The Theory of Reich's Fixed Point Theorem for Multivalued Operators

Tania Lazăr,Ghiocel Moţ,Gabriela Petruşel,Silviu Szentesi

doi:10.1155/2010/178421

Abstract

AbstractThe purpose of this paper is to present a theory of Reich's fixed point theorem for multivalued operators in terms of fixed points, strict fixed points, multivalued weakly Picard operators, multivalued Picard operators, data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, well-posedness of the fixed point problem, and the generated fractal operator.

Highlights

Let X, d be a metric space and consider the following family of subsets Pcl X : {Y ⊆ X | Y is nonempty and closed}
The purpose of this paper is to present a theory of Reich’s fixed point theorem for multivalued operators in terms of fixed points, strict fixed points, multivalued weakly Picard operators, multivalued Picard operators, data dependence of the fixed point set, sequence of multivalued operators and fixed points, Ulam-Hyers stability of a multivalued fixed point equation, wellposedness of the fixed point problem, and the generated fractal operator
Reich proved that any Reich-type multivalued a, b, c -contraction on a complete metric space has at least one fixed point see 3

Summary

Introduction

Let X, d be a metric space and consider the following family of subsets Pcl X : {Y ⊆ X | Y is nonempty and closed}. If X, d is a metric space, a multivalued operator T : X → Pcl X is said to be a Reich-type multivalued a, b, c -contraction if and only if there exist a, b, c ∈ R with a b c < 1 such that. Reich proved that any Reich-type multivalued a, b, c -contraction on a complete metric space has at least one fixed point see 3. In a recent paper Petrusel and Rus introduced the concept of “theory of a metric fixed point theorem” and used this theory for the case of multivalued contraction see 4. The purpose of this paper is to extend this approach to the case of Reich-type multivalued a, b, c -contraction.

Notations and Basic Concepts

A Theory of Reich’s Fixed Point Principle