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Iterative algorithm for multi-valued pseudocontractive mappings in Banach spaces. (English) Zbl 1198.47085
In 1967, {\it F. E.\thinspace 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 $u \in D$ and define a sequence $\{ x_n \}$ in $D$ by $$x_n = t_n T x_n + (1 - t_n)u, \quad n \in\Bbb N.$$ 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 $T : \overline{D} \rightarrow CB(E)$ 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 $t \in (0,1)$, there exists $y_t \in \overline{D}$ satisfying $y_t \in tT y_t + (1 - t)u$. If, in addition, $E$ is reflexive and has a uniformly Gâteaux differentiable norm, and is such that every closed convex bounded subset of $\overline{D}$ has the fixed point property for nonexpansive self mappings, then $T$ has a fixed point if and only if $\{ y_t \}$ remains bounded as $t \rightarrow 1$; moreover, in this case, $\{ y_t \}$ converges strongly to a fixed point of $T$ as $t \rightarrow 1-$.

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
Full Text: DOI
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