An Approximation Method for Continuous Pseudocontractive Mappings

LetK be a closed convex subset of a real Banach space E, T : K → K is continuous pseudocontractive mapping, and f : K → K is a fixed L-Lipschitzian strongly pseudocontractive mapping. For any t ∈ (0,1), let xt be the unique fixed point of t f + (1− t)T . We prove that if T has a fixed point and E has uniformly Gâteaux differentiable norm, such that every nonempty closed bounded convex subset of K has the fixed point property for nonexpansive self-mappings, then {xt} converges to a fixed point of T as t approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).