Abstract

Best proximity pair results are proved for noncyclic relatively u-continuous condensing mappings. In addition, best proximity points of upper semicontinuous mappings are obtained which are also fixed points of noncyclic relatively u-continuous condensing mappings. It is shown that relative u-continuity of is a necessary condition that cannot be omitted. Some examples are given to support our results.

1. Introduction

The concept of measure of noncompactness was first introduced by Kuratowski [1]. However, the interest in the concept was revived in 1955 when Darbo [2] proved a generalization of Schauder’s fixed point theorem using this concept. Sadovskii [3], in 1967, defined condensing mappings and extended Darbo’s theorem. Since then a lot of work has been done using this concept, and several interesting results have appeared, see, for instance, [4–9].

Let be a nonempty pair in a Banach space (that is, both and are nonempty sets). A mapping is called noncyclic provided and . If there exists which satisfies , , and , then we say that the noncyclic mapping has a best proximity pair. For a multivalued nonself mapping , a point is called a fixed point of if . The necessary condition for the existence of a fixed point for such is . If , then for each . Best proximity point theorems provide sufficient conditions for the existence of at least one solution for the minimization problem, . If , the point is called a best proximity point of . The existence results of best proximity points for multivalued mappings were obtained in [10–14] and [15]. Best proximity point theorems for relatively nonexpansive and relatively u-continuous were established by Elderd et al. in [16, 17] and by Markin and Shahzad in [18]. In recent years, the topics of best proximity points of single-valued and multivalued mappings have attracted the attention of many researchers, see, for example, the work in [6, 7, 19, 20] and the references cited therein. In this paper, we prove best proximity pair theorems for noncyclic relatively u-continuous condensing mappings. In addition, we obtain best proximity points of upper semicontinuous mappings which are fixed points of noncyclic relatively u-continuous condensing mappings. Also, we give examples to support our results and show by giving an example that relative u-continuity of is a necessary condition that cannot be omitted. Our results extend and complement results of [6, 7, 11].

2. Preliminaries

In this section, we present some notions and known results which will be used in the sequel.

Definition 1. Let be a bounded set in a metric space . The Kuratowski noncompactness measure (or simply, measure of noncompactness) is defined as follows:

Theorem 1. Let be a metric space. Then, for any nonempty bounded pair in (that is, both and are nonempty and bounded sets), the following hold:(1) if and only if is relatively compact(2) implies (3), where denotes the closure of (4)(5)For a normed space :(i)(ii)(iii), for any number (iv), where represents the convex hull of

Theorem 2. Let be a decreasing sequence of nonempty closed subsets of a complete metric space If as , then .

For more details about the measure of noncompactness, see [4].

Definition 2. Let be a nonempty pair in Banach space and a mapping. Then, is said to be noncyclic relatively u-continuous. If is noncyclic and for each , there is such thatfor each and .

Definition 3. Let be a nonempty convex pair in Banach space . A mapping is said to be affine if for each with and (respectively, ),

Definition 4. Let be a nonempty pair in Banach space and a multivalued mapping on , then is said to be upper semicontinuous if for each closed subset in , is closed in .

Lemma 1. (see [21]). Let be a nonempty, convex, and compact subset of a Banach space . If can be written as a finite composition of upper semicontinuous multivalued mappings of nonempty, compact, and convex values, then has a fixed point.

Definition 5. Let be a noncyclic relatively u-continuous mapping and be an upper semicontinuous multivalued mapping (here, denotes the collection of all nonempty, convex, and compact subsets of ), then by the commutativity of and , we mean that holds for each .
Given , a nonempty pair in Banach space, its proximal pair is given byMoreover, if is a nonempty, convex, and compact pair in , then is also a nonempty, convex, and compact pair.

Definition 6. Let be a normed space. For a nonempty subset of , the metric projection operator is given byFor a nonempty, convex, and compact subset of a strictly convex Banach space, is a single-valued mapping. Furthermore, for a nonempty, convex, and compact subset of a Banach space , is upper semicontinuous with nonempty, convex, and compact values.

Lemma 2. (see [11]). Let be a nonempty, convex, and compact pair in a strictly convex Banach space . Let be a noncyclic relatively u-continuous and be a mapping given byThen, for each .

Theorem 3. (see [18]). Let be a nonempty, convex, and compact pair in a strictly convex Banach space . If is a noncyclic relatively u-continuous mapping. Then, has best proximity pair.

In [6], Gabeleh and Markin introduced the class of noncyclic condensing operators.

Recall that a nonempty pair in a Banach space is called proximinal if and .

Definition 7. Let be a nonempty convex pair in a strictly convex Banach space . A mapping is called noncyclic condensing operator provided that, for any nonempty, bounded, closed, convex, proximinal, and -invariant pair with , there exists such that

Lemma 3. (see [11]). Let be a nonempty, convex, and compact pair in a strictly convex Banach space . If is a noncyclic relatively u-continuous mapping, then is continuous on and .

3. Main Results

Throughout this paper, we will assume that is a strictly convex Banach space and is the measure of noncompactness on .

Remark 1. Let be condensing in the sense of Definition 7 with . Then, for any bounded subset of , satisfiesTo see this, in (7), set and . Since , then

Theorem 4. Let be a nonempty, convex, and closed pair in such that is bounded and is nonempty. Suppose is a noncyclic relatively u-continuous, affine, and condensing mapping. Then, there exists such that , and . Moreover, if is an upper semicontinuous multivalued mapping, and commute, and for each , there exists such that and .

Proof. We follow [6, 11]. Clearly, is a nonempty, closed, convex, proximinal, and -invariant pair. Let be such that . Suppose is a family of nonempty, closed, convex, proximinal, and -invariant pairs such that , then is nonempty. Set , , and . So, and . Furthermore, and , that is, is noncyclic on . Also, for , , where for all with and , . Since is proximinal, there is such that , for each . Set . Then, . Moreover,So, one can conclude that . Similarly, , and hence, , that is, and . Notice thatBut , so . We conclude that is a nonempty, compact, and convex pair with . By Theorem 3, there exists such that , and .
Now, let , , and . By the above part, the pair is nonempty. Also, it is a convex pair. Indeed, for , with and (respectively, ):and by convexity of (respectively, ), we conclude that (respectively, ). Furthermore, since is condensing,which implies that the pair is compact.
For and , we havethat is, is invariant under . So, by the invariance of under , is invariant under . So, in view of Remark 3.1, Darbo’s fixed point theorem guarantees the existence of a fixed point for the continuous mapping : . Thus, , for . Define by , for each . Then, is an upper semicontinuous multivalued mapping with nonempty, compact, and convex values. Moreover, : is well-defined. Indeed, for , there is such that . So,By relative u-continuity of , we conclude that . Thus, and , by (15), . Then, . Consider , by Lemma 1, there is such that , that is, and . So, there is such that . We conclude that . But since , there is such that . Thus,Hence, .

Example 1. Consider the Hilbert space over with the basis (the canonical basis) and letThen, be a nonempty, convex, and closed pair of such that is bounded. Furthermore, andDefine the mapping by , for each Then, is a noncyclic relatively u-continuous, affine, and condensing mapping. Now, define by ; then, is an upper semicontinuous multivalued mapping, and commute, and for each , . For , we have and .

Example 2. Consider the Hilbert space over with the basis and letThen, be a nonempty, convex, and closed pair of such that is bounded with andDefine the mapping by for each . Then, is a noncyclic relatively u-continuous, affine, and condensing mapping. Furthermore, for , we have , , and . Now, let given by , then is an upper semicontinuous multivalued mapping, and commute, and for each , . For , we have and .