Quasi-Asymptotically Almost Periodic Functions and Applications

Abstract

The main aim of this paper is to consider the classes of quasiasymptotically almost periodic functions and Stepanov quasi-asymptotically almost periodic functions in Banach spaces. These classes extend the well known classes of asymptotically almost periodic functions, Stepanov asymptotically almost periodic functions and S-asymptotically ω-periodic functions with values in Banach spaces. We investigate the invariance of introduced properties under the action of finite and inifinite convolution products, providing also an illustrative application to abstract nonautonomous differential equations of first order.2010 Mathematics Subject Classification. 47A16, 47B37, 47D06. Key words and phrases. Quasi-asymptotically almost periodic functions, Stepanov quasiasymptotically almost periodic functions, convolution products, evolution systems, abstract nonautonomous differential equations of first order.

Almost periodic type functions and densities

“…and the function In Example 3, we will show that, for some concrete choices of sequences (τ n ) n∈N , the function f : R → R, given by (5), is Weyl p-almost automorphic for each finite exponent p ≥ 1. Since any Stepanov p-quasi-asymptotically almost periodic function is Weyl-p-almost periodic (p ≥ 1) in the sense of A. S. Kovanko's approach (see [32,Proposition 2.11]), it is quite reasonable to ask the following:…”

“…The (Stepanov) quasi-asymptotically almost periodic functions have been recently analyzed in [32]. For our further work, it will be necessary to recall the following definition: ap (I : E) for short, iff for each > 0 we can find two real numbers l > 0 and L > 0 such that any interval I ⊆ I of length L contains a point τ ∈ I such that…”

Generalized $c$-almost periodic type functions in ${\mathbb{R}}^{n}$

“…After that, we can apply the first part of proposition. The spaces introduced in Definition 3.1 do not form vector spaces under the pointwise addition of functions and these spaces are not closed under the pointwise multiplication with scalar-valued functions of the same type, as is well known in the one-dimensional case ( [30]). The introduced spaces are homogeneous and, under certain reasonable assumptions, these spaces are translation invariant, invariant under the homotheties with ratio b > 0 and the reflections at zero with respect to the first variable; details can be left to the interested readers.…”

“…Accepting the notation employed in [30] and [37], we have the following (I = R or I = [0, ∞); ω ∈ I):…”

Asymptotically Weyl almost periodic functions in Lebesgue spaces with variable exponents

“…The main purpose of paper [6] was to consider generalized almost periodicity that intermediate Stepanov and Bohr concept. On the other hand, the classes introduced by H. Weyl [27] and A. S. Kovanko [24] are enormously larger compared with the class of Stepanov almost periodic functions and the main purpose of papers [22]- [23] has been to initiate the study of generalized (asymptotical) almost periodicity that intermediate Stepanov and Weyl concept. In these papers, we have introduced the class of Stepanov p-quasi-asymptotically almost periodic functions and proved that this class contains all asymptotically Stepanov p-almost periodic functions and makes a subclass of the class consisting of all Weyl p-almost periodic functions (p ∈ [1, ∞)), taken in the sense of Kovanko's approach [24].…”