Stability of Mann’s iterates under metric regularity

We consider a Mann-like iteration for solving the inclusion x∈T(x)x∈T(x) where T:X⇉X is a set-valued mapping, defined from a Banach space X   into itself, which is metrically regular near a point (x¯,x¯) in its graph. We study the behavior of the iterates generated by our method and prove that they inherit the regularity properties of the mapping T. First we consider the case when the mapping T is metrically regular, then the case when it is strongly metrically regular. Finally, we present an inexact version of our method and we study its convergence when the mapping T is strongly metrically subregular.