Oscillation of solutions of non-linear neutral delay differential equations of higher order for $p(t) = \pm 1$

R. N. Rath, L. N. Padhy, N. Misra

  R. N. Rath, P. G. Department of Mathematics, Khallikote College, Berhampur 760001, Orissa, India

L. N. Padhy, Department of Mathematics, Konark Institute of Science and Technology, Jatni--752050, Bhubaneswar, Orissa, India

N. Misra, P. G. Department of Mathematics, Berhampur University, Berhampur 760007, Orissa, India

E-mail. rathanathmath@yahoo.co.in

In this paper, the oscillation criteria for solutions of the neutral delay differential equation (NDDE) \[ \left( {y(t)-p(t)\, y({t-\tau} )} \right)^{(n )}+ \alpha \,Q(t)\,\,G\left( {y({t-\sigma })} \right)= f(t) \] has been studied where $p(t) = 1$ or $p(t) \le 0$, $\alpha =\pm 1$, $Q\in C \left([0, \infty ), R^{+}\right)$, $f \in C([0, \infty ), R)$, $G \in C(R, R)$. This work improves and generalizes some recent results and answer some questions that are raised in [1].

AMSclassification. 34C10, 34C15, 34K40.

Keywords.  Oscillation, non-oscillation, neutral equations, asymptotic-behaviour.