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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 {α(n)} be a sequence of real numbers such that 0α(n)1 and lim supα(n)<1 when n. 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