## 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 $$0\leq \alpha(n)\leq 1$$ 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).

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H05 Monotone operators and generalizations
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### References:

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