]). Let A and B be nonempty subsets of a uniformly convex Banach space X such that A is closed and convex, and let T W A [ B ! A [ B be a cyclic '-contraction mapping. For x 0 2 A, define x nC1 WD T x n for each n 0. Then there exists a unique p 2 A such that x 2n ! p and kp Tpk D dist.A; B/.
In the current paper, we discuss sufficient conditions which ensure the existence and uniqueness of a solution for a nonlinear programming problem. Then we obtain a similar result of Theorem 0.6 for another class of cyclic mappings in uniformly convex Banach spaces. We also study the existence of best proximity pairs for noncyclic contractive mappings in strictly convex Banach spaces and so we present a generalization of Edelstein's fixed point theorem.

Preliminaries
In this section, we recall some notions which will be used in our main discussions. Definition 1.1. A Banach space X is said to be (i) uniformly convex if there exists a strictly increasing function ı W OE0; 2 ! OE0; 1 such that for every x; y; p 2 X; R > 0 and r 2 OE0; 2R, the following implication holds: kx pk Ä R; ky pk Ä R; kx yk r ) k x C y 2 pk Ä .1 ı. r R //RI (ii) strictly convex if for every x; y; p 2 X and R > 0, the following implication holds: ) k x C y 2 pk < R: It is well known that Hilbert spaces and l p spaces .1 < p < 1/ are uniformly convex Banach spaces. Also, the Banach space l 1 with the norm where, k:k 1 and k:k 2 are the norms on l 1 and l 2 , respectively, is strictly convex which is not uniformly convex (see [19] for more details Notice that if .A; B/ is a nonempty, bounded, closed and convex pair in a reflexive Banach space X , then .A 0 ; B 0 / is also nonempty, closed and convex pair in X . We say that the pair .A; B/ is proximinal if A D A 0 and B D B 0 . Also, the metric projection operator P A W X ! 2 A is defined as P A .x/ WD fy 2 A W kx yk D dist.fxg; A/g, where 2 A denotes the set of all subsets of A. It is well known that if A is a nonempty, bounded, closed and convex subset of a uniformly convex Banach space X , then the metric projection P A is single valued from X to A.  d.
It was announced in [21] that every nonempty, bounded, closed and convex pair in a uniformly convex Banach space X has the P-property.
Next two lemmas will be used in the sequel.
. Let A be a nonempty, closed and convex subset and B be a nonempty and closed subset of a uniformly convex Banach space X . Let fx n g and fz n g be sequences in A and let fy n g be a sequence in B such that (i) kz n y n k ! dist.A; B/, (ii) for every " > 0, there exists N 0 2 N so that for all m > n > N 0 , kx m y n k Ä dist.A; B/ C ". Then for every " > 0, there exists N 1 2 N such that kx m z n k Ä " for any m > n > N 1 .

Lemma 1.4 ([3]
). Let A be a nonempty, closed and convex subset and B be a nonempty and closed subset of a uniformly convex Banach space X . Let fx n g and fz n g be sequences in A and let fy n g be a sequence in B satisfying (i) kx n y n k ! dist.A; B/, (ii) kz n y n k ! dist.A; B/. Then kx n z n k ! 0. the optimal solution to the problem of

