Γ-CONVERGENCE OF DOUBLE SEQUENCES OF FUNCTIONS, AND MINIMIZERS

In the present paper, we introduce the concept of Gamma-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 Gamma-convergence. Later we deal with the problem of minimization. We prove that, under some additional assumptions, the Gamma-convergence of a double sequence (fkl) to a function f implies the convergence of the minimum values of fkl to the minimum value of f. Moreover, we prove that each limit point of the double sequence of the minimizers of fkl is a minimizer of f.