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

Analysis in Theory and Applications 2011年第3期27卷 224-238页

作者： ShaoyuanXu Afif Ben Amar Department of Mathematics and Information TechnologyHanshan Normal UniversityChaozhou 521041 Departement de MathematiquesUniversite de Gafsa Cite Universitaire Zarrouk Faculte des Sciences de Gafsa2112 Gafsa Tunisie

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.

关键词： measure of weak noncompactness weakly condensing and weakly nonexpansive weakly sequentially continuous weakly semi-closed at the origin fixedpoint theorem

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Fixed Point Theory for 1-Set weakly Contractive Operators in Banach Spaces

引用

Analysis in Theory and Applications 2013年第3期29卷 208-220页

作者： Shaoyuan Xu Afif Ben Amar Department of Mathematics and Statistics Hanshan Normal University De'partement de Mathe'matiques Faculte' des Sciences de SfaxUniversite' de SfaxSfaxTunisia

In this work, using an analogue of Sadovskii＇s fixed point result and several important inequalities we investigate and give new existence theorems for the nonlinear operator equation F（x） =μx, （μ≥1） for some weakly sequentially continuous, weakly condensing and weakly 1-set weakly contractive operators with different boundary conditions. Correspondingly, we can obtain some applicable fixed point theorems of Leray-Schauder, Altman and Furi-Pera types in the weak topology setting which generalize and improve the corresponding results of [3,15,16].

关键词： weakly condensing weakly sequentially continuous fixed point theorem operator equation.

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