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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 $\left\{{t}_{n}\right\}$ be a sequence in $\left(0,1\right)$ converging to 1. Fix $u\in D$ and define a sequence $\left\{{x}_{n}\right\}$ in $D$ by

${x}_{n}={t}_{n}T{x}_{n}+\left(1-{t}_{n}\right)u,\phantom{\rule{1.em}{0ex}}n\in ℕ·$

Then $\left\{{x}_{n}\right\}$ converges strongly to the element of $F\left(T\right)$ 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 $T:\overline{D}\to CB\left(E\right)$ be a continuous (relative to the Hausdorff metric) pseudocontractive mapping satisfying a weakly inward condition and let $u\in \overline{D}$ be fixed. Then, for each $t\in \left(0,1\right)$, there exists ${y}_{t}\in \overline{D}$ satisfying ${y}_{t}\in tT{y}_{t}+\left(1-t\right)u$. If, in addition, $E$ is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of $\overline{D}$ has the fixed point property for nonexpansive self mappings, then $T$ has a fixed point if and only if $\left\{{y}_{t}\right\}$ remains bounded as $t\to 1$; moreover, in this case, $\left\{{y}_{t}\right\}$ converges strongly to a fixed point of $T$ as $t\to 1-$.

MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties 54H25 Fixed-point and coincidence theorems in topological spaces