Fixed points and eigenvalues for countably asymptotically Φ -nonexpansive operators on convex sets under asymptotic contractive-type conditions

The purpose of this paper is to prove fixed point results for certain types of countably asymptotically Φ -nonexpansive (or countably Φ -condensing) operators on locally convex spaces and satisfying additional asymptotic contractive-type conditions. These results allow us to obtain generalizations of recent fixed point theorems of Ben Amar, Isac, Németh, O’Regan, and Touati to locally convex spaces. As an application, we obtain the existence of positive eigenvalues of countably asymptotically Φ -nonexpansive (or countably Φ -condensing) operators in locally convex spaces. Also, we present Krasnosel’skii fixed point theorems of two nonlinear operators acting on locally convex spaces. These results are based on a generalized notion of the semi-inner product in Lumer’s sense and the axiomatic measure of noncompactness.