Existence theory for semilinear evolution inclusions involving measures

Abstract

We provide existence results for semilinear differential inclusions involving measures: trueleftdu∈Au0.28em0.28emdt+Ffalse(t,ufalse)0.28emdg0.28em,1emt∈false[0,1false],leftu(0)=u0,where A is the infinitesimal generator of a C0‐semigroup {T(t),t≥0} of contractions on a separable Banach space X and g:[0,1]→R is a right‐continuous non‐decreasing function. The existence of mild solutions, as well as the compactness of the solution set are obtained via Kakutani–Ky Fan's fixed point theorem in the space of regulated functions endowed with weak, respectively strong topologies. Some examples of special cases covered by our existence results have been included.

Solvability for a Class of Integro-Differential Inclusions Subject to Impulses on the Half-Line

“…On the other hand, we can also use Proposition 2 and thus infer that the multimap F defined in (21) satisfies property (F). Therefore, we can apply Theorem 1 to problem (22) and obtain the existence of a function v…”

“…Proposition 2. Under assumptions (g0)-(g4) and (W0)-(W3), the multimap F defined in (21) satisfies (F).…”

“…The arbitrariness of (v 0 , w 0 ) leads to the continuity of f (t, •, •). As a consequence, the multimap F(t, •, •) is the sum of a continuous single-valued function and an upper semicontinuous multimap (see (21) and (W1)), so it is upper semicontinuous as well, i.e., F satisfies (F2).…”

“…Finally, let us fix t ≥ t 0 for which (g4) holds and any bounded Ω 1 , Ω 2 ⊂ L 2 ([0, 1]). Then, by (21), (18), and (W3), we have…”

“…in case we deal with the general integro-differential inclusion (1) or the model's parametric differential Equation (2), respectively. Problems involving instantaneous impulses have been extensively studied in the literature and are still a topic of considerable interest, as can be seen in recent articles [19][20][21][22][23][24][25]. For a first approach to the subject, we refer to the now classic monographs [26,27].…”