Multifunctions determined by integrable functions

Abstract

Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in the sense of Bochner, McShane or Birkhoff can be transferred to the generated multifunction while Henstock integrability does not guarantee it.keyword: Positive multifunction, gauge integral, selection, multifunction determined by a function, measure theory.[MSC 2010] 28B20, 26E25, 26A39, 28B05, 46G10, 54C60, 54C65

Decompositions of Weakly Compact Valued Integrable Multifunctions

“…Of course, if 0 ∈ Γ(t) for almost every t ∈ [0, 1], then Γ is positive. As regards other definitions of measurability and integrability that are treated here and are not explained and the known relations among them, we refer to [3,[15][16][17][18][19][20]26,36,38,[40][41][42], in order not to burden the presentation.…”

“…Proof. Let g : [0, 1] → X be Pettis but not McShane integrable and let G(t) := conv{0, g(t)} be the multifunction determined by g. Then G is positive and Pettis integrable (see [20] (Proposition 2.3)). But according to [20] (Theorem 2.7) G is not McShane integrable.…”

“…Below, we make usage of multifunctions determined by functions, that is the multifunctions of the shape G(t) = conv{0, g(t)}, where g is a Banach space valued function. We refere to [20], for the relations of integrability between g and G. At this stage we recall only that Henstock integrability of g, in general, does not imply Henstock integrability of G. In fact let g be a Henstock but non McShane integrable function. If, by contradiction, G is Henstock integrable then, by [18] (Proposition 3.1), G is McShane integrable and then, by [20] (Theorem 2.7), g is McShane integrable.…”

“…We refere to [20], for the relations of integrability between g and G. At this stage we recall only that Henstock integrability of g, in general, does not imply Henstock integrability of G. In fact let g be a Henstock but non McShane integrable function. If, by contradiction, G is Henstock integrable then, by [18] (Proposition 3.1), G is McShane integrable and then, by [20] (Theorem 2.7), g is McShane integrable. For the relations among different types of integrability for vector valued functions we refer also to [51].…”

On Regulated Solutions of Impulsive Differential Equations with Variable Times

“…where f : [a, b] × R → R, if not otherwise stated, is a continuous function; τ(x) ⊂ [a, b] is at most countable; and I r , I l : R → R and x 0 ∈ R. Our consideration is presented for single-valued problems, but it is still valid for multivalued problems, as can be observed in [3,11], eventually by using multivalued integration [12][13][14].…”