Doss ρ-Almost Periodic Type Functions in Rn

Abstract

In this paper, we investigate various classes of multi-dimensional Doss ρ-almost periodic type functions of the form F:Λ×X→Y, where n∈N,∅≠Λ⊆Rn, X and Y are complex Banach spaces, and ρ is a binary relation on Y. We work in the general setting of Lebesgue spaces with variable exponents. The main structural properties of multi-dimensional Doss ρ-almost periodic type functions, like the translation invariance, the convolution invariance and the invariance under the actions of convolution products, are clarified. We examine connections of Doss ρ-almost periodic type functions with (ω,c)-periodic functions and Weyl-ρ-almost periodic type functions in the multi-dimensional setting. Certain applications of our results to the abstract Volterra integro-differential equations and the partial differential equations are given.

Multi-dimensional Besicovitch almost periodic type functions and applications

“…Now we would like to recall that any Doss-p-almost periodic function F : [0, ∞) → Y , where p ∈ [1, ∞), can be extended to a Doss-p-almost periodic function F : R → Y defined by F (t) := 0, t < 0 (cf. [46] for the notion used in this paragraph). A similar type of extension can be achieved in a much more general situation; for example, we know that, under certain reasonable conditions, any Doss-(p, φ, F, B, Λ ′ , ρ)-almost periodic function F : Λ × X → Y can be extended to a Doss-(p, φ, F, B, Λ ′ , ρ 1 )-almost periodic function F : R n × X → Y, defined by F (t) := 0, t / ∈ Λ, F (t) := F (t), t ∈ Λ, with ρ 1 := ρ ∪ {(0, 0)} (the corresponding analysis from [46] contains small typographical errors that will be corrected in our forthcoming monograph [42]).…”

“…Here it worth noticing that the concept Besicovitch-p-almost periodicity has not been well explored for the functions of the form F : Λ → X, where ∅ = Λ ⊆ R n and Λ = R n (some particular results in the one-dimensional setting are given in the monograph [40], with Λ = [0, ∞)). This fact has strongly influenced us to write this paper, in which we continue the research studies [17,18,28,39,46] by investigating the multidimensional Besicovitch almost periodic type functions F : Λ × X → Y, where (Y, • Y ) is a complex Banach space and ∅ = Λ ⊆ R n . It is worth noting that this is probably the first research article which examines the existence and uniqueness of Besicovitch-p-almost periodic solutions for certain classes of PDEs on some proper subdomains of R n ; even the most simplest examples of quasi-linear partial differential equations of first order considered here vividly exhibit the necessity of further analyses of Besicovitch-p-almost periodic functions which are not defined on the whole Euclidean space R n .…”

Metrical approximations of functions

“…Let us note that the statement of Proposition 2.9 can be simply reformulated for Doss-(P, φ, F, B, Λ ′ , ρ)-almost periodic type functions (cf. also [25,Proposition 1]). Furthermore, the following result can be deduced following the lines of proof of [24, Proposition 2.4] (for simplicity, we assume here that the function F does not depend on the second argument): Proposition 3.12.…”

Multi-dimensional besicovitch almost periodic type functions and applications

“…Now we would like to recall that any Doss-p-almost periodic function F : [0, ∞) → Y , where p ∈ [1, ∞), can be extended to a Doss-p-almost periodic function F : R → Y defined by F (t) := 0, t < 0 (cf. [31] for the notion used in this paragraph). A similar type of extension can be achieved in a much more general situation; for example, we know that, under certain reasonable conditions, any Doss-(p, φ, F, B, Λ , ρ)-almost periodic function F : Λ × X → Y can be extended to a Doss-(p, φ, F, B, Λ , ρ 1 )-almost periodic function F : R n × X → Y, defined by F (t) := 0, t / ∈ Λ, F (t) := F (t), t ∈ Λ, with ρ 1 := ρ ∪ {(0, 0)} (the corresponding analysis from [31] contains small typographical errors that will be corrected in our forthcoming monograph [30]).…”

“…Here it worth noticing that the concept Besicovitch-p-almost periodicity has not been well explored for the functions of the form F : Λ → X, where ∅ = Λ ⊆ R n and Λ = R n (some particular results in the one-dimensional setting are given in the monograph [28], with Λ = [0, ∞)). This fact has strongly influenced us to write this paper, in which we continue the research studies [10,11,20,27,31] by investigating the multi-dimensional Besicovitch almost periodic type functions F : Λ × X → Y, where (Y, • Y ) is a complex Banach space and ∅ = Λ ⊆ R n . It is worth noting that this is probably the first research article which examines the existence and uniqueness of Besicovitch-p-almost periodic solutions for certain classes of PDEs on some proper subdomains of R n ; even the most simplest examples of quasi-linear partial differential equations of first order considered here vividly exhibit the necessity of further analyses of Besicovitch-p-almost periodic functions which are not defined on the whole Euclidean space R n .…”