Γ-CONVERGENCE OF DOUBLE SEQUENCES OF FUNCTIONS, AND MINIMIZERS

Abstract

In the present paper, we introduce the concept of Γ-convergence of a double sequence of functions defined from a metric space into real numbers. This convergence is useful as it is a convenient concept of convergence for approximating minimization problems in the field of mathematical optimization. First, we compare this convergence with pointwise and uniform convergence and obtain some properties of Γ-convergence. Later we deal with the problem of minimization. We prove that, under some additional assumptions, the Γ-convergence of a double sequence (f_{kl}) to a function f implies the convergence of the minimum values of f_{kl} to the minimum value of f. Moreover, we prove that each limit point of the double sequence of the minimizers of f_{kl} is a minimizer of f.