c-Almost periodic type functions and applications

Abstract

In this paper, we introduce several various classes of c-almost periodic type functions and their Stepanov generalizations, where c ∈ ℂ and |c| = 1. We also consider the corresponding classes of c-almost periodic type functions depending on two variables and prove several related composition principles. Plenty of illustrative examples and applications are presented.

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

“…This class of continuous functions has different ergodicity properties compared with the classes of ω-periodic functions and asymptotically ω-periodic functions, and it is not so easily comparable with the class of almost periodic functions since an S-asymptotically and facts about Lebesgue spaces with variable exponents L p(x) (Subsection 1.1), almost periodic type functions in R n (Subsection 1.2), (ω, c)-periodic functions and (ω j , c j ) j∈Nn -periodic functions (Subsection 1.3). Following our approach from [27]- [28] and [35], in Section 2 we introduce and analyze (S, D)-asymptotically (ω, c)-periodic type functions, S-asymptotically (ω j , c j , D j ) j∈Nn -periodic type functions and semi-(c j , B) j∈Nn -periodic functions (the last class of functions is investigated in Subsection 2.1); here, it is worth noting that the notion of (S, D)-asymptotical (ω, c)-periodicity seems to be new even in the one-dimensional setting. Various classes of multi-dimensional quasi-asymptotically c-almost periodic functions are examined in Section 3 following the approach obeyed in [26] and [37], while the Stepanov generalizations of multi-dimensional quasi-asymptotically c-almost periodic type functions are examined in Section 4 (the introduced classes seem to be new and not considered elsewhere even in the case that the exponent p(•) has a constant value).…”

“…Following our approach from [27]- [28] and [35], in Section 2 we introduce and analyze (S, D)-asymptotically (ω, c)-periodic type functions, S-asymptotically (ω j , c j , D j ) j∈Nn -periodic type functions and semi-(c j , B) j∈Nn -periodic functions (the last class of functions is investigated in Subsection 2.1); here, it is worth noting that the notion of (S, D)-asymptotical (ω, c)-periodicity seems to be new even in the one-dimensional setting. Various classes of multi-dimensional quasi-asymptotically c-almost periodic functions are examined in Section 3 following the approach obeyed in [26] and [37], while the Stepanov generalizations of multi-dimensional quasi-asymptotically c-almost periodic type functions are examined in Section 4 (the introduced classes seem to be new and not considered elsewhere even in the case that the exponent p(•) has a constant value). The main aim of Section 5 is to continue our analysis of Weyl c-almost periodic type functions from [26] in the multi-dimensional setting.…”

“…Various classes of multi-dimensional quasi-asymptotically c-almost periodic functions are examined in Section 3 following the approach obeyed in [26] and [37], while the Stepanov generalizations of multi-dimensional quasi-asymptotically c-almost periodic type functions are examined in Section 4 (the introduced classes seem to be new and not considered elsewhere even in the case that the exponent p(•) has a constant value). The main aim of Section 5 is to continue our analysis of Weyl c-almost periodic type functions from [26] in the multi-dimensional setting. Some applications of our results to the abstract Volterra integro-differential equations are presented in Section 6; we also provide numerous illustrative examples henceforth.…”

“…M.T. Khalladi, M. Kostić, M. Pinto, A. Rahmani and D. Velinov [26,Definition 3.1], where the authors have considered the case in which X = {0} and…”

Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions

“…For c = 1 they are reduced to ω-periodic functions, for c = e irt they are reduced to Bloch functions, for c = −1 they are reduced to antiperiodic functions and so on. Furthermore, various generalizations such as c-semiperiodic, c-almost periodic functions were considered in [12,13]. Many other extensions to impulsive, discrete or fractional differential equations have been investigated in [5,6], Agaoglou et al [2] studied the existence and uniqueness of (ω, c)-periodic solutions of impulsive evolution equations in complex Banach spaces, Li et al [16] studied (ω, c)-periodic solutions of impulsive differential with matrix coefficients, Liu et al [17], [18] considered noninstantaneous impulsive differential equations establishing existence and uniqueness of (ω, c)-periodic solutions for semilinear problems.…”