Selections of bounded variation under the excess restrictions

AbstractLet 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)=supx∈Finfy∈Gd(x,y) be the Hausdorff excess of F over G. The excess variation of a multifunction F:[a,b]→c(X), which generalizes the ordinary variation V of single-valued functions, is defined by V+(F,[a,b])=supπ∑i=1me(F(ti−1),F(ti)) where the supremum is taken over all partitions π={ti}i=0m of the interval [a,b]. The main result of the paper is the following selection theorem: If F:[a,b]→c(X), V+(F,[a,b])<∞, t0∈[a,b] and x0∈F(t0), 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], f(t0)=x0, V(f,[a,t0))⩽V+(F,[a,t0)) and V(f,[t0,b])⩽V+(F,[t0,b]). We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1–82]. In contrast to this, a multifunction F satisfying e(F(s),F(t))⩽C(t−s) for some constant C⩾0 and all s,t∈[a,b] with s⩽t (Lipschitz continuity with respect to e(⋅,⋅)) admits a Lipschitz selection with a Lipschitz constant not exceeding C if t0=a and may have only discontinuous selections of bounded variation if a