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Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces. (English) Zbl 1198.47085

In 1967, F. E. Browder [Arch. Ration. Mech. Anal. 24 82–90 (1967; Zbl 0148.13601)] established the following important theorem:

Theorem B. Let D be a bounded closed convex subset of a real Hilbert space H and T a nonexpansive self-mapping of D. Let {t n } be a sequence in (0,1) converging to 1. Fix uD and define a sequence {x n } in D by

x n =t n Tx n +(1-t n )u,n·

Then {x n } converges strongly to the element of F(T) nearest to u.

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 D be an open nonempty convex subset of a real Banach space E, let T:D ¯CB(E) be a continuous (relative to the Hausdorff metric) pseudocontractive mapping satisfying a weakly inward condition and let uD ¯ be fixed. Then, for each t(0,1), there exists y t D ¯ satisfying y t tTy 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 the fixed point property for nonexpansive self mappings, then T has a fixed point if and only if {y t } remains bounded as t1; moreover, in this case, {y t } converges strongly to a fixed point of T as t1-.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
54H25Fixed-point and coincidence theorems in topological spaces
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