Necessary and sufficient conditions for oscillations of neutral equations with several coefficients

AbstractConsider the neutral equation with several positive and negative coefficients dd[tx(t)+∑Ipix(t−τi)−∑Jrjx(t−ϱj)]+∑Kqkx(t−σk)=0, (1) where I, J, and K are initial segments of natural numbers, pi, τi, rj, ϱj, qk are positive numbers, and σk ⩾ 0 for i ϵ I, j ϵ J, and k ϵ K. Then a necessary and sufficient condition for the oscillation of all solutions of (1) is that its characteristic equation λ+λ∑Ipie−λτi−λ∑Irje−λϱj+∑Kqke−λσk=0 has no real roots