Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces

Let D be nonempty open convex subset of a real Banach space E. Let T : (D) over bar -> KC(E) be a continuous pseudocontractive mapping satisfying the weakly inward condition and let u is an element of (D) over bar be fixed. Then for each t is an element of (0,1) there exists y(t) is an element of (D) over bar satisfying y(t) is an element of tTy(t) + (1 - t)u. If, in addition, E is reflexive and has a uniformly Gateaux differentiable norm, and is such that every closed convex bounded subset of (D) over bar has fixed point property for nonexpansive self-mappings, then T has a fixed point if and only if {y(t)} remains bounded as t -> 1; in this case, {y(t)} converges strongly to a fixed point of T as t -> 1(-). Moreover, an explicit iteration process which converges strongly to a fixed point of T is constructed in the case that T is also Lipschitzian. (C) 2010 Elsevier Inc. All rights reserved.