Levitan/Bohr almost periodic and almost automorphic solutions of second order monotone differential equations

Caraballo, Tomás; Cheban, David

doi:10.1016/j.jde.2011.04.021

Cited by 12 publications

(21 citation statements)

References 19 publications

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“…In our earlier articles [1,2], we have proved some partial results on these problems. However, our main aim in the present paper is to analyze the same kind of questions but for dissipative almost periodic difference equations.…”

Section: Introductionmentioning

confidence: 84%

“…Under appropriate assumptions, every non-autonomous differential/difference equation generates a cocycle (a non-autonomous dynamical system). Several examples can be found, for instance, in [1].…”

Section: 2mentioning

confidence: 99%

“…We can now analyze the questions related to Seifert's problem mentioned in the Introduction of this paper. To be more precise, consider again the initial differential equation (1) x ′ = f (t, x), where f ∈ C(R × R n , R n ). Assume that the right-hand side of (1) satisfies hypotheses ensuring the existence, uniqueness and extendability of solutions of (1), i.e., for all (t 0 , x 0 ) ∈ R × R n there exists a unique solution x(t; t 0 , x 0 ) of (1) with initial data t 0 , x 0 , and defined for all t ≥ t 0 .…”

Section: Analysis Of Seifert's Problem For Dissipative Difference Equmentioning

confidence: 99%

“…We present our results within the framework of general non-autonomous dynamical systems (cocycles) and we apply our abstract theory already developed in some previous papers (see, e.g. [1,2] and the references therein) to study some classes of difference equations.…”

Section: Introductionmentioning

confidence: 99%

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On the structure of the global attractor for non-autonomous difference equations with weak convergence

Caraballo

Cheban

2012

Journal of Difference Equations and Applications

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Abstract. The aim of this paper is to describe the structure of global attractors for non-autonomous difference systems of equations with recurrent (in particular, almost periodic) coefficients. We consider a special class of this type of systems (the so-called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers to study the almost periodic (almost automorphic, recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent) solutions of difference equations.

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Section: Introductionmentioning

confidence: 84%

Section: 2mentioning

confidence: 99%

Section: Analysis Of Seifert's Problem For Dissipative Difference Equmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the structure of the global attractor for non-autonomous difference equations with weak convergence

Caraballo

Cheban

2012

Journal of Difference Equations and Applications

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“…Later the works of B. A. Shcherbakov were extended and generalized by many authors: I. Bronshtein [15,ChIV], T. Caraballo and D. Cheban [16,17,18,19], D. Cheban [20,21], D. Cheban and C. Mammana [23], D. Cheban and B. Schmalfuss [24], and others.…”

Section: Introductionmentioning

confidence: 99%

Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations

Cheban

Liu

2020

Journal of Differential Equations

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The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudorecurrence, Poisson stability) of solutions for semi-linear stochastic equationwith exponentially stable linear operator A and Poisson stable in time coefficients f and g. We prove that if the functions f and g are appropriately "small", then equation ( * ) admits at least one solution which has the same character of recurrence as the functions f and g.Definition 2.4. Let ε > 0. A number τ ∈ R is called ε-almost period of the function ϕ if ρ(ϕ(t + τ ), ϕ(t)) < ε for all t ∈ R. Denote by T (ϕ, ε) the set of ε-almost periods of ϕ.Definition 2.5. A function ϕ ∈ C(R, X) is said to be Bohr almost periodic if the set of ε-almost periods of ϕ is relatively dense for each ε > 0, i.e. for each ε > 0 there exists l = l(ε) > 0 such that T (ϕ, ε) ∩ [a, a + l] = ∅ for all a ∈ R.Definition 2.6. A function ϕ ∈ C(R, X) is said to be pseudo-periodic in the positive (respectively, negative) direction if for each ε > 0 and l > 0 there exists a ε-almost period τ > l (respectively, τ < −l) of the function ϕ. The function ϕ is called pseudo-periodic if it is pseudo-periodic in both directions. Definition 2.7. For given ϕ ∈ C(R, X), denote by ϕ h the h-translation of ϕ, i.e. ϕ h (t) = ϕ(h + t) for t ∈ R. The hull of ϕ, denoted by H(ϕ), is the set of all the limits of ϕ hn in C(R, X), i.e. H(ϕ) := {ψ ∈ C(R, X) : ψ = lim n→∞ ϕ hn for some sequence {h n } ⊂ R}.

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Sensitivity analysis for a fractional stochastic differential equation with $$S^{p}$$-weighted pseudo almost periodic coefficients and infinite delay

Zhang

2022

Fract Calc Appl Anal

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Levitan/Bohr almost periodic and almost automorphic solutions of second order monotone differential equations

Cited by 12 publications

References 19 publications

On the structure of the global attractor for non-autonomous difference equations with weak convergence

On the structure of the global attractor for non-autonomous difference equations with weak convergence

Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations

Sensitivity analysis for a fractional stochastic differential equation with $$S^{p}$$-weighted pseudo almost periodic coefficients and infinite delay

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