Attouch-Wets convergence and a differential operator for convex functions

Abstract

The purpose of this note is to characterize Attouch-Wets convergence for sequences of proper lower semicontinuous convex functions defined on a Banach space X in terms of the behavior of an operator Δ \Delta defined on the space of such functions with values in X × R × X ∗ X \times R \times {X^ \ast } , defined by Δ ( f ) = { ( x , f ( x ) , y ) : ( x , y ) ∈ ∂ f } \Delta (f) = \{ (x,f(x),y):(x,y) \in \partial f\} . We show that ⟨ f n ⟩ \langle {f_n}\rangle is Attouch-Wets convergent to f if and only if points of Δ ( f ) \Delta (f) lying in a fixed bounded set can be uniformly approximated by points of Δ ( f n ) \Delta ({f_n}) for large n. The operator Δ \Delta is a natural carrier of the Borwein variational principle, which is a key tool in both directions of our proof.