On solutions of differential inclusions in homogeneous spaces of functions

Abstract

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.

On uniqueness of fractional powers of multi-valued linear operators and the incomplete Cauchy problem

“…Moreover, in that case, we do not need (6) to prove the previous theorem, since then (α − A) (α − A) −1 = 1, α ∈ ρ(A), and A t,i are single-valued, whenever 0 < t < 1 and i = 1, 2. Thus, the preceding theorem admits a simpler formulation for single-valued operators.…”

“…In [7], a wide coverage of general issues of the theory of this kind of operator could be found. For applications of multi-valued methods to the study of degenerate differential equations, we refer to the monograph [11] (see also [17,Section 6.1]; for more recent studies in the same direction see [3][4][5][6]12]). …”

Теорема Герхарда - Прюсса Для Некоторого Класса Вырожденных Полугрупп Операторов

“…Дифференциальное отношение ℒ 0 рассматривалось в [11]. Для случая полугруп-пы класса ( 0 ) оператор ℒ 0 рассматривался в [8], [9], [12], [13].…”

Controllability for a Class of Degenerate Functional Differential Inclusions in a Banach Space

“…We begin with some necessary definitions and results from the theory of multivalued linear operators. Details can be found in [4,5], and [11].…”