The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Zeitschrift für Analysis und ihre Anwendungen


Full-Text PDF (251 KB) | Metadata | Table of Contents | ZAA summary
Volume 19, Issue 2, 2000, pp. 381–393
DOI: 10.4171/ZAA/957

Published online: 2000-06-30

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

Hong Thai Nguyen[1], M. Juniewicz[2] and J. Zieminska[3]

(1) Szczecin University, Poland
(2) Szczecin University, Poland
(3) Szczecin University, Poland

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.

Keywords: Joint, continuous and $\epsilon$-approximate selectors, H-upper and lower semicontinuous multifunctions, multifunctions satisfying one-side estimates, Dirichlet elliptic inclusions, multi-valued elliptic systems, problems with strong non-linearities, with lack of compactness and with critical exponents

Nguyen Hong Thai, Juniewicz M., Zieminska J.: $CM$-Selectors for Pairs of Oppositely Semicontinuous Multifunctions and Some Applications to Strongly Nonlinear Inclusions. Z. Anal. Anwend. 19 (2000), 381-393. doi: 10.4171/ZAA/957