Selections of bounded variation under the excess restrictions

Abstract

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 a multifunction F : [ a , b ] → c ( X ) , which generalizes the ordinary variation V of single-valued functions, is defined by V + ( F , [ a , b ] ) = sup π ∑ i = 1 m e ( F ( t i − 1 ) , F ( t i ) ) where the supremum is taken over all partitions π = { t i } i = 0 m 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 ] ) < ∞ , t 0 ∈ [ a , b ] and x 0 ∈ F ( t 0 ) , 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 ( t 0 ) = x 0 , V ( f , [ a , t 0 ) ) ⩽ V + ( F , [ a , t 0 ) ) and V ( f , [ t 0 , b ] ) ⩽ V + ( F , [ t 0 , 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 t 0 = a and may have only discontinuous selections of bounded variation if a < t 0 ⩽ b . The same situation holds for continuous selections of F : [ a , b ] → c ( X ) when it is excess continuous in the sense that e ( F ( s ) , F ( t ) ) → 0 as s → t − 0 for all t ∈ ( a , b ] and e ( F ( t ) , F ( s ) ) → 0 as s → t + 0 for all t ∈ [ a , b ) simultaneously.