New Results and Generalizations for Approximate Fixed Point Property and Their Applications

Wei-Shih Du,Farshid Khojasteh

doi:10.1155/2014/581267

Abstract

We first introduce the concept of manageable functions and then prove some new existence theorems related to approximate fixed point property for manageable functions andα-admissible multivalued maps. As applications of our results, some new fixed point theorems which generalize and improve Du's fixed point theorem, Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and Nadler's fixed point theorem and some well-known results in the literature are given.

Highlights

Introduction and PreliminariesIn 1922, Banach established the most famous fundamental fixed point theorem which has played an important role in various fields of applied mathematical analysis
By Theorem 15, we prove F(T) ≠ 0
(S1) T is H-continuous; (S2) T is closed; (S3) the map g : X → [0, ∞) defined by g(x) = d(x, Tx) is l.s.c.; (S4) the function α has the property (B), T admits a fixed point in X

Summary

Introduction

Introduction and PreliminariesIn 1922, Banach established the most famous fundamental fixed point theorem (so-called the Banach contraction principle [1]) which has played an important role in various fields of applied mathematical analysis. Since φ is an MT-function, by Theorem 2, there exist ra ∈ [0, 1) and εa > 0 such that φ(s) ≤ ra for all s ∈ [a, a + εa).

Results

Conclusion