Multiselectors Under the Excess Restrictions

The present research is devoted to studying the most popular in the current period theory of set-valued functions of bounded variation. Let X be a metric space with metric d, c(X) denote the family of all nonempty compact subsets of X and, given F,G∈c(X), let e(F,G)=sup_(x∈F) inf_(y∈G) d(x,y) be the Hausdorff excess of F over G. The excess variation of the multifunction F:[a,b]→c(X), which generalizes the ordinary variation V of single-valued function f, is defined by V_+ (F,[a,b]). According to these designations the main result of the research is the following selection theorem: if F – multifunction and V_+- Jordan’ variation to the right, then there exists a single-valued function f:[a,b]→X of bounded variation such that f(t)∈F(t) for all t∈[a,b] is called as selection of bounded variation. It is necessary to mention, the desired selection preserves all properties of the given multifunction and generally can have discontinuities.