Bounded approximants to monotone operators on Banach spaces

Abstract

It is shown that for any maximal monotone set-valued operator T on a real Banach space E, there is a sequence {Tn} of bounded maximal monotone operators which have nonempty values at each point and which converge to T in a reasonable sense. Better convergence properties are shown to hold when T is in a new proper subclass of maximal monotone operators (the “locally” maximal monotone operators), a subclass which coincides with the entire class in reflexive spaces. The approximation method is patterned on the one which results when the (maximal monotone) subdifferential ∂f of a proper lower semicontinuous convex function f is approximated by a sequence of bounded subdifferentials {∂fn}, where each fn is the (continuous and convex) inf-convolution of f with the function n ‖ · ‖. The main advantage of this approximation scheme over the classical Moreau-Yosida approximation method is that it exists in non-reflexive Banach spaces.