Asymptotic centers and fixed points for multivalued nonexpansive mappings

Abstract

Let X be a nearly uniformly convex Banach space, C a convex closed bounded subset of X and T : C → 2 C a multivalued nonexpansive mapping with convex compact values. We prove that T has a fixed point. This result improves former results in Domínguez Benavides, T., P. Lorenzo, Fixed point theorems for multivalued nonexpansive mappings without uniform convexity, Abstr. Appl. Anal. 2003:6 (2003), 375–386 and solves an open problem appearing in Xu, H.K., Metric fixed point theory for multivalued mappings, Dissertationes Math. (Rozprawy Mat.) 389 (2000), 39 pp.