We present a new approximate joint selection theorem which unifies Michael’s theorem (1956) on continuous selections and Cellina’s theorem (1969) on continuous -approximate selections. More precisely, we show that, given a convex-valued -upper semicontinuous multifunction and a convex-closed-valued lower semicontinuous multifunction with , one can find a continuous function which is both a selection of and an -approximate selection of . We also prove a parametric version of this theorem for multifunctions and of two variables where 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.
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Hong Thai Nguyen, M. Juniewicz, J. Zieminska, -Selectors for Pairs of Oppositely Semicontinuous Multifunctions and Some Applications to Strongly Nonlinear Inclusions. Z. Anal. Anwend. 19 (2000), no. 2 pp. 381–393