A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION OF HIGHER-ORDER NEUTRAL EQUATIONS WITH SEVERAL DELAYS

Consider the higher-order neutral delay differential equationd~t/dt~n(x(t)+sum from i=1 to lp_ix(t-τ_i)-sum from j=1 to mr_jx(t-ρ_j))+sum from k=1 to Nq_kx(t-u_k)=0,(A)where the coefficients and the delays are nonnegative constants with n≥2 even. Then anecessary and sufficient condition for the oscillation of (A) is that the characteristicequationλ~n+λ~nsum from i=1 to lp_ie~(-λτ_i-λ~n)sum from j=1 to mr_je~(-λρ_j)+sum from k=1 to Nq_ke~(-λρ_k)=0has no real roots.

A Necessary and sufficient condition for the oscillation of higher-order neutral equations with several delays

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.