Space of Quasi-Periodic Limit Functions and Its Applications

Xie, Rui; Xia, Zhinan; Liu, Junwei

doi:10.3390/math7111132

Cited by 3 publications

(1 citation statement)

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“…For multi-periodic solutions of various classes of ordinary differential equations and partial differential equations, we also refer the reader to [8,9,22,27,36,37,43,52,53,55,56,58]. Especially, we would like to mention the investigations of G. Nadin [44]- [46] concerning the space-time periodic reaction-diffusion equations, G. Nadin-L.Rossi [47] concerning transition waves for Fisher-KPP equations, L. Rossi [51] concerning Liouville type results for almost periodic type linear operators, the investigation of B. Scarpellini [54] concerning the space almost periodic solutions of reaction-diffusion equations, and the recent investigation of R. Xie, Z. Xia, J. Liu [65] concerning the quasi-periodic limit functions, (ω 1 , ω 2 )-(quasi)-periodic limit functions and their applications, given only in the two-dimensional setting.…”

Section: Introductionmentioning

confidence: 99%

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

Fečkan,

Khalladi,

Kostić

et al. 2021

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%

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

Fečkan,

Khalladi,

Kostić

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

Multi-dimensional ρ-almost periodic type functions and applications

Fečkan

Khalladi

Kostić

et al. 2022

Applicable Analysis

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Multi-dimensional almost periodic type functions and applications

Chávez¹,

Khalil²,

Kostić³

et al. 2020

Preprint

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In this paper, we analyze multi-dimensional (R X , B)-almost periodic type functions and multi-dimensional Bohr B-almost periodic type functions. The main structural characterizations and composition principles for the introduced classes of almost periodic functions are established. Several applications of our abstract theoretical results to the abstract Volterra integrodifferential equations in Banach spaces are provided, as well. Examples and applications to the abstractVolterra integro-differential equations 3.1. Application to nonautonomous retarded functional evolution equations 4. Appendix 4.1. n-Parameter strongly continuous semigroups 4.2. Multivariate trigonometric polynomials and approximations of periodic functions of several real variables References 2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. (R, B)-Multi-almost periodic type functions, (R X , B)-multi-almost periodic type functions, Bohr B-almost periodic type functions, composition principles, abstract Volterra integro-differential equations. Marko Kostić is partially supported by grant 451-03-68/2020/14/200156 of Ministry of Science and Technological Development, Republic of Serbia. Manuel Pinto is partially supported by Fondecyt 1170466.The Euler Gamma function is denoted by Γ(•). If t 0 ∈ R n and ǫ > 0, then we set B(t 0 , ǫ)Now we are ready to briefly explain the organization and main ideas of this paper. In Subsection 1.1, we recall the basic facts and definitions about vectorvalued almost periodic functions of several real variables; in Subsection 1.2, we recall some applications of vector-valued almost periodic functions of several real variables made so far. Definition 2.1 and Definition 2.2 introduce the notion of (R, B)-multi-almost periodicity and the notion of (R X , B)-multi-almost periodicity for a continuous function F : I × X → Y, respectively. The convolution invariance of space consisting of all (R X , B)-multi-almost periodic functions is stated in Proposition 2.5, while the supremum formula for the class of (R, B)-multi-almost periodic functions is stated in Proposition 2.6.The notion of Bohr B-almost periodicity and the notion of B-uniform recurrence for a continuous function F : I × X → Y are introduced in Definition 2.9, provided that the region I satisfies the semigroup property I + I ⊆ I. Numerous illustrative examples of Bohr B-almost periodic functions and B-uniformly recurrent functions are presented in Example 2.12 and Example 2.13. In Definition 2.14, we introduce the notion of Bohr (B, I ′ )-almost periodicity and (B, I ′ )-uniform recurrence, provided that ∅ = I ′ ⊆ I ⊆ R n , F : I × X → Y is a continuous function and I + I ′ ⊆ I. After that, we provide several examples of Bohr (B, I ′ )-almost periodic functions and (B, I ′ )-uniformly recurrent functions in Example 2.15. The relative compactness of range F (I × B) for a Bohr B-almost periodic function F : I × X → Y,where B ∈ B, is analyzed in Proposition 2.16. The Bochner criterion for Bohr B-almost periodic functions is sta...

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Space of Quasi-Periodic Limit Functions and Its Applications

Cited by 3 publications

References 19 publications

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

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

Multi-dimensional ρ-almost periodic type functions and applications

Multi-dimensional almost periodic type functions and applications

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