Strong and weak convergence of Ishikawa iterations for best proximity pairs

Abstract Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.


Introduction
Let X be a normed linear space. A self-mapping T : X → X is said to be nonexpansive provided that Tx−Ty ≤ x − y . It is well known that if A is a nonempty, compact and convex subset of a Banach space X, then every nonexpansive mapping of A into itself has a xed point.
In 1965, Kirk proved that if A is a nonempty, weakly compact and convex subset of a Banach space X with a geometric property, called normal structure, then every nonexpansive self-mapping T : A → A has a xed point (Kirk's xed point theorem [1]). Now, suppose that (A, B) is a nonempty pair of subsets of a normed linear space X. A mapping T : A∪B → A ∪ B is said to be noncyclic relatively nonexpansive if T is noncyclic, that is, T(A) ⊆ A, T(B) ⊆ B, and Tx − Ty ≤ x − y for all (x, y) ∈ A × B. Under this weaker assumption over T w.r.t. nonexpansiveness, the existence of the so-called best proximity pair, that is, a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B), was rst studied in [2] as below.

Theorem 1.1. (See Theorem 2.2 of [2]) Let (A, B) be a nonempty, bounded, closed and convex pair of subsets of a uniformly convex Banach space X, and suppose T : A ∪ B → A ∪ B is a noncyclic relatively nonexpansive mapping. Then T has a best proximity pair.
An interesting observation about Theorem 1.1 is that the mapping T may not be continuous whereas the existence of two xed points of T which estimates the distance between two sets A and B is guaranteed (see also [3,4] for more information).
In addition to the existence result of best proximity pairs for noncyclic relatively nonexpansive mappings, the convergence of Krasnoselskii's iteration process for such mappings was discussed as follows. The current paper is organized as follows: in Section 2, we recall some notions and notations which will be used in our coming discussion. We also introduce a geometric concept of proximal Opial's condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. In Section 3, we improve Theorem 1.2 and prove strong and weak convergence theorems for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. Finally, in Section 4, we establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces. We also present some appropriate examples to illustrate our main conclusions.

Preliminaries
To describe our results, we need some de nitions and notations.
(ii) strictly convex if the following implication holds x, y, p ∈ X and R > : It is well known that Hilbert spaces and l p spaces ( < p < ∞) are uniformly convex Banach spaces. Also, the Banach space l with the norm where, . and . are the norms on l and l , respectively is strictly convex but not uniformly convex (see [5] for more details).
The following lemma gives a suitable property for characterization of uniformly convex Banach spaces.
for all λ ∈ [ , ] and all x, y ∈ X such that x ≤ r and y ≤ r.
We also refer to the following auxiliary lemma. Another appropriate property of uniformly convex Banach spaces will be explained in the next lemma. The closed and convex hull of a set A will be denoted by conv(A). Also, B(p, r) will denote the closed ball in the space X centered at p ∈ X with radius r > . Given (A, B) a pair of nonempty subsets of a Banach space, if x − y = dist(A, B), x ∈ A, y ∈ B then y (x) is called a proximal point of the point x (y). Also, the proximal pair of (A, B) will be denoted by (A , B ) which is given by Proximal pairs may be empty but, in particular, if (A, B) is a nonempty, bounded, closed and convex pair in a re exive Banach space X, then (A , B ) is also nonempty, closed and convex. For a noncyclic mapping T : A ∪ B → A ∪ B the set of all best proximity pairs of T will be denoted by Prox A×B (T).
Suppose A is a nonempty and convex subset of X. A self-mapping T : A → A is said to be a ne if for any x, y ∈ A and < λ < . Also, a mapping T : A ∪ B → A ∪ B is called a ne provided that both T| A and T| B are a ne, where (A, B) is a convex pair in X.

De nition 2.5. A pair (A, B) in a Banach space is said to be proximinal if A = A and B = B .
For a nonempty subset A of X a metric projection operator P A : X → A is de ned as where A denotes the set of all subsets of A. It is well known that if A is a nonempty, closed and convex subset of a re exive and strictly convex Banach space X, then the metric projection P A is single valued from X to A. Next result will be used in the sequel.
Then the following statements hold:

De nition 2.7. [9] Let (A, B) be a pair of nonempty subsets of a metric space (X, d) with A ≠ ∅. The pair (A, B) is said to have the P-property if and only if
Next theorem plays an important role in our next conclusions.
Lemma 2.8. [10,11] Every, nonempty, bounded, closed and convex pair in a uniformly convex Banach space X has the P-property.
We nish this section by introducing a notion of proximal Opial's condition. We recall that a Banach space X is said to satisfy Opial's condition ( [12] It is well known that nite dimensional Banach spaces, Hilbert spaces and l p spaces ( < p < ∞) satisfy Opial's condition.

De nition 2.9. Let (A, B) be a nonempty, closed, convex and proximinal pair in a strictly convex Banach space X. Then (A, B) is said to satisfy proximal Opial's condition if for every sequence {xn} in A(respectively in B) with xn
We mention that the condition of proximinality of the closed and convex pair (A, B) in the above de nition is essential even if the considered Banach space X is Hilbert. Let us illustrate this fact with the following example.

