Almost periodic and almost automorphic solutions of linear
differential equations

Caraballo, Tomás; Cheban, David

doi:10.3934/dcds.2013.33.1857

Cited by 6 publications

(1 citation statement)

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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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