Fixed point theorems in generating spaces of quasi-norm family and applications

Jian-Zhong Xiao,Xing-Hua Zhu

doi:10.1155/fpta/2006/61623

Abstract

Some new concepts of generating spaces of quasi-norm family are introduced and their linear topological structures are studied. These spaces are not necessarily locally convex. By virtue of some properties in these spaces, several Schauder-type fixed point theorems are proved, which include the corresponding theorems in locally convex spaces as their special cases. As applications, some new fixed point theorems in Menger probabilistic normed spaces and fuzzy normed spaces are obtained.

Highlights

The Schauder fixed point theorem and its generalizations which were obtained by Krasnoselskii et al, play important role in nonlinear analysis
Many interesting extensions and important applications of these theorems were presented by Fan [1] and others. Several extensions of these theorems in Menger probabilistic normed spaces were given under some conditions by Zhang-Guo [11] and Lin [6]
We introduce some new concepts of generating spaces of quasi-norm family, and establish some new unified versions of Schauder-type fixed point theorems in more general setting

Summary

Introduction

The Schauder fixed point theorem and its generalizations which were obtained by Krasnoselskii et al (see [3, 5], we call them the Schauder-type fixed point theorems), play important role in nonlinear analysis. Several extensions of these theorems in Menger probabilistic normed spaces were given under some conditions by Zhang-Guo [11] and Lin [6]. We introduce some new concepts of generating spaces of quasi-norm family, and establish some new unified versions of Schauder-type fixed point theorems in more general setting. We study the existence problems concerning the fixed points for operators on Menger probabilistic normed space and fuzzy normed space. Our results contain the former versions of the Schauder-type fixed point theorems and the corresponding theorems in Menger probabilistic normed spaces and fuzzy normed spaces as their special cases. 2. Fixed point theorems in generating spaces of quasi-norm family Throughout this paper we denote the set of all positive integers by Z+ and the field of real or complex numbers by E.

Fixed point theorems in generating spaces

Applications: fixed point theorems in probabilistic case and fuzzy case