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 Γ-convergence to the case of vector-valued mappings and specify notion of ΓΛ,μ -convergence similar to the one previously introduced in [12]. In particular, we show that ΓΛ,μ -convergence concept introduced in this paper possesses a compactness property whereas this property was failed in [12]. In spite of the fact this paper contains another definition of ΓΛ,μ -limits for vector-valued mapping and we show that the ΓΛ,μ -lower limit in the new version coincides with the previous one, whereas the ΓΛ,μ -upper limit leas 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 ΓΛ,μ -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 ΓΛ,μ -limits. The main results are illustrated by numerous examples.