Characterizations of variational convergence of bifunctions defined on products of two sets

Abstract

We present definitions of types of variational convergence of finite-valued bifunctions defined on rectangular domains and establish characterizations of these convergences. In the introduction, we present the origins of the research on variational convergence and then we lead to the specific problem of this paper. The content of the paper consists of 3 parts: variational convergance of fucntion; variational convergance of bifunction; and characterizations of variational convergence of bifunction, this part is the main results of this paper. In section 2, we presented the definition of epi convergence and presented a basic property problem that will be used to extend and develop the next two sections. In section 3, we start to present a new definition, the definition of convergence epi / hypo, minsup and maxinf. To clearly understand of these new definitions we have provided comments (remarks) and some examples which reader can check these definitions. The above contents serve the main result of this paper will apply in part 4. Now, we will explain more detail for this part as follows. Firstly, variational convergence of bifunctions is characterized by the epi- and hypo-convergence of related unifunctions, which are slices sup- and inf-projections. The second characterization expresses the equivalence of variational convergence of bifunctions and the same convergence of the so-called proper bifunctions defined on the whole product spaces. In the third one, the geometric reformulation, we establish explicitly the interval of all the limits by computing formulae of the left- and right-end limit bifunctions, and this is necessary and sufficient conditions of the sequence bifunctions to attain epi / hypo, minsup and maxinf convergence.