Journal of Fixed Point Theory and Applications Fixed point theorems for the sum of (ws)-compact and asymptotically Φ-nonexpansive mappings Afif Ben Amar, Donal O’Regan and Amel Touati Abstract. In this paper, we introduce the concept of an asymptotically Φ-nonexpansive operator. In addition, we establish some Krasnoselskii- type 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 Φ-nonexpansive operator on an unbounded closed convex subset of a Banach space. Also we present Leray–Schauder al- ternatives and Furi–Pera-type fixed point theorems for the sum of two (ws)-compact mappings. Mathematics Subject Classification. 47H10, 47J05, 47J10. Keywords. (ws)-compact, asymptotically Φ-nonexpansive, weakly com- pact, measure of weak noncompactness, demiclosed, fixed point theo- rems. 1. Introduction Many nonlinear problems involve the study of nonlinear equations of the form Ax + Bx = x, x ∈ K, where K is a closed convex subset of a Banach space X (see [22]). A mapping T defined on a nonempty closed convex subset C of a Banach space X is said to be nonexpansive if ∥Tx − Ty∥≤∥x − y∥ ∀x, y ∈ C. The mapping T is called asymptotically nonexpansive [17] if there exists a sequence {k n }∈ (0, ∞) with lim n→∞ k n = 0 such that for all x, y ∈ C, ∥T n x − T n y∥≤ k n ∥x − y∥ ∀n ≥ 1. J. Fixed Point Theory Appl. 17 (2007) 1–32 DOI 10.1007/s11784-016-0326-8 © Springer International Publishing 2016