Selections of bounded variation under the excess restrictions

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) = supx∈F infy∈G d(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 singlevalued functions, is defined by V+(F, [a, b]) = supπ m i=1 e(F (ti−1),F(ti )) where the supremum is taken over all partitions π = {ti }m i=0 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 singlevalued 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