Fixed point theorems for countably asymptotically $\Phi-$nonexpansive maps in locally convex spaces and application

Abstract

In this paper, we introduce the concept of a countably asymptotically $\Phi-$nonexpansive operator. In addition, we establish new fixed point results for some countably asymptotically $\Phi-$nonexpansive and sequentially continuous maps, fixed-point results of Krasnosel'skii type in locally convex spaces. Moreover, we present Leray-Schauder-type fixed point theorems for countably asymptotically $\Phi-$nonexpansive maps in locally convex spaces. Apart from that we show the applicability of our results to the theory of Volterra integral equations in locally convex spaces. The main condition in our results is formulated in terms of the axiomatic measure of noncompactness. Our results improve and extend in a broad sense recent ones obtained in literature.