Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

M De La Sen

doi:10.1155/2013/310106

Abstract

This paper is devoted to investigating the limit properties of distances and the existence and uniqueness of fixed points, best proximity points and existence, and uniqueness of limit cycles, to which the iterated sequences converge, of single-valued, and so-called, contractive precyclic self-mappings which are proposed in this paper. Such self-mappings are defined on the union of a finite set of subsets of uniformly convex Banach spaces under generalized contractive conditions. Each point of a subset is mapped either in some point of the same subset or in a point of the adjacent subset. In the general case, the contractive condition of contractive precyclic self-mappings is admitted to be point dependent and it is only formulated on a complete disposal, rather than on each individual subset, while it involves a condition on the number of iterations allowed within each individual subset before switching to its adjacent one. It is also allowed that the distances in-between adjacent subsets can be mutually distinct including the case of potential pairwise intersection for only some of the pairs of adjacent subsets.

Highlights

A relevant attention has been recently devoted to the research of existence and uniqueness of fixed points of self-mappings as well as to the investigation of associate relevant properties like, for instance, stability of the iterations [1,2,3]
Relevant properties on the existence and uniqueness of fixed points and best proximity points for multivalued cyclic self-mappings have been studied under general contractive-type conditions based on the Hausdorff metric between subsets of a metric space
For instance, [4, 7,8,9], including as a relevant particular case the contractive condition on contractive single-valued selfmappings, [1, 4,5,6,7,8,9,10], as well as concerns related to their extension to cyclic self-mappings

Summary

Introduction

A relevant attention has been recently devoted to the research of existence and uniqueness of fixed points of self-mappings as well as to the investigation of associate relevant properties like, for instance, stability of the iterations [1,2,3]. Precyclic contractive self-mappings allow the generation of iterated sequences under constraints of the form Tj(Ai) ⊆ Ai ∪ Ai+1 for j = j(i, x) being less than a prescribed positive integer number j = j (i, x); for all x ∈ ⋃i∈p Ai, for all i ∈ p which can be set and point dependent, while Tj(Ai) ⊆ Ai+1; for all i ∈ p. In this case, depending on conditions on the parameterization sequence {an}, the convergence of the solution in one of the subsets could be possible, even if ε ≠ 0, when j∗ is infinity in at least one of the sets A1 and A2 for some subset of values of the solution so that the solution enters such a set and remains in it for all later iterations. A1 and A2 so that the 2-precyclic self-mapping is a 2cyclic one

Convergence of Iterated Sequences to Fixed Points

Convergence to Best Proximity Points