Fixed point theorems of convex-power 1-set-contraction operators in Banach spaces

Zhao Lvhuizi,Sun Jingxian

doi:10.1186/1687-1812-2012-56

Abstract

In this article, we give the definition of a class of new operators, namely, convex-power 1-set-contraction operators in Banach spaces, and study the existence of fixed points of this class of operators. By using methods of approximation by operators, we obtain fixed point theorems of convex-power 1-set-contraction operators, which generalize fixed point theorems of 1-set-contraction operators in Banach spaces. By using the fixed point theorem, the existence of solutions of nonlinear Sturm-Liouville problems in Banach spaces is investigated under more general conditions than those used in former literatures.Mathematics Subject Classification 2010: 47H10.

Highlights

0 Introduction For the need of studying differential equations and integral equations, Sun and Zhang [1] gave the definition of convex-power condensing operators and obtained the fixed point theorem of this class of operators
Li [2] gave the fixed point theorem of semiclosed 1-set-contraction operators
By combinating the definitions of convex-power condensing operators and 1-set-contraction operators, we give the definition of convex-power 1-set-contraction operators in Banach spaces and study the existence of fixed points of this class of new operators

Summary

Introduction

For the need of studying differential equations and integral equations, Sun and Zhang [1] gave the definition of convex-power condensing operators and obtained the fixed point theorem of this class of operators. A is said to be convex-power 1-set-contraction in Banach spaces. Suppose that A: D ® D is semi-closed and convex-power 1-set-contraction, A has at least one fixed point in D. Since A is convex-power 1-set-contraction, there exist x0 Î D and a positive integer n0, such that for any bounded subset S ⊂ D, α A(n0,x0) (S) ≤ α (S). By the definition of the convex-power 1-set-contraction operator and the properties of the measure of noncompactness, we have α An(n0,x0) (S) ≤. Generalizes the fixed point theorem of semi-closed 1-set-contraction operators. Let E be a Banach space, D ⊂ E bounded, convex and closed. Suppose that A: D ® D is semi-compact and convex-power 1-set-contraction, A has at least one fixed point in D

By the proof of Theorem

Proof of