Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces

Let E be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, C a nonempty closed convex subset of E, and T : C -> K(E) a nonexpansive mapping. For u epsilon C and t epsilon (0, 1), let x(t) be a fixed point of a contraction G(t) : C -> K(E), defined by G(t)x := tTx + (1 - t)u, x epsilon C. It is proved that if C is a nonexpansive retract of E, {x(t)} is bounded and Tz = {z} for any fixed point z of T, then the strong lim(t -> 1) x(t) exists and belongs to the fixed point set of T. Furthermore, we study the strong convergence of {x} with the weak inwardness condition on T in a reflexive Banach space with a uniformly Gateaux differentiable norm. (c) 2006 Elsevier Ltd. All rights reserved.