*(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 $\left\{\alpha \right(n\left)\right\}$ be a sequence of real numbers such that $0\le \alpha \left(n\right)\le 1$ and $lim\; sup\alpha \left(n\right)<1$ when $n\to \infty $. When the set $F\left(T\right)$ of the fixed points of $T$ is nonempty, then the sequence $x\left(n\right)$ constructed by the hybrid method converges strongly to the point that is the generalised projection from $C$ onto $F\left(T\right)$.

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 (nonlinear operator equations) |

47H09 | Mappings defined by “shrinking” properties |

47H05 | Monotone operators (with respect to duality) and generalizations |