OSCILLATIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we obtain some sufficient conditions for the oscillation and nonoscillation of some first order linear neutral functional differential equations. A mistake in the literature is also presented at the end of the paper.     

OSCILLATIONS OF NEUTRAL DIFFERENTIAL DIFFERENCE EQUATIONS

Recently the authors have studied the oscillations of some neutral differential difference equations and obtained very good results (see [1—4]). In this paper we consider the oscillations of the neutral differential difference equationd/dt[x(t)+sum from i=1 px(t-τ)+sum form j=1 to n qx(t-σ)=0, t≥t,] (*)where p, τ, q and σ (i=1, 2, …, m; j=1, 2, …, n) are positive constants. Some sufficient conditions for all solutions of (*) to oscillate are obtained. And in some ease we give neeessaxy and sufficient conditions for (*) to oscillate.     

OSCILLATIONS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH IMPULSES

１ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｐａｐｅｒ，ｗｅｃｏｎｓｉｄｅｒｔｈｅｓｅｃｏｎｄ－ｏｒｄｅｒｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈｉｍｐｕｌｓｅｓａｓ（ｒ（ｔ）ｘ′）′＋ｆ（ｔ，ｘ）＝０，ｔ≥ｔ０，ｔ≠０，ｋ＝１，２，…，ｘ（ｔ＋ｋ）...     

OSCILLATIONS OF FIRST ORDER NONLINEAR RETARDED DIFFERENTIAL EQUATIONS

ＯＳＣＩＬＬＡＴＩＯＮＳＯＦＦＩＲＳＴＯＲＤＥＲＮＯＮＬＩＮＥＡＲＲＥＴＡＲＤＥＤＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＷａｎｇＱｉｒｕ（王其如）（ＬｕｏｙａｎｇＴｅａｃｈｅｒｓ＇Ｃｏｌｌｅｇｅ，洛阳师范高等专科学校，邮编：４７１０２２）Ａｂｓ...     

OSCILLATION OF EVEN ORDER NEUTRAL PARTIAL DIFFERENTIAL EQUATIONS

ＯＳＣＩＬＬＡＴＩＯＮＯＦＥＶＥＮＯＲＤＥＲＮＥＵＴＲＡＬＰＡＲＴＩＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥ＬｉｕＢｉｎｇ（刘斌）（ＨｕｂｅｉＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒｔｈｅｓｕｆｆｉｃｉｅｎｔ...     

LINEARIZED OSCILLATIONS FOR NEUTRAL DIFFERENTIAL EQUATIONS

Consider the nonlinear neutral delay differential equationwhere P, Q ∈ C([t0,∞),R), τ > 0, δ > 0. We obtain a linearized oscillation result by an associate linear equation in the case when liminfP(t) = p0 ∈[-1,0).     

GENERIC OSCILLATIONS OF FIRST ORDER IMPULSIVE DELAY DIFFERENTIAL EQUATIONS

In this paper, we will study generic oscillation and generic nonoscillation of first order impulsive delay differential equations. A necessary and sufficient condition and some sufficient conditions are obtained for both phenomena based on the root of characteristic equation.     

OSCILLATIONS OF NEUTRAL LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS

We consider L. N. F. D. E with oscillating coefficients. We obtain some sufficient conditionsfor oscillations of equations. Results in [3] are generalized.     

OSCILLATIONS OF SOLUTIONS OF NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS AND DELAYS

ＯＳＣＩＬＬＡＴＩＯＮＳ　ＯＦ　ＳＯＬＵＴＩＯＮＳ　ＯＦ　ＮＥＵＴＲＡＬ　ＤＩＦＦＥＲＥＮＴＩＡＬ　ＥＱＵＡＴＩＯＮＳ　ＷＩＴＨ　ＶＡＲＩＡＢＬＥ　ＣＯＥＦＦＩＣＩＥＮＴＳ　ＡＮＤ　ＤＥＬＡＹＳ　ＯＳＣＩＬＬＡＴＩＯＮＳＯＦＳＯＬＵＴＩＯＮＳＯＦＮＥ...     

OSCILLATION IN FIRST ORDER NEUTRAL DIFFERENTIAL EQUATIONS

A new oscillation result is established for the first order neutral differential equationx(t) - px(t - (?)) + qx(t - σ) = 0.The results extends and improves some existing results in the literature.     

