## Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces.(English)Zbl 1168.47047

The purpose of the paper is to study the existence and approximation of fixed points of firmly nonexpansive mappings in Banach spaces. The following definition is used. Let $$E$$ be a smooth Banach space, $$C$$ be a nonempty closed convex subset of $$E$$, and $$J$$ be the normalized duality mapping from $$E$$ into $$E^{\ast}$$. The mapping $$T$$ is of firmly nonexpansive type if, for all $$x, y\in C$$,
$\langle T x - T y, J T x - J T y\rangle \leq \langle T x - T y, J x - J y \rangle.$
A fixed point theorem for firmly nonexpansive-type mappings in Banach spaces is obtained. It is shown that every nonexpansive-type mapping which has a fixed point is strongly relatively nonexpansive. A weak convergence theorem is presented. The obtained results are applied to the proximal point algorithm for monotone operators satisfying the range condition in Banach spaces.

### MSC:

 47H10 Fixed-point theorems 47H05 Monotone operators and generalizations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J25 Iterative procedures involving nonlinear operators
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