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→KC(E) be a continuous pseudocontractive mapping satisfying the weakly inward condition and let u∈D be fixed. Then for each t∈(0,1) there exists yt∈D satisfying yt∈tTyt+(1-t)u. If, in addition, E is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of D has fixed point property for nonexpansive self-mappings, then T has a fixed point if and only if {yt} remains bounded as t→1; in this case, {yt} 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.