Principle of weakly contractive maps in Hilbert spaces.

*(English)* Zbl 0897.47044
Gohberg, I. (ed.) et al., New results in operator theory and its applications: the Israel M. Glazman memorial volume. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 98, 7-22 (1997).

Summary: We introduce a class of contractive maps on closed convex sets of Hilbert spaces, called weakly contractive maps, which contains the class of strongly contractive maps and which is contained in the class of nonexpansive maps. We prove the existence of fixed points for the weakly contractive maps which are a priori degenerate in general case. We establish then the convergence in norm of classical iterative sequences to fixed points of these maps, give estimates of the convergence rate and prove the stability of the convergence with respect to some perturbations of these maps. Our results extend Banach principle previously known for strongly contractive maps only.

##### MSC:

47H09 | Mappings defined by “shrinking” properties |

47J25 | Iterative procedures (nonlinear operator equations) |

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