A Necessary and sufficient condition for the oscillation of higher-order neutral equations with several delays
Citation：
Wang Zhicheng.A Necessary and sufficient condition for the oscillation of higher-order neutral equations with several delays[J].Chinese Annals of Mathematics B,1991,12(3):243~254
Page view： 822Net amount： 975
Authors：
Wang Zhicheng;
Foundation：
Projects supported by the National Natural Science Foundstion.
Abstract：
Consider the higher-order neutral delay differential equation
$$\[\frac{{{d^n}}}{{d{t^n}}}(x(t) + \sum\limits_{i = 1}^l {{p_i}x(t - {\tau _i}) - \sum\limits_{j = 1}^m {{r_j}x(t - {\rho _j})} } ) + \sum\limits_{k = 1}^N {{q_k}x(t - {u_k})} = 0\]$$
where the coefficients and the delays are nonnegative constants with $\[n \ge 2\]$ even. Then a necessary and sufficient condition for the oscillation of (A) is that the characteristic equation
$$\[{\lambda ^n} + {\lambda ^n}\sum\limits_{i = 1}^l {{p_i}{e^{ - \lambda {\tau _i}}}} - {\lambda ^n}\sum\limits_{j = 1}^m {{r_j}{e^{ - \lambda {\rho _j}}}} + \sum\limits_{k = 1}^N {{q_k}{e^{ - \lambda {u_k}}}} = 0\]$$
has no real roots.
