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

Let E be a uniformly convex Banach space with a uniformly Gâteaux differentiable norm, C a nonempty closed convex subset of E, and T:C→K(E) a nonexpansive mapping. For u∈C and t∈(0,1), let xt be a fixed point of a contraction Gt:C→K(E), defined by Gtx≔tTx+(1−t)u,x∈C. It is proved that if C is a nonexpansive retract of E, {xt} is bounded and Tz={z} for any fixed point z of T, then the strong limt→1xt exists and belongs to the fixed point set of T. Furthermore, we study the strong convergence of {xt} with the weak inwardness condition on T in a reflexive Banach space with a uniformly Gâteaux differentiable norm.