Bounded Oscillation of Nonlinear Neutral Differential Equations of Arbitrary Order

Abstract

The paper is concerned with oscillation properties of n-th order neutral differential equations of the form $$[x(t) + cx(\tau (t))]^{(n)} + q(t)f\left( {x(\sigma (t))} \right) = 0,{\text{ }}t \geqslant t_0 >0,$$ where c is a real number with \(\left| c \right| \ne 1,q \in C\left( {\left[ {t_0 ,\infty } \right),\mathbb{R}} \right),f \in C(\mathbb{R},\mathbb{R}),\tau ,\sigma \in C\left( {\left[ {t_0 ,\infty } \right),\mathbb{R}_ + } \right)\) with τ(t) < t and \(\mathop {\lim }\limits_{t \to \infty } \tau (t) = \mathop {\lim }\limits_{t \to \infty } \sigma (t) = \infty \).Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which c is a function of t and a certain type of a forcing term is present.