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A strong convergence theorem for relatively nonexpansive mappings in a Banach space. (English) Zbl 1071.47063
The present paper is concerned with the problem of finding a fixed point of a relatively nonexpansive mapping defined on a closed convex subset of a Banach space. To this end, the authors investigate under what conditions a sequence constructed by the so-called “hybrid method in mathematical programming” converges strongly to a fixed point of the mapping. The main result (Theorem 3.1) goes as follows: Theorem. Let $E$ be a uniformly convex and uniformly smooth Banach space, let $C$ be a nonempty closed convex subset of $E$, let $T$ be a relatively nonexpansive mapping from $C$ into itself, and let $\{\alpha(n)\}$ be a sequence of real numbers such that $0\le \alpha(n)\le 1$ and $\limsup\alpha(n)< 1$ when $n\to\infty$. When the set $F(T)$ of the fixed points of $T$ is nonempty, then the sequence $x(n)$ constructed by the hybrid method converges strongly to the point that is the generalised projection from $C$ onto $F(T)$. As special cases, they obtain analogous strong convergence results for a nonexpansive mapping on a Hilbert space (using the metric projection) and for a maximal monotone operator on a Banach space (using the generalised projection).

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H05Monotone operators (with respect to duality) and generalizations
Full Text: DOI
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