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 ¯ → K C ( 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 y t ∈ D ¯ satisfying y t ∈ t T y t + ( 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 { 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.