JournalszaaVol. 19 , No. 2pp. 381–393

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

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\epsilon-approximate selections. More precisely, we show that, given a convex-valued HH-upper semicontinuous multifunction FF and a convex-closed-valued lower semicontinuous multifunction GG with F(x)G(x)F(x) \bigcap G(x) \neq \emptyset, one can find a continuous function ff which is both a selection of GG and an ϵ\epsilon-approximate selection of FF. We also prove a parametric version of this theorem for multifunctions FF and GG of two variables (s,u)Ω×X(s, u) \in \Omega \times X where Ω\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, CMCM-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