Multi-dimensional (ω, c)-almost periodic type functions and applications

M Kostić

doi:10.1515/msds-2020-0130

Abstract

Abstract In this paper, we analyze various classes of multi-dimensional (ω, c)-almost periodic type functions with values in complex Banach spaces. The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our abstract theoretical results to the abstract Volterra integro-differential equations, as well.

Highlights

Introduction and preliminariesThe notion of an almost periodic function was introduced by H
We provide certain applications of our abstract theoretical results to the abstract Volterra integro-di erential equations, as well
We have recently extended the notion of (ω, c)-periodicity by examining various classes of (ω, c)-almost periodic type functions

Summary

Introduction and preliminaries

The notion of an almost periodic function was introduced by H. Suppose that ωj ∈ R \ { }, cj ∈ C \ { }, M > , ωj ej + I ⊆ I ( ≤ j ≤ n), the set I is closed, the function F : I → X is (ωj , cj)j∈Nn -periodic, |cj| ≤ for all j ∈ Nn and, for every t = (t , t , · · ·, tn) ∈ I, there exist a point η = (η , η , · · ·, ηn) ∈ IM and integers kj ∈ N ( ≤ j ≤ n) such that tj = kj ωj + ηj ( ≤ j ≤ n). Following our idea from [16, De nition 2.1], we can introduce and analyze several various generalizations of the class of multi-dimensional (ωj , cj)j∈Nn -periodic functions with the help of Proposition 2.11. Suppose that ωj ∈ R\{ }, cj ∈ C\{ } and ωj ej+I ⊆ I ( ≤ j ≤ n); if a function F : I → X is (ωj , cj)j∈Nn -periodic, for each j

Nn and for every k

After multiplication with c t j j