Existence Results of best Proximity Pairs for a Certain Class of Noncyclic Mappings in Nonreflexive Banach Spaces Polynomials 

Introduction

Let be a nonempty subset of a normed linear space . A self-mapping is said to be nonexpansive provided that for all . In 1965, Browder showed that every nonexpansive self-mapping defined on a nonempty, bounded, closed and convex subset of a uniformly convex Banach space , has a fixed point. In the same year, Kirk generalized this existence result by using a geometric notion of normal structure. We recall that a nonempty and convex subset of a Banach space is said to have normal structure if for any nonempty, bounded, closed and convex subset of with , there exists a point for which . The well-known Kirk’s fixed point theorem states that if is a nonempty, weakly compact and convex subset of a Banach space which has the normal structure and is a nonexpansive mapping, then has at least one fixed point. In view of the fact that every nonempty, bounded, closed and convex subset of a uniformly convex Banach space has the normal structure, the Browder’ fixed point result is an especial case of Kirk’s theorem.

Material and methods

Let be a nonempty pair of subsets of a normed linear space . is said to be a noncyclic mapping if . Also the noncyclic mapping is called relatively nonexpansive whenever for any . Clearly, if , then we get the class of nonexpppansive self-mappings. Moreover, we note the noncyclic relatively nonexpansive mapping may not be continuous, necessarily. For the noncyclic mapping , a point is called a best proximity pair provided that



In the other words, the point is a best proximity pair for if and are two fixed points of which estimates the distance between the sets and .

The first existence result about such points which is an interesting extension of Browder’s fixed point theorem states that if is a nonempty, bounded, closed and convex pair in a uniformly convex Banach space and if is a noncyclic relatively nonexpansive mapping, then has a best proximity pair. Furthermore, a real generalization of Kirk’s fixed point result for noncyclic relatively nonexpansive mappings was proved by using a geometric concept of proximal normal structure, defined on a nonempty and convex pair in a considered Banach space.

Results and discussion

Let be a nonempty and convex pair of subsets of a normed linear space and be a noncyclic mapping. The main purpose of this article is to study of the existence of best proximity pairs for another class of noncyclic mappings, called noncyclic strongly relatively C-nonexpansive. To this end, we use a new geometric notion entitled -uniformly semi-normal structure defined on in a Banach space which is not reflexive, necessarily. To illustrate this geometric property, we show that every nonempty, bounded, closed and convex pair in uniformly convex Banach spaces has -uniformly semi-normal structure under some sufficient conditions.

Conclusion

The following conclusions were drawn from this research.

We introduce a geometric notion of -uniformly semi-normal structure and prove that: Let be a nonempty, bounded, closed and convex pair in a strictly convex Banach space such that is nonempty and . Let be a noncyclic strongly relatively C-nonexpansive mapping. If has the -uniformly semi-normal structure, then has a best proximity pair.

In the setting of uniformly convex in every direction Banach space , we also prove that: Let be a nonempty, weakly compact and convex pair in and be a noncyclic mapping such that for all with . If



where is a projection mapping defined on then has -semi-normal structure.

We present some examples showing the useability of our main conclusions.

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