On $\Gamma$-convergence of vector-valued mappings

Abstract

This paper deals with a new concept of limit for sequences of vector-valued mappings in normed spaces. We generalize the well-known concept of $\Gamma$-convergence to the case of vector-valued mappings and specify notion of $\Gamma ^{\Lambda ,\mu }$-convergence similar to the one previously introduced in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In particular, we show that $\Gamma ^{\Lambda ,\mu }$-convergence concept introduced in this paper possesses a compactness property whereas this property was failed in Dovzhenko et al. (Far East J Appl Math 60:1–39, 2011). In spite of the fact this paper contains another definition of $\Gamma ^{\Lambda ,\mu }$-limits for vector-valued mapping we prove that the $\Gamma ^{\Lambda ,\mu }$-lower limit in the new version coincides with the previous one, whereas the $\Gamma ^{\Lambda ,\mu }$-upper limit leads to a different mapping in general. Using the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their coepigraphs, we establish the connection between $\Gamma ^{\Lambda ,\mu }$-convergence of the sequences of mappings and $K$-convergence of their epigraphs and coepigraphs in the sense of Kuratowski and study the main topological properties of $\Gamma ^{\Lambda ,\mu }$-limits. The main results are illustrated by numerous examples.