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

### Hong Thai Nguyen

Szczecin University, Poland### M. Juniewicz

Szczecin University, Poland### J. Zieminska

Szczecin University, Poland

## 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 $\epsilon$-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 $F(x) \bigcap G(x) \neq \emptyset$, one can find a continuous function $f$ which is both a selection of $G$ and an $\epsilon$-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.

## Cite this article

Hong Thai Nguyen, M. Juniewicz, J. Zieminska, $CM$-Selectors for Pairs of Oppositely Semicontinuous Multifunctions and Some Applications to Strongly Nonlinear Inclusions. Z. Anal. Anwend. 19 (2000), no. 2, pp. 381–393

DOI 10.4171/ZAA/957