A Lyusternik-Graves theorem for the proximal point method

We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y ∈ T (x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point point (x̄, 0) in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T , and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular.