Abstract

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair in a reflexive Banach space satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping on satisfying and , to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping on satisfying and , Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of relative to . Some illustrative examples are provided to support our results.

1. Introduction and Preliminaries

Let and be nonempty subsets of a Banach space . A mapping satisfying (respectively, ) for all is called relatively nonexpansive mapping (respectively, relatively isometry mapping). For more results on relatively nonexpansive (respectively, relatively isometry) mappings, readers can see the research papers in [1, 2] and references therein.

For any two nonempty bounded subsets and of a Banach space , we denote some notations as follows: where . Here, it is to note that if , then .

Let be a nonempty convex subset of a normed linear space , and let be a mapping with Fix, where Fix. A set is said to have approximate fixed point property (AFPP) if the nonexpansive mapping has an approximate fixed point sequence, that is, a sequence in satisfies .

Definition 1 [3]. A normed space is said to be uniformly convex (or uniformly rotund) if and only if for every there exists such that whenever implies , , and .

Definition 2 [4]. A nonempty convex subset of a Banach space is said to have normal structure if for any nonempty convex closed bounded subset of with there exists such that , where .

Eldred et al. [1] introduced the notions of proximal pair and proximal normal structure.

Definition 3 [1]. A nonempty pairof a normed linear spaceis known as a proximal pair if, for every,there existssuch that

A nonempty convex pair in a Banach space is said to have proximal normal structure if is a closed bounded convex pair for which and , there exists such that

Here, it is to note that every nonempty convex weakly compact pair in a uniformly convex Banach space has proximal normal structure. If , then proximal normal structure becomes normal structure of Definition 2.

Definition 4 [5]. A proximal pairin a Banach spaceis known as a proximal parallel pair if(1)for every element in , there exists a unique element in such that and(2), where is a unique element in

Further, Espinola [5] proved the following lemma.

Lemma 5 [5]. Ifis a nonempty proximal pair in a strictly convex Banach space, then proximal pairis a proximal parallel pair.

Definition 6 [6]. The nonempty proximal parallel pairin a Banach spaceis said to have rectangle property if for any, where and .

Eldred et al. [1] proved the following result.

Theorem 7 [1]. Letbe a nonempty closed bounded convex pair in a uniformly convex Banach space. Letbe a relatively nonexpansive self-mapping onsatisfyingand. Letbe an initial point, and define a sequence (Krasnoselskii’s iteration formula) by, . Then,. Ifis a subset of some compact set in, then the limit point ofunder the norm topology is the best proximity point of.

It is ascertained that the geometric property, that is, proximal normal structure, was used in the following result of Eldred et al. [1].

Theorem 8 [1]. Letbe a strictly convex Banach space, and letbe a nonempty weakly compact convex pair having proximal normal structure. Letbe a self-mapping onsatisfyingthen has fixed points , , and .

Definition 9 [7]. Letbe a nonempty convex subset of a real Hilbert space, and letbe a self-mapping on. Letbe an initial point, andis a sequence defined bywhere , .

The iterative sequence defined in (6) is called Ishikawa’s iteration. If , then Ishikawa’s iteration sequence reduces to Mann’s iteration sequence. Eldred and Praveen [8] generalized and extended Theorem 7 of Eldred et al. [1] by using Mann’s iteration method.

Definition 10 [9]. Letbe a nonempty convex subset of a real Hilbert space, and letbe a self-mapping on. Fix. Letbe an initial point, and a sequenceis defined by

The iterative sequence defined in (7) is called Halpern’s iteration.

The following interesting result will be used extensively in the sequel.

Proposition 11 [10]. Letbe a uniformly convex normed linear space,, and. For any, ifare such that, , then there existssuch that

Almezel et al. [11] modified the result of Xu [12] in the following way.

Lemma 12 [11, 12]. Letbe a sequence of nonnegative real numbers satisfyingwhere and satisfy the following conditions. (1) is a sequence in , where (2) is a sequence in ; either or Then, as .

Let and be nonempty bounded subsets of a Banach space . The number is the Chebyshev radius of relative to and is the set of all Chebyshev centers of relative to . Since the function is convex and continuous on , is lower semicontinuous with respect to the weak topology. Consequently, if is a nonempty weakly compact convex set, then is a nonempty convex weakly compact subset of . Rajesh and Veeramani [2] proved the following proposition.

Proposition 13 [2]. Letbe a nonempty convex weakly compact proximal parallel pair in a Banach space. Let the nonempty pairhave the rectangle property. Then,for, andfor. Moreover,.

Definition 14 [13]. Letbe a Banach space. We say thatsatisfies Opial’s condition if for any sequenceinconverges weakly to some, thenfor all. If a reflexive Banach spacesatisfies Opial’s condition, thenhas a normal structure.

Proposition 15 (demiclosed principle [13]). Letbe a Banach space, and letbe a nonempty weakly compact subset of. Also, letbe a nonexpansive self-mapping onwith Fix. If a sequenceinconverges weakly toand a sequenceconverges strongly to, then. Moreover, if, thenis demiclosed at zero.

We need the following result of Dutta and Veeramani [14] to prove Proposition 17.

Theorem 16 [14]. If a nonempty convex pairin a Banach spacedoes not have a proximal normal structure, then there exist sequences, such thatfor all, andor where .

2. Opial’s Condition and Ishikawa’s Iteration for Relatively Nonexpansive Mappings

The geometrical property, that is, the proximal normal structure, is the sufficient condition for the existence of the best proximity [1]. For details about the best proximity point, one can see research papers in [1, 2, 5, 15–19]. We now prove the following result, which shows that the above condition can be dropped if a reflexive Banach space satisfies Opial’s condition.

Proposition 17. Every closed bounded convex pairin reflexive Banach spacesatisfying Opial’s condition has proximal normal structure.

Proof. Suppose the pair does not have a proximal normal structure. Then, by Theorem 16, there exist sequences such that for all , for some , and . Let the sequence converges weakly to . Therefore, .
Suppose , then , and the same holds as . Therefore, when taking , we get , and , which is a contradiction, hence the result.

After analyzing the theorems, definitions, lemma, and propositions mentioned above, we have some impressive new results herewith.

Theorem 18. Letbe a nonempty convex closed bounded proximal pair of, a uniformly convex Banach space. Letbe a relatively nonexpansive self-mapping onsatisfyingand. Letbe an initial point, and a sequenceis defined asThen, . If is a subset of a compact set, then the limit point of under the norm topology is the best proximity point of .

Proof. If , then it is not necessary to discuss. Suppose , then by applying the result of Theorem 8, there exists such that . Since is a nonincreasing sequence, there exists such that .
Suppose , then there exists a subsequence of such that Let and such that and . Since is a uniformly convex Banach space, the modulus of convexity function is strictly increasing and continuous. Hence, . So, we can choose a small positive number such that , where .
Let and for some . Now, where . Further where
By choosing as small as we wish, we get which is a contradiction. Hence, and .
If is compact, then the sequence has a subsequence such that . Since is a proximal pair, there exists such that .
Now, we have , and is a nonincreasing sequence; it implies that . This shows that . By strict convexity of the norm, and as give . Since , it follows that .

We obtain the following result from Theorem 18 by taking for .

Corollary 19 [8]. Letbe a nonempty convex closed bounded proximal pair of, a uniformly convex Banach space, and letbe a relatively nonexpansive self-mapping onsatisfyingand. Letbe an initial point, and a sequenceis defined as