Abstract
In this paper, we introduce the concept of an asymptotically -nonexpansive operator. In addition, we establish some Krasnoselskiitype fixed point theorems for the sum of two operators A and B, where the operator A is assumed to be (ws)-compact, and B is a (ws)-compact and asymptotically -nonexpansive operator on an unbounded closed convex subset of a Banach space. Also we present Leray-Schauder alternatives and Furi-Pera-type fixed point theorems for the sum of two (ws)-compact mappings.