Józef Banaś, Afif Ben~Amar
Measures of noncompactness in locally convex spaces and fixed point theory for the sum of two operators on unbounded convex sets

Comment.Math.Univ.Carolin. 54,1 (2013) 21-40.

Abstract: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.

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
AMS Subject Classification: 47H10