Selection Properties and Set-Valued Young Integrals of Set-Valued Functions

Abstract

The paper deals with some selection properties of set-valued functions and different types of set-valued integrals of a Young type. Such integrals are considered for classes of Hölder continuous or with bounded Young p-variation set-valued functions. Two different cases are considered, namely set-valued functions with convex values and without convexity assumptions. The integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion.

On a Set-Valued Young Integral with Applications to Differential Inclusions

“…This allows to establish the compactness of the integral, to get the upper semicontinuity of the integral with respect to F , and then to establish the existence of solutions to some differential inclusions. Note that our integral converges to that of Michta and Motyl [26] when the tuning parameter r goes to +∞. However, our integral is always compact-valued, whereas that of [26] may be unbounded, see Example 3.11 below.…”

“…Note that our integral converges to that of Michta and Motyl [26] when the tuning parameter r goes to +∞. However, our integral is always compact-valued, whereas that of [26] may be unbounded, see Example 3.11 below.…”

“…and then J T,α,r (F, w) is a closed subset of R e . [26] is equal to R, whereas, by Proposition 3.10, J T,α,r (F, w) is a compact interval. Assume that β 1 2 , and let w(t) = t 2β cos(π/t) and…”

“…Remark 3.9. Michta and Motyl define a larger Aumann-Young integral in [26]. For convex-valued F , their integral is…”

“…In the case of a deterministic signal w with possibly infinite variation, Michta and Motyl [26,27] are the only references so far defining a set-valued Young integral à la Aumann of the form (1), for convex as well as nonconvex-valued multifunctions. In their approach, the set of selections S(F ) is large, namely, in the case of our setting, S(F ) is the set of all α-Hölder continuous selections of F .…”