Oscillation and nonoscillation criteria for delay differential equations

Elbert, Árpád; Stavroulakis, I. P.

doi:10.1090/s0002-9939-1995-1242082-1

Cited by 53 publications

(37 citation statements)

References 2 publications

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“…In contrast to the autonomous case (4.4), the converse of Corollary 4.3 in general is not true. Under the hypotheses of the corollary, it may happen that all solutions of (4.1) are oscillatory and the characteristic equation (4.5) of the limiting equation (4.4) has a real root (see the example in [12, p. 320] or [4,Remark 3]). …”

Section: Proposition 42 Consider the Equationmentioning

confidence: 99%

Asymptotic behavior and oscillation of functional differential equations

Pituk

2006

Journal of Mathematical Analysis and Applications

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Asymptotic relations between the solutions of a linear autonomous functional differential equation and the solutions of the corresponding perturbed equation are established. In the scalar case, it is shown that the existence of a nonoscillatory solution of the perturbed equation often implies the existence of a real eigenvalue of the limiting equation. The proofs are based on a recent Perron type theorem for functional differential equations.

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Section: Proposition 42 Consider the Equationmentioning

confidence: 99%

Asymptotic behavior and oscillation of functional differential equations

Pituk

2006

Journal of Mathematical Analysis and Applications

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“…On the basis of this method, Koplatadze and Chanturiya [9] have constructed the oscillation theory for a wide class of second and higher order nonlinear functional-differential equations. In the present paper, using the above mentioned method, new and optimal in a certain sense conditions guarantee, respectively, the oscillation of all proper solutions of equation (1) and the existence of both remote from zero and vanishing at infinity proper Kneser solutions of that equation are obtained. The next proposition follows immediately from condition (2).…”

Section: Msc 2010: 34c15 34k11 34k12mentioning

confidence: 96%

“…A continuous function u : [a, +∞[ → ℝ is said to be a solution of the differential equation (1) if it is continuously differentiable in some interval ]a , +∞[ ⊂ ]a, +∞[, and in this interval it satisfies equation (1). • A solution u of the differential equation (1), defined on the interval [a, +∞[, is said to be proper if it is not identically zero in any neighborhood of +∞, i.e., there exists a sequence t k ∈ [a , +∞[ (k = , , .…”

Section: Msc 2010: 34c15 34k11 34k12mentioning

confidence: 99%

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Oscillatory and monotone solutions of first-order nonlinear delay differential equations

Partsvania

Sokhadze

2016

Georgian Mathematical Journal

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For first order nonlinear delay differential equations, necessary and sufficient conditions are established for the oscillation of all proper solutions as well as for the existence of at least one vanishing at infinity proper Kneser solution.MSC 2010: 34C15, 34K11, 34K12In the interval ℝ + = [ , +∞[, we consider the differential equationwhere f : ℝ + × ℝ → ℝ and τ : ℝ + → ℝ are continuous functions. Moreover, the function τ satisfies the following conditions:We use the following definitions: • Let a ≥ . A continuous function u : [a, +∞[ → ℝ is said to be a solution of the differential equation (1) if it is continuously differentiable in some interval ]a , +∞[ ⊂ ]a, +∞[, and in this interval it satisfies equation (1). • A solution u of the differential equation (1), defined on the interval [a, +∞[, is said to be proper if it is not identically zero in any neighborhood of +∞, i.e., there exists a sequence t k ∈ [a , +∞[ (k = , , . . .) such that lim k→+∞ t k = +∞ and u(t k ) ̸ = (k = , , . . . ).• A proper solution of equation (1) is said to be oscillatory if it has a sequence of zeros tending to +∞; otherwise it is said to be nonoscillatory. • A proper solution u of equation (1) is said to be a Kneser solution if there exists t > such that• A Kneser solution u of equation (1) is said to be vanishing at infinity if lim t→+∞ u(t) = , and it is said to be remote from zero if lim t→+∞ u(t) ̸ = .

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“…Condition (2) shows that the oscillation of all solutions of the sublinear equation (1) is determined only by the coefficient p(t), and is independent of the delay τ(t).…”

Section: Introductionmentioning

confidence: 99%