The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+ b(t) x([t-k])=0 is considered,where a(t) and b(t) are continuous functions on [-k,∞),b(t)≥0,k is a positive integer...The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+ b(t) x([t-k])=0 is considered,where a(t) and b(t) are continuous functions on [-k,∞),b(t)≥0,k is a positive integer and [·] denotes the greatest integer function.Some new oscillation and nonoscillation conditions are obtained.展开更多
Consider the n-th order neutral differential equation where n > 1 is an odd integer, p(t) ∈ C([t0, ∞ ), R), Q(t), R(t)∈ C([t0, ∞), R+), τ > 0, σ, r ∈ R+. In this paper we will survey some recent results o...Consider the n-th order neutral differential equation where n > 1 is an odd integer, p(t) ∈ C([t0, ∞ ), R), Q(t), R(t)∈ C([t0, ∞), R+), τ > 0, σ, r ∈ R+. In this paper we will survey some recent results on the oscillation of Eq.() without the usual requirement that We also give some interesting open problems on this topic.展开更多
Zhang and Yan in ref. [1] gave some sufficient conditions for the oscillation of eqs.(1) and (2) by using the methods as in difference equations with discrete arguments, and thereby revealed certain oscillation relati...Zhang and Yan in ref. [1] gave some sufficient conditions for the oscillation of eqs.(1) and (2) by using the methods as in difference equations with discrete arguments, and thereby revealed certain oscillation relation between difference equations with continuous arguments and discrete ones. However, the oscillation results in ref. [1] need the hypothesis liminf p_i(t)】0, which is an essential condition.t→∞ In this note, we compare eq. (1) with certain delay differential equation, and thereby establish some new sufficient conditions for the oscillation of eqs. (1) and (2). These conditions are integral conditions which do not need the hypothesis liminf p_i(t)】0.展开更多
Our aim in this paper is to obtain a necessary and sufficient condition characterized by coefficients and delays under which every solution of higher-order neutral equation will be oscillating, where p ≠ 0, τ > ...Our aim in this paper is to obtain a necessary and sufficient condition characterized by coefficients and delays under which every solution of higher-order neutral equation will be oscillating, where p ≠ 0, τ > 0, σ> 0 and q are all constants.展开更多
1986年张炳根给出了关于一阶时滞差分微分方程(RDDE) x(t)+sum from i=1 to n p_ix(t—τ_1)=0(P_i,τ_i,均为正、实常数,i=1,2,…,n)所有解均振动的充分性条件,本文指出,该条件实际上是充分必要条件;本文还指出,在某些条件下,该充分条...1986年张炳根给出了关于一阶时滞差分微分方程(RDDE) x(t)+sum from i=1 to n p_ix(t—τ_1)=0(P_i,τ_i,均为正、实常数,i=1,2,…,n)所有解均振动的充分性条件,本文指出,该条件实际上是充分必要条件;本文还指出,在某些条件下,该充分条件中的不等式改为等式时也是方程振动的充分必要条件。展开更多
文摘The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+ b(t) x([t-k])=0 is considered,where a(t) and b(t) are continuous functions on [-k,∞),b(t)≥0,k is a positive integer and [·] denotes the greatest integer function.Some new oscillation and nonoscillation conditions are obtained.
文摘Consider the n-th order neutral differential equation where n > 1 is an odd integer, p(t) ∈ C([t0, ∞ ), R), Q(t), R(t)∈ C([t0, ∞), R+), τ > 0, σ, r ∈ R+. In this paper we will survey some recent results on the oscillation of Eq.() without the usual requirement that We also give some interesting open problems on this topic.
基金Project supported by the National Natural Science Foundation of China.
文摘Zhang and Yan in ref. [1] gave some sufficient conditions for the oscillation of eqs.(1) and (2) by using the methods as in difference equations with discrete arguments, and thereby revealed certain oscillation relation between difference equations with continuous arguments and discrete ones. However, the oscillation results in ref. [1] need the hypothesis liminf p_i(t)】0, which is an essential condition.t→∞ In this note, we compare eq. (1) with certain delay differential equation, and thereby establish some new sufficient conditions for the oscillation of eqs. (1) and (2). These conditions are integral conditions which do not need the hypothesis liminf p_i(t)】0.
文摘Our aim in this paper is to obtain a necessary and sufficient condition characterized by coefficients and delays under which every solution of higher-order neutral equation will be oscillating, where p ≠ 0, τ > 0, σ> 0 and q are all constants.
文摘1986年张炳根给出了关于一阶时滞差分微分方程(RDDE) x(t)+sum from i=1 to n p_ix(t—τ_1)=0(P_i,τ_i,均为正、实常数,i=1,2,…,n)所有解均振动的充分性条件,本文指出,该条件实际上是充分必要条件;本文还指出,在某些条件下,该充分条件中的不等式改为等式时也是方程振动的充分必要条件。
