Y. G. Sficas

Necessary and sufficient conditions for oscillations of neutral differential equations

Abstract Consider the neutral delay differential equation x′(t) + px′(t–τ) + qx(t-σ) = 0, t ≧ t 0 , (∗) where τ, q, and σ are positive constants while p is a real parameter, Then a necessary and sufficient condition for all solutions of (∗) to oscillate is that the characteristic equation λ + pλe−λτ + qe−λσ = 0 of (∗) has no real roots.

Stable steady state of some population models

Applying an analytical method and several limiting equations arguments, some sufficient conditions are provided for the existence of a unique positive equilibriumK for the delay differential equationx=−γx+D(xt), which is the general form of many population models. The results are concerned with the global attractivity, uniform stability, and uniform asymptotic stability ofK. Application of the results to some known population models, which shows the effectiveness of the methods applied here, is also presented.

Global attractivity in a nonlinear difference equation

Abstract We consider the (nonlinear) difference equation x n = a + ∑ k=1 m b k x n −k , n = 0,1,2… where a and bk (k = 1, 2, …, m) are nonnegative numbers with B ≡ ∑mk = 1bk > 0, and we are interested in whether all positive solutions are attracted by the positive equilibrium L = ( a 2 ) + ( a 2 ) 2 + B .

An oscillation criterion for first order linear delay differential equations

A new oscillation criterion is given for the delay differential equation x0(t) + p(t)x (t ú(t)) = 0, where p, ú 2 C ([0Ò1)Ò [0Ò1)) and the function T defined by T(t) = t ú(t), t 1⁄2 0 is increasing and such that limt!1 T(t) = 1. This criterion concerns the case where lim inf t!1 R t T(t) p(s) ds e . Received by the editors November 27, 1996. AMS subject classification: 34K15.

Necessary and Sufficient Conditions for Oscillations of Second-Order Neutral Equations

Consider the neutral delay differential equation d2dt2 [y(t) + Py(t − τ)] + qy(t − σ) = 0, (1) where p, q ϵ R and τ, σ ϵ R+. We proved that Eq. (1) oscillates if and only if the characteristic equation λ2 + λ2pe−λτ + qe−λσ = 0 has no real roots.

Oscillations in higher-order neutral differential equations

Consider the n-th order (n ≥ 1) neutral differential equation ... (formule)... where δ ∈ {0,+1,−1}, ζ ∈ {+1,−1}, −∞ < τ 1 < τ 2 < ∞ WITH τ 1 t 2 ¬= 0, −∞ < σ 1 < σ 2 < ∞ AND μ AND η ARE INCREASING REAL-VALUED FUNCTIONS ON [τ 1 , τ 2 ] and [σ 1 , σ 2 ] respectively. The function μ is assumed to he not constant on [τ 1 , τ] and [τ, τ 2 ] for every τ ∈ (τ 1 , τ 2 ); similarly, for each σ ∈ (σ 1 , σ 2 ), it is supposed that η is not constant on [σ 1 , σ] and [σ, σ 2 ]

Oscillations in neutral equations with periodic coefficients

Therefore if Q is not identically zero and if Q is (»periodic with r and aintegral multiples of co, then (7) reduces to (5) where t, and ax are defined by(3). The fact that (5) characterizes the oscillatory behavior of ( 1 ) is a remarkableresult that is not obvious.The oscillatory behavior of neutral differential equations has been the subjectof many recent investigations. See, for example [l]-[5], [8], and [12] and thereferences cited therein. The technique that we employ in the proof of Theorem1 was initiated in [9] and [ 10] and had been used successfully in linear neutralautonomous equations. See, for example, [4], [5], and [13]. It is, however,surprising that the same technique may be modified to also apply to equationswith periodic coefficients.Let y = max[r, a). By a solution of (1) we mean a function x where x £C[[t0 - y, oc), R], for some t0 = 0, such that x(t) +px(t - t) is continuouslydifferentiable on [t0, oo) and (1) is satisfied for t = t0 .

ON THE BEHAVIOUR OF THE OSCILLATORY SOLUTIONS OF SECOND-ORDER LINEAR UNSTABLE TYPE DELAY DIFFERENTIAL EQUATIONS

Second-order linear (non-autonomous as well as autonomous) delay differential equations of unstable type are considered. In the non-autonomous case, sufficient conditions are given in order that all oscillatory solutions are bounded or all oscillatory solutions tend to zero at

On the nature of the nonoscillatory solutions of a class of neutral delay differential equations

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Sharp Conditions for the Oscillation of Delay Difference Equations

. In the case where the equations are autonomous, necessary and sufficient conditions are established for all oscillatory solutions to be bounded or all oscillatory solutions to tend to zero at