$CM$-Selectors for Pairs of Oppositely Semicontinuous Multifunctions and Some Applications to Strongly Nonlinear Inclusions

Abstract

We present a new approximate joint selection theorem which unifies Michael’s theorem (1956) on continuous selections and Cellina’s theorem (1969) on continuous $ϵ$-approximate selections. More precisely, we show that, given a convex-valued $H$-upper semicontinuous multifunction $F$ and a convex-closed-valued lower semicontinuous multifunction $G$ with $F(x)\bigcap G(x)\ne \varnothing$, one can find a continuous function $f$ which is both a selection of $G$ and an $ϵ$-approximate selection of $F$. We also prove a parametric version of this theorem for multifunctions $F$ and $G$ of two variables $(s,u)\in \Omega \times X$ where $\Omega$­ is a measure space. Using this selection theorem, we obtain an existence result for elliptic systems involving a vector Laplacian and a strongly nonlinear multi-valued right-hand side, subject to Dirichlet boundary conditions.