Differential inclusions and multivalued integrals

Abstract

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x (t) ∈ F (t, x(t)), x(0) = g(x), t ∈ [0, T ] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented by a short survey about multivalued integration including Pettis and Henstock-Kurzweil-Pettis multivalued integrals.

On the solutions of Caputo–Hadamard Pettis-type fractional differential equations

“…Recall that (see e.g. [8][9][10][11][12][13][14][15]) the weakly measurable function x : I → E is said to be ψ-Dunford (where ψ is a Young function) integrable on I if and only if ϕx ∈ L ψ (I ) for each ϕ ∈ E * .…”

“…Let us briefly explain how to extend the above results for multivalued problems. We extend both the case of classical Pettis integrals (i.e., α = 1) [12]) and the case of Caputo (and Riemann-Liuoville) integrals [9]. We should mention that as the Hadamard integral is not of a convolution of a real-valued integrable function with the Pettis integrable vector-valued one, we are unable to apply the property of such integrals proved in [9], But still we can apply Proposition 5 instead.…”

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].…”

Multifunctions determined by integrable functions

“…In particular the use of intervals to represent uncertainty in the area of decision and information theory has been suggested by several authors. At the same time, positive interval-valued multifunctions have also played an important role in applications and they arise quite naturally, for example, in the context of fractal image coding, as shown in [34] or in differential inclusions (see for example [17,22,23]).…”