Let A and B be two nonempty subsets of a Banach space X. A mapping T : is said to be cyclic relatively nonexpansive if T(A) and T(B) and for all ( ) . In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings. 1. Introduction Let be a Banach space and . Recall that a mapping is nonexpansive provided that for all . A closed convex subset of a Banach space has normal structure in the sense of Brodskii and Milman [1] if for each bounded, closed, and convex subset of which contains more than one point, there exists a point which is not a diametral point; that is, where is the diameter of . The set is said to have fixed point property (FPP) if every nonexpansive mapping has a fixed point. In 1965, Kirk proved the following famous fixed theorem. Theorem 1 (see [2]). Let be a nonempty, weakly compact, and convex subset of a Banach space . If has normal structure, then has the FPP. We mention that every compact and convex subset of a Banach space has normal structure (see [3]) and so has the FPP. Moreover, every bounded, closed, and convex subset of a uniformly convex Banach space has normal structure (see [4]) and then by Theorem 1 has the FPP. It is interesting to note that there exists a weakly compact and convex subset of which does not have the fixed point property (see [5] for more information). In particular, cannot have normal structure. In the current paper, we introduce a geometric notion of seminormal structure on a nonempty, closed, and convex pair of subsets of a Banach space and present a new fixed point theorem which is an extension of Kirk’s fixed point theorem. We also study the stability of fixed points by using this geometric property. Finally, we establish a best proximity point theorem for a new class of mappings. 2. Preliminaries In [6], Kirk et al. obtained an interesting extension of Banach contraction principle as follows. Theorem 2 (see [6]). Let and be nonempty closed subsets of a complete metric space . Suppose that is a cyclic mapping, that is, and . If for some and for all , , has a unique fixed point in . An interesting feature
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