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

Abstract

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