Doss <i>ρ</i>-Almost Periodic Type Functions in <inline-formula><math display="inline"><semantics><mrow><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>^<i>n</i>

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.

Keywords

• Doss <i>ρ</i>-almost periodic type functions in ℝ^<i>n</i>
• Lebesgue spaces with variable exponents
• abstract Volterra integro-differential equations