In 1967, F. E. Browder [Arch. Ration. Mech. Anal. 24 82–90 (1967; Zbl 0148.13601)] established the following important theorem:
Theorem B. Let be a bounded closed convex subset of a real Hilbert space and a nonexpansive self-mapping of . Let be a sequence in converging to 1. Fix and define a sequence in by
Then converges strongly to the element of nearest to .
The main purpose of the paper under review is to answer the question: is Browder’s theorem valid for the class of multi-valued Lipschitz pseudocontractive mappings in Banach spaces more general than Hilbert spaces?
The main result of the paper offers an affirmative answer to this question.
Theorem 3.3. Let be an open nonempty convex subset of a real Banach space , let be a continuous (relative to the Hausdorff metric) pseudocontractive mapping satisfying a weakly inward condition and let be fixed. Then, for each , there exists satisfying . If, in addition, is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of has the fixed point property for nonexpansive self mappings, then has a fixed point if and only if remains bounded as ; moreover, in this case, converges strongly to a fixed point of as .