Krasnosel’skii type fixed point theorems under weak topology features

In this paper we prove the following Krasnosel’skii type fixed point theorem: Let MM be a nonempty bounded closed convex subset of a Banach space XX. Suppose that A:M→XA:M→X and B:X→XB:X→X are two weakly sequentially continuous mappings satisfying: (i)AMAM is relatively weakly compact;(ii)BB is a strict contraction;(iii)(x=Bx+Ay,y∈M)⇒x∈M. Then A+BA+B has at least one fixed point in MM.This result is then used to obtain some new fixed point theorems for the sum of a weakly compact and a nonexpansive mapping. The results presented in this paper encompass several earlier ones in the literature.