Almost Periodic Solutions of Abstract Impulsive Volterra Integro-Differential Inclusions

Abstract

In this paper, we introduce and systematically analyze the classes of (pre-)(B,ρ,(tk))-piecewise continuous almost periodic functions and (pre-)(B,ρ,(tk))-piecewise continuous uniformly recurrent functions with values in complex Banach spaces. We weaken substantially, or remove completely, the assumption that the sequence (tk) of possible first kind discontinuities of the function under consideration is a Wexler sequence (in order to achieve these aims, we use certain results about Stepanov almost periodic type functions). We provide many applications in the analysis of the existence and uniqueness of almost periodic type solutions for various classes of the abstract impulsive Volterra integro-differential inclusions.

Stepanov Almost Periodic Type Functions and Applications to Abstract Impulsive Volterra Integro-Differential Inclusions

“…The subsequent result follows from [30] (Theorem 3) and the argument contained in the proof of [30] (Theorem 2). The only thing worth noting is that if 0 < p < 1, then we should replace the number η p k with the number η k throughout the proof of Theorem 2, and assume that η k ∈ (0, ( /4) p δ) for all k ∈ N:…”

“…The following result provides, even for the usually considered exponents p ≥ 1, an extension of [30] (Theorem 1) for pre-(B, T, (t k ))-piecewise continuous almost periodic functions (the extension for pre-(B, T, (t k ))-piecewise continuous uniformly recurrent functions can be deduced in a similar manner):…”

“…In our recent joint research article [30] with W.-S. Du and D. Velinov, we recently analyzed certain relations between piecewise continuous almost periodic functions (piecewise continuous uniformly recurrent functions) and Stepanov almost periodic functions (Stepanov uniformly recurrent functions). We start this subsection by recalling the following notion: Definition 6.…”

“…(cf. [30] (Definition 6 (i))) Suppose that ρ is a binary relation on Y, the function F : R × X → Y [F : [0, ∞) × X → Y] satisfies that for every x ∈ X, the function t → F(t; x), t ∈ R is piecewise continuous with the possible first kind discontinuities at the points of a fixed sequence…”

“…Section 3.1 investigates the relations between piecewise continuous almost periodic functions and metrically Stepanov-p-almost periodic functions (p > 0). In this subsection, we provide proper generalizations of [30] (Theorem 1, Theorem 2) concerning the relations between the class of pre-(B, T, (t k ))-piecewise continuous almost periodic functions and the class S (F,T,P t ,P ) Ω,Λ Section 3.2 investigates the invariance of Stepanov-p-almost periodicity under the actions of the infinite convolution products (0 < p < 1) and provides certain applications of the introduced notion to the abstract Volterra integro-differential equations. The main result of this subsection are Theorem 3 and Proposition 3 (concerning applications, we thought it necessary to emphasize at the very beginning that the situation is very complicated in the case where 0 < p < 1, since the reverse Hölder inequality is valid in our new framework (for more detailed information, see, e.g., [31] (Proposition 3))).…”