Almost periodic solutions of Volterra difference systems

Koyuncuoğlu, Halis Can; Adıvar, Murat

doi:10.1515/dema-2017-0030

Cited by 6 publications

(4 citation statements)

References 23 publications

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“…The notion of bi-almost periodicity plays an incredible role in the research study [72] by H. C. Koyuncuoǧlu and M. Adıvar, where the authors have analyzed the existence of almost periodic solutions for a class of discrete Volterra systems and the research study [93] by M. Pinto and C. Vidal, where the authors have used the notion of integrable bi-almost-periodic Green functions of linear homogeneous differential equations and the Banach contraction principle to show the existence of almost and pseudoalmost periodic mild solutions for a class of the abstract differential equations with constant delay (see also the research article [26], where A. Chávez, S. Castillo and M. Pinto have used the notion of bi-almost automorphy in their investigation of almost automorphic solutions of abstract differential equations with piecewise constant arguments). The notion of k-bi-almost periodicity was introduced by M. Pinto in [92] and further analyzed in [32,Section 4], where the authors have analyzed the existence and uniqueness of weighted pseudo almost periodic solutions for a class of abstract integrodifferential equations.…”

Section: ) Holds and For Any Sequence Whichmentioning

confidence: 99%

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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Section: ) Holds and For Any Sequence Whichmentioning

confidence: 99%

Multi-dimensional almost periodic type functions and applications

Chávez¹,

Khalil²,

Kostić³

et al. 2020

Preprint

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“…Indeed, almost periodic solutions of Volterra difference equations have taken prominent attention in the existing literature, and there is a vast literature based on the existence of discrete almost periodic solutions for numerous kind of Volterra difference equations. In pioneering paper of S. Elaydi (see [6]) the investigation of sufficient conditions for the existence of discrete almost periodic solutions was stated as an open problem, and [12] (2018) provided a solution to this open problem by using the discrete variant of exponential dichotomy and the fixed point theory. It is clear that the space RDAP (Z : R n ) is a much more larger space than the space of discrete almost periodic functions.…”

Section: Generalized Almost Periodic Functions and Applicationsmentioning

confidence: 99%

Generalized almost periodic solutions of Volterra difference equations

Kostic,

Koyuncuoğlu

2023

Malaya J. Mat.

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“…In the recent past, the theories of almost periodic and almost automorphic functions have taken prominent attention from scholars, and the existence of almost periodic and almost automorphic solutions of dynamic equations has become a hot research topic on time domains with continuous, discrete, and hybrid structures. We refer readers to the monographs [10,[12][13][14][15], papers [16][17][18][19][20][21][22][23][24][25][26][27], and references therein. Analysis of the linkage between the existence of bounded and periodic solutions of dynamic equations has always been an interesting research topic in applied mathematics.…”

Section: Introductionmentioning

confidence: 99%

Almost Automorphic Solutions to Nonlinear Difference Equations

Kostić,

Koyuncuoğlu,

Fedorov

2023

Mathematics

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In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the contraction mapping principle. As the second objective of the manuscript, we reconsider one of the landmark results, namely the Bohr–Neugebauer theorem, in the qualitative theory of dynamical equations, and we investigate the relationship between the existence of almost automorphic solutions and the existence of solutions with a relatively compact range for the proposed difference equation type. Thus, we provide a discrete counterpart of the Bohr–Neugebauer theorem due to the almost automorphy notion under some technical conditions.

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Almost periodic solutions of Volterra difference systems

Cited by 6 publications

References 23 publications

Multi-dimensional almost periodic type functions and applications

Multi-dimensional almost periodic type functions and applications

Generalized almost periodic solutions of Volterra difference equations

Almost Automorphic Solutions to Nonlinear Difference Equations

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