This paper deals with a new concept of limit for sequences of locally compact vector-valued mappings in normed spaces. We generalize the well-known concept of Γ-convergence to the so-called -convergence in vector-valued case. To this aim, we study the link between the lower semicontinuity property of vector-valued mappings and the topological properties of their coepigraphs. We show that, if the objective space is partially ordered by a pointed cone with nonempty interior, then coepigraphs are stable with respect to their closure and, moreover, the locally semicompact vector-valued mappings with closed coepigraphs are lower continuous. Using these results, we establish the relationship between -convergence of the sequences of mappings and K-convergence of their coepigraphs in the sense of Kuratowski and study the main topological properties of -limits.