# Mosco stability of proximal mappings in reflexive Banach spaces

In this paper we establish criteria for the stability of the proximal mapping \textrm{Prox} $_{\varphi }^{f}=(\partial \varphi +\partial f)^{-1}$ associated to the proper lower semicontinuous convex functions $\varphi$ and $f$ on a reflexive Banach space $X.$ We prove that, under certain conditions, if the convex functions $\varphi _{n}$ converge in the sense of Mosco to $\varphi$ and if $\xi _{n}$ converges to $\xi ,$ then \textrm{Prox} $_{\varphi _{n}}^{f}(\xi _{n})$ converges to \textrm{Prox} $_{\varphi }^{f}(\xi ).$

preprint, 2006