Mosco Stability of Proximal Mappings in Reflexive Banach Spaces

In this paper we establish criteria for the stability of the proximal mapping Prox φ = (∂φ+∂f)−1 associated to the proper lower semicontinuous convex functions φ and f on a reflexive Banach space X.We prove that, under certain conditions, if the convex functions φn converge in the sense of Mosco to φ and if ξn converges to ξ, then Prox f φn(ξn) converges to Prox f φ(ξ). 1. Preliminaries Let X be a real reflexive Banach space with the norm k·k and let X∗ be its dual with the norm denoted k·k∗. Let f : X → (−∞,+∞] be a proper, lower semicontinuous, convex function. Then the function f∗, the Fenchel conjugate of f, is also a proper, lower semicontinuous, convex function and f∗∗ = f (see, e.g., [12, pp. 78-79]). We assume that f is a Legendre function in the sense given to this term in [8, Definition 5.2]. This implies that both functions f and f∗ have domains with nonempty interior, are differentiable on the interiors of their respective domains, (1.1) dom∇f = int dom f = dom ∂f, (1.2) ran∇f = dom ∇f∗ = int dom f∗,