Necessary and sufficient conditions for oscillations of higher order delay differential equations

Abstract:

Consider the $n{\text {th}}$ order delay differential equation (1) \[ {x^{(n)}}(t) + {( - 1)^{n + 1}}\sum \limits _{i = 0}^k {{p_i}x(t - {\tau _i}) = 0, \qquad t \geq {t_0}},\] where the coefficients and the delays are constants such that $0 = {\tau _0} < {\tau _{1}} < \cdots < {\tau _k};{p_0} \geq 0,{p_i} > 0,i = 1,2,\ldots ,k;k \geq 1$ and $n \geq 1$. The characteristic equation of (1) is (2) \[ {\lambda ^n} + {( - 1)^{n + 1}}\;\sum \limits _{i = 0}^k {{p_i}{e^{ - \lambda {\tau _i}}} = 0}. \] We prove the following theorem. Theorem. (i) For $n$ odd every solution of (1) oscillates if and only if (2) has no real roots. (ii) For $n$ even every bounded solution of (1) oscillates if and only if (2) has no real roots in $( - \infty ,0]$. The above results have straightforward extensions for advanced differential equations.