*(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 ${E}^{*}$. The mapping $T$ is of firmly nonexpansive type if, for all $x,y\in C$,

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 for nonlinear operators on topological linear spaces |

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

47H09 | Mappings defined by “shrinking” properties |

47J25 | Iterative procedures (nonlinear operator equations) |