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

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-$au$-compact operator and $au$-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 $au$-sequentially continuous and $au$-compact while $S$ is $au$-sequentially continuous (and $Phi_{au}$-condensing, $Phi_{au}$-nonexpansive or nonlinear contraction or nonexpansive). The main condition in our results is formulated in terms of axiomatic $au$-measures of noncompactness. Apart from that we show the applicability of some our results to the theory of integral equations in the Lebesgue space.