Fixed point variational solutions for uniformly continuous pseudocontractions in Banach spaces

Abstract

Let be a reflexive Banach space with a uniformly Gâteaux differentiable norm, let be a nonempty closed convex subset of , and let be a uniformly continuous pseudocontraction. If is any contraction map on and if every nonempty closed convex and bounded subset of has the fixed point property for nonexpansive self-mappings, then it is shown, under appropriate conditions on the sequences of real numbers , , that the iteration process , , , strongly converges to the fixed point of , which is the unique solution of some variational inequality, provided that is bounded.