A Selection Principle for Mappings of Bounded Variation

Abstract

E. Helly's selection principle states that an infinite bounded family of real functions on the closed inter¨al, which is bounded in¨ariation, contains a pointwise con¨ergent sequence whose limit is a function of bounded¨ariation. We extend this theorem to metric space valued mappings of bounded variation. Then we apply the extended Helly selection principle to obtain the existence of regular selections of Ž . non-convex set-valued mappings: any set-¨alued mapping from an inter¨al of the real line into nonempty compact subsets of a metric space, which is of boundedariation with respect to the Hausdorff metric, admits a selection of bounded¨ariation. Also, we show that a compact-valued set-valued mapping which is Lipschitzian, absolutely continuous, or of bounded Riesz ⌽-variation admits a selection which is Lipschitzian, absolutely continuous, or of bounded Riesz ⌽-variation, respectively.

On the extension and generation of set-valued mappings of bounded variation

“…. , m. Denote by BV(E; X) the set of all mappings f : E → X for which V (f, E) < ∞; these mappings are called of bounded Jordan variation on E. A mapping g : E → X is said to be Lipschitzian on E if its (minimal) Lipschitz constant, defined by L(g, E) = sup{d(g(t), g(s))/|t − s| | t, s ∈ E, t = s}, (2) is finite. We set Lip(E; X) = {g : E → X | L(g, E) < ∞}.…”

“…When E is a (closed) interval, particular cases of this theorem are contained in [17] (part (a), X = R n ), [18] ((b) and (c), X = R n , F convex and nonconvex valued), [29] [28] ((b), X a metric space), [5] ((a), X a Banach space, F continuous and Gr(F ) compact), [6] ((a)-(c), X a Banach space and Gr(F ) compact), [7] ((a)-(c), X a Banach space) and [2], [10] ((a)-(c), X a metric space).…”

A note on tameness of families having bounded variation

“…In the particular case of (X, ≤) = [a, b] Definition 5.4 and Theorem 5.5 are well known, [1,4]. For a direct proof of Helly's Theorem 1.1 see, for example, [25].…”

“…In Section 5 we give two generalized versions of Helly's theorem for functions defined on abstract linearly ordered sets. Namely, Theorems 5.3 and 5.5 which are partial generalizations of Fuchino-Plewik [8,Theorem 7] (full generalization under s = ℵ 1 where s denotes the splitting number [8], in particular this is the case under the Continuum Hypothesis) and Belov-Chistyakov [1,Theorem 1].…”

On certain convex compactifications for relaxation in evolution problems

“…The following historical notes are worth: Generalization of the classical Helly theorem [21] to vector-valued bounded-variation mappings is by [3], to metric-space-valued bounded-variation mappings is by [8,9] (assuming, in addition, continuity over [0, T ]) and also [5]. A generalization to topological-space-valued mappings by using the general dissipation distance is by [28,Sec.3].…”