Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets

Commentationes mathematicae Universitatis Carolinae

Józef Banaś¹ 

关键词: $\tau$-measure of noncompactness;    $\tau$-sequential continuity;    $\Phi_{\tau}$-condensing operator;    $\Phi_{\tau}$-nonexpansive operator;    nonlinear contraction;    fixed point theorem;    demi-$\tau$-compactness;    operator $\tau$-semi-closed at origin;    Lebesgue space;    integral equation;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

In this paper we prove a collection of new fixed point theorems for operators of the form $T+S$ on an unbounded closed convex subset of a Hausdorff topological vector space $(E,\Gamma )$. We also introduce the concept of demi-$\tau$-compact operator and $\tau$-semi-closed operator at the origin. Moreover, a series of new fixed point theorems of Krasnosel'skii type is proved for the sum $T+S$ of two operators, where $T$ is $\tau$-sequentially continuous and $\tau$-compact while $S$ is $\tau$-sequentially continuous (and $\Phi_{\tau}$-condensing, $\Phi_{\tau}$-nonexpansive or nonlinear contraction or nonexpansive). The main condition in our results is formulated in terms of axiomatic $\tau$-measures of noncompactness. Apart from that we show the applicability of some our results to the theory of integral equations in the Lebesgue space.

【 授权许可】

CC BY