A nonlinear programming problem: common best proximity point
will be the one for which the value dist.A; B/ is attained. Thereby, a point p 2 A [ B is a common best proximity point for the cyclic pair .T I S / if and only if that is a solution of the minimization problem (2). In this section, we provide some sufficient conditions in order to study the existence of a solution for (2). We begin with the following result. Proof. Choose x 0 2 A. Since S.A/ Â T .A/, there exists x 1 2 A such that S x 0 D T x 1 . Again, by the fact that S.A/ Â T .A/, there exists x 2 2 A such that S x 1 D T x 2 . Continuing this process, we can find a sequence fx n g in A such that S x n D T x nC1 . It follows from the conditions .i i / and .i i i / that that is, fkS x n S S x n k g is a decreasing sequence of nonnegative real numbers and hence it converges. Let r be the limit of kS x n S S x n k . We claim that r D 0. Suppose that r > 0. Then By a similar argument we conclude that Let us prove that for any " > 0 there exists N 0 2 N such that for all m > n > N 0 , kS x m S S x n k < ": Suppose the contrary. Then there exists > 0 such that for all k 2 N there exist m k > n k k for which kS x m k S S x n k k "; kS x m k 1 S S x n k k < ": .kS x m k S S x n k k /; and from the upper semi-continuity of we have " Ä ."/ which is a contradiction. Using Lemma 1.3, fS x n g is a Cauchy sequence and converges to q 2 B. So T x n ! q. By this reality that T j A is continuous, S T x n D T S x n ! T q and so T T x n ! T q. We have kS T x n S x n k Ä .kT T x n T x n k /: Letting lim sup in above relation when n ! 1, then by the fact that is upper semi-continuous from the right, we obtain kT q qk Ä .kT q qk /, which implies that kq T qk D 0. Also, and so q 2 B is a common best proximity point for the cyclic pair .T I S /. Now assume that q 0 2 B is another common best proximity point for the cyclic pair .T I S /. Then kq 0 T q 0 k D dist.A; B/ D kq 0 S q 0 k. By the fact that .A; B/ has the P-property, T q D S q and T q 0 D S q 0 . We have which implies that kS T q S q 0 k D 0. Equivalently, kS T q S qk D 0. Therefore, Again since .A; B/ has the P-property, q D S T q D q 0 and the proof is complete.
The following corollary is the main result of [22]. Remark 2.3. We mention that Theorem 2.1 can be proved in complete metric spaces by using a geometric notion of property UC on closed pairs, which is a property for closed and convex pairs in uniformly convex Banach spaces (see Theorem 3.9 of [22]). Since we will use the other geometric notions of uniformly convex Banach spaces, we prefer to prove Theorem 2.1 in uniformly convex Banach spaces.
The following best proximity point theorem is a different version of Theorem 0.6. Proof. As we mentioned, .A 0 ; B 0 / is nonempty, closed and convex. Note that the mapping S is cyclic on It is clear that P is cyclic on A 0 [ B 0 . We have two following observations. 1 Ä r: p Therefore, all of the assumptions of Theorem 2.1 hold and so, the cyclic pair .T I S / has a unique common best proximity point in B and this point is p D e 2 which is a common fixed point of the mappings T and S in this case.
The following example shows that the uniformly convexity condition of the Banach space X in Theorem 2.4 is sufficient but not necessary.
Example 2.6. Let X be the real Banach space l 2 renormed according to where, kxk 1 denotes the l 1 -norm and kxk 2 the l 2 norm. Assume fe n g is a canonical basis of l 2 . Note that for any x 2 X we have kxk 2 Ä kxk Ä p 2kxk 2 which implies that k:k is equivalent to k:k 2 and so, .X; k:k/ is a reflexive Banach space. Moreover, in view of the fact that l 1 is not strictly convex, X is not uniformly convex. Put For all x 2 A and r 2 .0; 1/ we have kSx Syk D ke 2 1 2 e 1 k D maxf that is, S is cyclic contraction. We note that y is a unique best proximity point of S in B.

A generalization of Edelstein fixed point theorem
We begin the main results of this section by stating the well known Edelstein's fixed point theorem.
The existence of best proximity pairs was first studied by Eldred et al. in [24] using a geometric notion of proximal normal structure on nonempty, weakly compact and convex pairs in strictly convex Banach spaces for noncyclic relatively nonexpansive mappings.

Definition 3.2 ([24]).
A convex pair .A; B/ in a Banach space X is said to have proximal normal structure if for any bounded, closed, convex and proximinal pair .
Since every nonempty, compact and convex pair in a Banach space X has proximal normal structure (Proposition 2.2 of [24]), the following result concludes.

Theorem 3.3 (Theorem 2.2 of [24]
). Let .A; B/ be a nonempty, compact and convex pair in a strictly convex Banach space X and T be a noncyclic relatively nonexpansive mapping, that is, T is noncyclic and kT x T yk Ä kx yk for all .x; y/ 2 A B. Then T has a best proximity pair.
Motivated by Theorem 3.3, we study the convergence results of best proximity pairs for noncyclic contractive mappings in strictly convex Banach spaces. Next lemma describes the relation between noncyclic relatively nonexpansive mappings and noncyclic contractive mappings in uniformly convex Banach spaces. Proof. We only have to prove that kT x T yk D dist.A; B/ whenever kx yk D dist.A; B/. So let kx yk D dist.A; B/. Choose a sequence .fx n g; fy n g/ in A B such that kx n y n k > dist.A; B/ and x n ¤ x; y n ¤ y for any n 2 N. By the compactness condition of the pair .A; B/, we may assume that lim n!1 x n D x 2 A and lim n!1 y n D y 2 B. Then lim n!1 kx n y n k D dist.A; B/. Notice that if kx n 0 yk D dist.A; B/ for some n 0 2 N, then by the strictly convexity of X we must have x n 0 D x which is a contradiction. Thus dist.A; B/ Äk P A .T y/ T y kÄ kT x n T yk <k x n y k : Therefore, k T x n T y k! dist.A; B/. Since k P A .T y/ T y kÄk T x n T y k and T y 2 B 0 , T x n ! P A .T y/: Similarly we can see that T y n ! P B .T x/. In view of the fact that kT x n T y n k ! dist.A; B/, we obtain kP A .T y/ P B .T x/k D dist.A; B/. Again, using the strict convexity of X , T x D P A .T y/; and T y D P B .T x/: Thereby, kT x T yk D dist.A; B/ and the result follows.
Next example shows that the strictly convexity of the Banach space X in Lemma 3.5 is a necessary condition. For .x; y/ 2 A B if kx yk 1 > dist.A; B/, then kT x T yk 1 D k.0; which implies that T is noncyclic contractive. Besides, kT  that is, T is not a noncyclic relatively nonexpansive mapping.
The following theorem is an extension of Edelstein's fixed point theorem in strictly convex Banach spaces.
Theorem 3.7. Let .A; B/ be a nonempty, compact and convex pair in a strictly convex Banach space X and T W A [ B ! A [ B be a noncyclic contractive mapping. Then T has a unique best proximity pair. Moreover, for any .x 0 ; y 0 / 2 A 0 B 0 if we define x nC1 WD T x n and y nC1 WD T y n then the sequence f.x n ; y n /g converges to the best proximity pair of T .
Proof. It follows from Lemma 3.5 that T is a noncyclic relatively nonexpansive mapping. Since the pair .A; B/ is compact and convex, the existence of a best proximity pair for the mapping T is concluded from Theorem 3.3. Suppose .p; q/ 2 A B is a best proximity pair of the mapping T . Then p D Tp; q D T q and kp qk D dist.A; B/. It is worth noticing that the fixed point sets of T in A 0 and B 0 are singleton. Indeed, if p 0 2 A 0 such that p 0 D Tp 0 and p ¤ p 0 then from the strictly convexity of X we have kp 0 qk > dist.A; B/. Therefore, which is impossible. Equivalently, we can see that the fixed point set in B 0 is singleton. This implies that T has a unique best proximity pair in A B. Let x 0 2 A 0 and x nC1 D T x n . Assume that fx n k g is a subsequence of fx n g such that x n k ! z 2 A 0 . Thus d.x n ; P B p/ D d. which is a contradiction and so we must have d.z; P B p/ D dist.A; B/. Then z D p. Since any convergent subsequence of fx n k g converges to p, the sequence itself converges to p. Similarly we can prove the convergence of fy n g to the point q and this competes the proof. Then the existence result of a unique best proximity pair for such mappings was established using a notion of projectional property (Theorem 4.6 of [25]). It is remarkable to note that under the assumptions of Theorem 3.7 the condition .i i / on the noncyclic mapping T holds naturally.
At the end of this section, we study the existence of a unique common best proximity point for a cyclic pair of commuting mappings under a contractive condition. We begin with the following lemma.