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

Abstract

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 x t be a fixed point of a contraction G t : C → K ( E ) , defined by G t x ≔ t T x + ( 1 − t ) u , x ∈ C . It is proved that if C is a nonexpansive retract of E , { x t } is bounded and T z = { 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 t } with the weak inwardness condition on T in a reflexive Banach space with a uniformly Gâteaux differentiable norm.