*(English)*Zbl 1028.47049

Let $G$ be a closed convex subset of a Banach space $B$. A mapping $A:G\to B$ is said to be (i) nonexpansive if for all $x,y\in G$, $\parallel Ax-Ay\parallel \le \parallel x-y\parallel $, (ii) weakly contractive of class ${C}_{\psi \left(t\right)}$ on $G$ if there exists a continuous and increasing function $\psi \left(t\right)$ defined on ${\mathbb{R}}^{+}$ such that $\psi $ is positive on ${\mathbb{R}}^{+}\setminus \left\{0\right\}$, $\psi \left(0\right)=0$, ${lim}_{t\to +\infty}\psi \left(t\right)=+\infty $, and for all $x,y\in G$, $\parallel Ax-Ay\parallel \le \parallel x-y\parallel -\psi (\parallel x-y\parallel )$.

The authors study descent-like approximation methods and proximal methods of retraction type for solving fixed-point problems with nonself-mappings in Hilbert and Banach spaces. They prove strong and weak convergence for weakly contractive and nonexpansive maps, respectively. They also establish the stability of these methods with respect to perturbations of the operators and the constraint sets.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H06 | Accretive operators, dissipative operators, etc. (nonlinear) |

47H09 | Mappings defined by “shrinking” properties |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |