STEPANOV $\rho$-ALMOST PERIODIC FUNCTIONS IN GENERAL METRIC

Abstract

In this paper, we analyze various classes of multi-dimensional Stepanov $\rho$-almost periodic functions in general metric. The main structural properties for the introduced classes of Stepanov almost periodic type functions are established. We also provide an illustrative application to the abstract degenerate semilinear fractional differential equations.

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

“…We continue by stating the following result, which is not so easily comparable to Theorem 3 or [33] (Proposition 2.3): Proposition 3. Suppose that p ∈ [1, ∞), 1/p + 1/q = 1, there exists a finite real constant c > 0, such that ν(t) ≥ c > 0, t ∈ R, that ν : R → (0, ∞) is a Lebesgue measurable function and there exists a function ω : R → [0, ∞), such that ν(x + y) ≤ ν(x)ω(y) for all x, y ∈ R. Suppose further that f : R → X is Stepanov-(p, T, ν)-almost periodic, where ρ = T ∈ L(Y), and (R(t)) t>0 ⊆ L(X, Y) is a strongly continuous operator family satisfying the following conditions:…”

“…Proof. We will provide the main details of the proof since it can be given with the help of the argumentation employed for proving [5] (Proposition 2.6.11) and [33] (Proposition 2.3). Due to (ii), the function F(•) is well defined.…”

“…19), where the Hölder inequality is essentially employed in the proofs, cannot be clarified for Stepanov-p-almost periodic functions with the exponent p ∈ (0, 1). Especially, we would like to emphasize that the composition principles established in [40] (Theorem 2.2), [33] (Theorem 2.2) and [6] (Theorem 6.2.32, Theorem 6.2.33) cannot be clarified for Stepanov-p-almost periodic functions with the exponent p ∈ (0, 1).…”

Asymptotically ρ-almost periodic type functions in general metric

“…Further on, in our recent research study [13], we have initiated the study of multi-dimensional ρ-almost periodic functions in general metric. The Stepanov class of metrical multi-dimensional ρ-almost periodic functions and the Weyl class of metrical multi-dimensional ρ-almost periodic functions have recently been analyzed in [14] and [15], respectively.…”

“…The main aim of this research paper is to continue the research studies [13,14,15] by investigating various classes of multi-dimensional asymptotically ρ-almost periodic type functions in general metric. More precisely, we analyze here the following classes of multi-dimensional asymptotically ρ-almost periodic type functions:…”