Existence and Approximation of Fixed Points for Set-Valued Mappings

Simeon Reich,Alexanderj Zaslavski

doi:10.1155/2010/351531

Simeon Reich, Alexanderj Zaslavski

Open Access

https://doi.org/10.1155/2010/351531

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Abstract

AbstractTaking into account possibly inexact data, we study both existence and approximation of fixed points for certain set-valued mappings of contractive type. More precisely, we study the existence of convergent iterations in the presence of computational errors for two classes of set-valued mappings. The first class comprises certain mappings of contractive type, while the second one contains mappings satisfying a Caristi-type condition.

Highlights

The study of the convergence of iterations of mappings of contractive type has been an important topic in Nonlinear Functional Analysis since Banach’s seminal paper 1 on the existence of a unique fixed point for a strict contraction 2–5
One of the methods used for proving the classical Banach theorem is to show the convergence of Picard iterations, which holds for any initial point
In the case of set-valued mappings, we do not have convergence of all trajectories of the dynamical system induced by the given mapping

Summary

Introduction

The study of the convergence of iterations of mappings of contractive type has been an important topic in Nonlinear Functional Analysis since Banach’s seminal paper 1 on the existence of a unique fixed point for a strict contraction 2–5. We require xt 1 to approximate the best approximation up to a positive number t, such that the sequence { t}∞t 0 is summable This method allowed Nadler 7 to obtain the existence of a fixed point of a strictly contractive set-valued mapping and the authors of 6 to obtain more general. In view of this state of affairs, it is important to study convergence of the iterates of set-valued mappings in the presence of errors. We study the existence of convergent iterations in the presence of computational errors for two classes of set-valued mappings. We first show that a set-valued mapping satisfies a Caristi-type condition if and only if there exists an iterative sequence {xi}∞i 1 such that the sum of the distances between xi and xi 1, when i runs from zero to infinity, is finite. We prove an analog of the Caristi-type result in 6 , replacing the closedness of the graph of the mapping with a lower semicontinuity assumption as in Caristi’s original theorem 13

Set-Valued Mappings of Contractive Type

Caristi-Type Theorems for Set-Valued Mappings