Bounded Oscillation of a Forced Nonlinear Neutral Differential Equation

Abstract

This paper is concerned with the $n$th-order forced nonlinear neutral differential equation $\left[x\right(t)-p(t\left)x\right(\tau \left(t\right)){]}^{\left(n\right)}+{\sum }_{i=1}^m{q}_i(t\left)f_i\right(x\left({\sigma }_{i1}\right(t\left)\right),x\left({\sigma }_{i2}\right(t\left)\right),\dots ,x\left({\sigma }_{i{k}_i}\right(t\left)\right))=g(t),t\ge t_0$. Some necessary and sufficient conditions for the oscillation of bounded solutions and several sufficient conditions for the existence of uncountably many bounded positive and negative solutions of the above equation are established. The results obtained in this paper improve and extend essentially some known results in the literature. Five interesting examples that point out the importance of our results are also included.