OSCILLATORY AND ASYMPTOTIC BEHAVIOUR OF n ORDER NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

This paper considers oscillatory and asymptotic behaviour of following n order neutral functional differential equation:d~n/dt~n[x(t)-cx(t-τ)] (-1)~(n-1) integral from n=-τ~* to 0 x(t θ)dη(θ)=0, (1)_nwhere τ>0, τ~*>0, 1>c>0, η(θ) is nondecreasing function with bounded variation on[-τ~*, 0].In this paper the author obtains some results for any integer n and c∈[0, 1). When c=0 or n=1, these results coincide with the results in G. Ladas's paper [4] and the author's papers [1, 2].     

OSCILLATION CRITERIA FOR A CLASS OF HIGHER ORDER NEUTRAL DIFFERENTIAL EQUATIONS

We establish a new oscillation condition for even order neutral type differential equation of the following formwhere f is continuous and exists 1 ≤ k ≤ m(?)(t) ≡ Tk(t), a, (?)i ∈ C([0, ∞], R), i = 1, …,m. such that 0 ≤ a(t) ≤ L and     

STABILITY OF SOLUTIONS OF FIRST ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS WITH OSCILLATORY COEFFICIENTS

ＳＴＡＢＩＬＩＴＹＯＦＳＯＬＵＴＩＯＮＳＯＦＦＩＲＳＴＯＲＤＥＲＮＯＮＬＩＮＥＡＲＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＯＳＣＩＬＬＡＴＯＲＹＣＯＥＦＦＩＣＩＥＮＴＳ￥ＷａｎｇＱｉｒｕ(王其如）（ＬｕｏｙａｎｇＴｅａｃｈｅｒ?..     

OSCILLATION OF HYPERBOLIC DIFFERENTIAL EQUATIONS OF NEUTRAL TYPE

Neutral hyperbolic differential equations with continuous distributed devia-ting arguments are studied, and sufficient conditions for each solution of the boundary value problem to be oscillatory in a cylindrical domain are obtained.     

NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH SE

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＳＦＯＲＯＳＣＩＬＬＡＴＩＯＮＳＯＦＮＥＵＴＲＡＬＤＥＬＡＹＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＳＥＶＥＲＡＬＰＯＳＩＴＩＶＥＡＮＤＮＥＧＡＴＩＶＥＣＯＥＦＦＩＣＩＥＮＴＳ￥...     

OSCILLATION OF HIGHER ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS

ＯＳＣＩＬＬＡＴＩＯＮＯＦＨＩＧＨＥＲＯＲＤＥＲＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷＩＴＨＰＯＳＩＴＩＶＥＡＮＤＮＥＧＡＴＩＶＥＣＯＥＦＦＩＣＩＥＮＴＳＬｉＷａｎｔｏｎｇ（李万同）（ＺｈａｎｇｙｅＴｅａｃｈｅｒｓ＇Ｃｏｌｌｅｇｅ...     

POSITIVE SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

ＰＯＳＩＴＩＶＥＳＯＬＵＴＩＯＮＳＯＦＮＥＵＴＲＡＬＦＵＮＣＴＩＯＮＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳ￥（黄振勋，高国柱）ＨｕａｎｇＺｈｅｎｘｕｎ（Ｄｅｐｔ．ｏｆＭａｔｈｅｍａｔｉｃｓ，ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ２０...     

ON OSCILLATIONS OF PERTURBED SECOND ORDER DIFFERENTIAL EQUATION

A perturbed second order linear differential equation is considered. A criteria concerning the oscillatory behaviour of solutions of the perturbed equation based on the perturbing function and the solutions of the associated unperturbed equation is given.     

THE COMPARISON THEOREMS OF OSCILLATION OF NEUTRAL DIFFERENTIAL EQUATIONS

ＴＨＥＣＯＭＰＡＲＩＳＯＮＴＨＥＯＲＥＭＳＯＦＯＳＣＩＬＬＡＴＩＯＮＯＦＮＥＵＴＲＡＬＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＳＷａｎｇＬｉａｎｗｅｎ（王连文）（ＳｈａｎｄｏｎｇＵｎｉｖｅｒｓｉｔｙ，山东大学，邮编：２５０１００）Ａｂｓｔｒａｃｔ：Ｉ...     

ON OSCILLATIONS OF PERTURBED SECOND ORDER DIFFERENTIAL EQUATION

A perturbed second order linear differential eqtuation is considered. A criteria concerning the oscillatory behaviour of solutions of the perturbed equation based on the perturbing function and the solutions of the associated unperturbed equation is given.