Relatively Nonexpansive Operators with Respect to Bregman Distances

Abstract

The following problem often occurs in functional analysis and optimization theory, as well as in other fields of pure and applied mathematics: given a nonempty, closed and convex subset K of a Banach space X and an operator T:K→K, do the sequences iteratively generated in K by the rule x k+1=Tx k converge to a fixed point of T no matter how the initial point x 0∈K is chosen? It is well known that this indeed happens, in some sense, for “standard” classes of operators (e.g., certain nonexpansive operators and operators of contractive type). In this chapter we show that the question asked above has an affirmative answer even if the operator T is not contractive in any standard sense, but is relatively nonexpansive with respect to a Bregman distance induced by a convex function f. More precisely, we prove that in appropriate complete metric spaces of operators which are relatively nonexpansive with respect to Bregman distances there exists a subset which is a countable intersection of open and everywhere dense sets such that for any operator belonging to this subset, all its orbits converge strongly.