Convergence results
We begin our results of this section by improving Theorem 1.2 as follows. Proof. It follows from the proof of Theorem 1.2 that xn − Txn → and that the sequence {xn} converges to a xed point of T in A, say p ∈ A. Besides, Equivalently, and by induction we obtain xn − yn = dist(A, B) for all n ∈ N. Also for each n, It now follows from the lower semi-continuity of norm that By Lemma 2.8 we conclude that xn − p = yn − q for all n ∈ N. Thereby, yn → q. We nish the proof by showing that q is a xed point of T in B. Indeed, Again by the fact that (A, B) has the P-property, we conclude that q = Tq.
In what follows we prove the strong and weak convergence results of Ishikawa iteration scheme for noncyclic mappings. For more information regarding Ishikawa iteration scheme we refer to [14,15].
Here, we establish the rst main result of this section. From the above, we can deduce the following inequalities: We consider two following cases.

CASE I:
Suppose αn and βn satisfy (C). From (2) we conclude that Letting m → ∞, we obtain ∞ n= αn( − αn)φ( Tyn − xn ) < ∞. In view of the fact that αn( − αn) ≥ ϵ, it results that limn φ( Tyn − xn ) = . Therefore, Tyn − xn → . Since P is a ne and isometry and PT = TP on A ∪ B ,  A, B). We now have Continuing this process and by induction we obtain xn − x n = dist(A, B) for each n ∈ N. By a similar argument of aforesaid discussion we conclude that On the other hand, which deduces that (u, v) ∈ Prox A×B (T). Due to the fact that (A , B ) has the proximal Opial's condition, we must have Tx = x. Equivalently, we can prove that Ty = y. Moreover, that is, (x, y) ∈ Prox A×B (T).
Next theorem guarantees the weak convergence of the iterative sequence de ned in (1) to nding best proximity pairs of noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces.

Noncyclic contraction type mappings
We begin our main results of this section with the following well known xed point theorem which is the main idea for coming discussions. In fact, the condition on T ensures that {T n (x)} is a Cauchy sequence for each x ∈ X, and continuity does the rest.
Here, we state the following existence theorems of best proximity pairs for two di erent classes of noncyclic mappings. In this section, we establish a best proximity pair result under some su cient conditions. For this purpose, we introduce a new class of noncyclic contraction mappings as below.
It is clear that every noncyclic contraction mapping is noncyclic contraction type mapping in the sense of De nition 4.5. The next example shows that the reverse implication does not hold.
Example 4.1. Let X be the real Banach space l renormed according to where, x ∞ denotes the l ∞ -norm and x the l norm. Let {en} be the canonical basis of l . Note that this norm is equivalent to · and so, (X, · ) is a re exive Banach space. Consider Then T is noncyclic and for each α ∈ [ , ) and y ∈ B we have T y = v and so, that is, T is noncyclic contraction type mapping. We also note that if y = u, then Therefore, T is not a generalized noncyclic relatively nonexpansive mapping.
Next lemma will be used in our main result of this section.
The following theorem guarantees the existence of best proximity pairs for noncyclic contraction type mappings which are a ne. Proof. Assume that F denotes the collection of all nonempty, closed and convex pair (E, F) ⊆ (A, B) such that T is noncyclic on E ∪ F. Then (A, B) ∈ F ≠ ∅. By using Zorn's Lemma we get an element say (K , K ) which is minimal with respect to being nonempty, closed, convex and T-invariant. Note that (conv(T(K )), conv(T(K ))) ⊆ (K , K ) is a nonempty, bounded, closed and convex pair in X. Further, T(conv(T(K ))) ⊆ T(K ) ⊆ conv(T(K )), and also, T(conv(T(K ))) ⊆ conv(T(K )), that is, T is noncyclic on conv(T(K )) ∪ conv(T(K )). It now follows from the minimality of (K , K ) that conv(T(K )) = K , conv(T(K )) = K .
Since T(K ) ⊆ K , we have T (K ) ⊆ T(K ) and so, conv(T (K )) ⊆ conv(T(K )) = K . Then Then there exists an element u ∈ conv(T(K )) such that u = Tu . By the fact that u ∈ conv(T(K )), we have u = n i= α i Tx i , where ≤ α i ≤ and n i= α i = and x i ∈ K for i = , , ..., n. Since T| A is a ne, we obtain u = Tu = T( n i= α i Tx i ) = n i= α i T(Tx i ) = n i= α i T x i , which implies that u ∈ conv(T (K )). Therefore, T[conv(T (K ))] ⊆ conv(T (K )). By the similar manner, we have T[conv(T (K ))] ⊆ conv(T (K )), that is, T is noncyclic on conv(T (K ))∪ conv(T (K )). Minimality of (K , K ) implies that conv(T (K )) = K and conv(T (K )) = K . Let x ∈ K . Then for each y ∈ K we have Therefore, δ(K , K ) = dist(A, B)(= dist(K , K )).
Strictly convexity of the Banach space X yields that both K and K must be singleton and so Prox A×B (T) ≠

∅.
Next xed point result is concluded from Theorem 4.7. That is, T is a noncyclic contraction type mapping. Now, by Corollary 4.8, T has a xed point. Note that existence of xed point for T cannot be obtained from Theorem 4.1. Indeed, if z := ze + ze , then