OSCILLATION THEOREMS RELATED TO AVERAGING TECHNIQUE FOR SECOND ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS

Abstract

Some oscillation theorems are established by the averaging technique for the second order nonlinear neutral delay differential equation $$ \begin{array}{l} (r(t) |x^{\prime}(t)|^{\gamma-1} x^{\prime}(t))^{\prime} + q_1(t) |y(t-\sigma_1)|^{\alpha-1} y(t-\sigma_1) \\ \hspace{5mm} + q_2(t) |y(t-\sigma_2)|^{\beta-1} y(t-\sigma_2) = 0, \quad t \geq t_0, \end{array} $$ where $x(t) = y(t) + p(t) y(t-\tau)$, $\tau$, $\sigma_1$ and $\sigma_2$ are nonnegative constants, $\alpha$, $\beta$ and $\gamma$ are positive constants, and $r$, $p$, $q_1$, $q_2 \in C([t_0, \infty), \mathbb{R})$. The results obtained here essentially improve some known results in the literature. In particular, two interesting examples that point out the applications of our results are also included.