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

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