Eberlein-weakly almost periodic (in Stepanov-like sense) functions and application

Abstract

In this article, we prove a number of properties concerning a (new) class of (Stepanov-like) Eberlein-weakly almost periodic (S P -E.w. a. p.) functions with values in a Banach space. We use the results obtained to study the asymptotic behaviour of solutions to the evolution equation:where A is the generator of an integral resolvent family in a Banach space X, a 2 L 1 (R), and f is a given X-valued function on R. The objective is to deduce Eberlein-weakly almost periodicity (in Stepanov-like sense) of the solution u from corresponding properties on the part f.

Stepanov Eberlein almost periodic functions and applications

“…This is a Banach space with respect to the norm ‖.‖ S p . Definition 4 (Ait Dads et al [23]). Let 𝜓 ∈ BS p (R; ).…”

“…For more works related Stepanov space, we refer to the literature [20–22] and the references therein. Later, Ait Dads et al [23] considered the concept of Stepanov Eberlein‐weakly almost periodic functions ( $$$ {S}^p $$$‐E.w.a.p for short) which contain both E‐w.a.p. functions and almost periodic in the sense of Stepanov functions ( $$$ {S}^p $$$‐a.p.…”

“…Whence, the value of our results in theory likewise application. On the other hand, they studied the following Volterra equation [23] $$$$ \frac{d}{dt}z(t)=\int_{-\infty}^ta\left(t-s\right)\left[ Bz(s)+h(s)\right] ds,\mathrm{for}\kern5.14pt t\in \mathrm{\mathbb{R}}, $$$$ where $$$ B $$$ is the generator of an integral resolvent family in a Banach space, $$$ a\in {L}^1\left(\mathrm{\mathbb{R}}\right) $$$. They proved that the solution of Equation $$$ (1) $$$ is $$$ {S}^p $$$‐E.w.a.p.…”

“…of the solution. Indeed, we can apply the method used in Theorem 4.4 to improve Theorem 4.2 [23]. Furthermore, this approach open the way for similar improvement in the literature of E.w.a.p.…”

“…Whence, the value of our results in theory likewise application. On the other hand, they studied the following Volterra equation [23]…”