Multi-dimensional $ρ$-almost periodic type functions and applications

Abstract

In this paper, we analyze various classes of multi-dimensional ρalmost periodic type functions F : I × X → Y and multi-dimensional (ω, ρ)almost periodic type functions F : I ×X → Y, where n ∈ N, ∅ = I ⊆ R n , X and Y are complex Banach spaces and ρ is a binary relation on Y. The proposed notion is new even in the one-dimensional setting, for the functions of the form F : I → Y. The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations, as well.2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. Multi-dimensional ρ-almost periodic functions, multi-dimensional (ω, ρ)-almost periodic functions, multi-dimensional (ω j , ρ j ) j∈Nn -periodic functions, abstract Volterra integro-differential equations.The third named author is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia.

Metrical almost periodicity and applications

“…We will prove the statement for Bohr (B, I ′ , ρ, P)-almost periodic functions. As in [13], we have I + (I ′ − I ′ ) ⊆ I. Further on, let ǫ > 0 and B ∈ B be given.…”

“…Concerning some subjects not considered in this paper, we would like to mention that we have recently analyzed, in [5]- [6] and [13], various notions of D-asymptotical periodicity and D-asymptotical automorphy in the multi-dimensional setting. We will not analyze these topics here.…”

“…The organization and main ideas of this paper, which reconsiders and continues our former research studies A. Chávez et al [5]- [6] and M. Fečkan et al [13], can be briefly described as follows. In Subsection 1.1, we recall the basic definitions and results from the theory of Lebesgue spaces with variable exponents.…”

“…The notion of Bohr (B, I ′ , ρ)-almost periodic function and the notion of (B, I ′ , ρ)unformly recurrent function have recently been introduced by M. Fečkan et al in [13,Definition 2.1]:…”

“…Concerning Bohr (B, I ′ , ρ, P)-almost periodic functions in the finite-dimensional spaces, we will only note that the statement of [13,Proposition 2.20] admits a reformulation in our framework provided that P has a linear vector structure and the metric d is translation invariant.…”