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

Banaś, Józef ; Ben Amar, Afif

Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 1, pp. 21-40.

Résumé

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.

Détail

Classification : 47H10
Keywords: $\tau$-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