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.