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

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).

Reviewer: Gilbert Crombez (Gent)

### MSC:

47J25 | Iterative procedures involving nonlinear operators |

47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |

47H05 | Monotone operators and generalizations |

### Keywords:

relatively nonexpansive mapping; nonexpansive mapping; asymptotic fixed point; generalized projection; maximal monotone operator; strong convergence; hybrid method
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\textit{S.-y. Matsushita} and \textit{W. Takahashi}, J. Approx. Theory 134, No. 2, 257--266 (2005; Zbl 1071.47063)

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