MEASURES OF WEAK NONCOMPACTNESS AND FIXED POINT THEORY FOR 1-SET WEAKLY CONTRACTIVE OPERATORS ON UNBOUNDED DOMAINS

• Published in: Analysis in theory & applications Vol. 27; no. 3; pp. 224 - 238
• Main Authors: Xu, Shaoyuan, Ben Amar, Afif
• Format: Journal Article
• Language: English
• Published: Nanjing Editorial Board of Analysis in Theory and Applications 01.09.2011; Hanshan Normal University, China%Université de Gafasa, Cié Universitaire Zarrouk, Tunisie
• Subjects: Analysis; Approximations and Expansions; Banach空间; Mathematics; Schauder; 不动点定理; 不动点理论; 弱紧性; 收缩; 无界域; 非线性算子方程; measure of weak noncompactness; 47J10; 47H10; weakly sequentially continuous; weakly condensing and weakly nonexpansive; 47J05; weakly semi-closed at the origin; fixed point theorem; fixedpoint theorem
• ISSN: 1672-4070; 1573-8175
• DOI: 10.1007/s10496-011-0224-2

The main purpose of this paper is to prove a collection of new fixed point theorems and existence theorems for the nonlinear operator equation F（x） = αx （α≥〉 1） for so-called 1-set weakly contractive operators on unbounded domains in Banach spaces. We also introduce the concept of weakly semi-closed operator at the origin and obtain a series of new fixed point theorems and the existence theorems for the nonlinear operator equation F（x） = αx （α≥1） for such class of operators. As consequences, the main results general- ize and improve the relevant results, which are obtained by O＇Regan and A. Ben Amar and M. Mnif in 1998 and 2009 respectively. In addition, we get the famous fixed point theorems of Leray-Schauder, Altman, Petryshyn and Rothe type in the case of weakly sequentially continuous, 1-set weakly contractive （μ-nonexpansive） and weakly semi-closed operators at the origin and their generalizations. The main condition in our results is formulated in terms of axiomatic measures of weak compactness.