On Gamma -convergence of vector-valued mappings

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.