Approximation of Fixed Points of Metrically Regular Mappings

Abstract

We consider two different algorithms for solving the inclusion T(x) ∋ x, where T is a metrically regular set-valued mapping acting from a Banach space X to itself. The first one is given by the following Mann-type iteration: where λ n is a sequence of positive scalars in (0, 1) such that λ n ↑ 1. The second one is the classical proximal point algorithm adapted to metrically regular mappings: where μ n is a sequence of positive scalars such that μ n ↓ 0. We prove that if the modulus of regularity of T is sufficiently small then both algorithms (*) and (**) converge locally superlinearly to a fixed point of T. Similar convergence results are obtained for the cases when the mapping T is strongly subregular and strongly regular.