Almost periodic homogeneous linear difference systems without almost periodic solutions

Veselý, Michal

doi:10.1080/10236198.2011.585984

Cited by 24 publications

(5 citation statements)

References 24 publications

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“…Before proceeding to the next subsection, we would like to recommend for the readers the doctoral dissertation of M. Veselý [14], where we have located a great number of results about the existence and uniqueness of almost periodic solutions of the abstract difference equation u(k + 1) = A k u(k) + f (k), k ∈ Z, where (A k ) k∈Z is a sequence of closed linear operators satisfying certain properties.…”

Section: On the Abstract Difference Equationmentioning

confidence: 99%

“…It is not difficult to prove that, for every almost periodic sequence (x k ) k∈N in X, there exists a unique almost periodic sequence ( xk ) k∈Z in X such that xk = x k for all k ∈ N, so that a sequence (x k ) k∈N in X is almost periodic if and only if there exists an almost periodic function f : [0, ∞) → X such that x k = f (k) for all k ∈ N. The class of almost periodic sequences is essentially important in the analysis of the qualitative properties of solutions for various classes of impulsive Volterra integro-differential equations, Volterra integro-difference equations and ordinary differential equations; cf. the research monographs [10,12,13] and the doctoral dissertation [14] for some results obtained in this direction.…”

Section: Introductionmentioning

confidence: 99%

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Generalized ρ-Almost Periodic Sequences and Applications

Kostić

Chaouchi

et al. 2023

Fractal Fract

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In this paper, we analyze the Bohr ρ-almost periodic type sequences and the generalized ρ-almost periodic type sequences of the form F:I×X→Y, where ∅≠I⊆Zn, X and Y are complex Banach spaces and ρ is a general binary relation on Y. We provide many structural results, observations and open problems about the introduced classes of ρ-almost periodic sequences. Certain applications of the established theoretical results to the abstract Volterra integro-difference equations are also given.

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Section: On the Abstract Difference Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized ρ-Almost Periodic Sequences and Applications

Kostić

Chaouchi

et al. 2023

Fractal Fract

View full text Add to dashboard Cite

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“…Note that, in Hasil and Veselý, 45 there is mentioned that Equation (1.2) is oscillatory for some functions R,S also satisfying (5.4). It is based on constructions published in Veselý 55 (see also studies [56][57][58][59] in the discrete case). However, the existence of such coefficients R,S follows from results about equations with perturbations as well (see Elbert and Schneider 60 and also other studies [61][62][63][64] ).…”

Section: Corollaries and Commentsmentioning

confidence: 99%

Non‐oscillation of linear and half‐linear differential equations with unbounded coefficients

Šišoláková

2020

Math Methods in App Sciences

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We deal with Euler-type half-linear second-order differential equations, and our intention is to derive conditions in order their non-trivial solutions are non-oscillatory. This paper connects to the article P. Hasil, J. Šišoláková, M. Veselý: Averaging technique and oscillation criterion for linear and half-linear equations, Appl. Math. Lett. 92 (2019), 62-69, where the corresponding oscillatory counterpart is studied and an oscillation criterion is established. The used effective technique for this investigation is the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This paper is completed by two corollaries and an example concerning the linear case when we obtain new results as well.

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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 homogeneous linear difference systems without almost periodic solutions

Cited by 24 publications

References 24 publications

Generalized ρ-Almost Periodic Sequences and Applications

Generalized ρ-Almost Periodic Sequences and Applications

Non‐oscillation of linear and half‐linear differential equations with unbounded coefficients

Almost Automorphic Solutions to Nonlinear Difference Equations

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