## Taiwanese Journal of Mathematics

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

Zhiting Xu

#### 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.

#### Article information

Source
Taiwanese J. Math., Volume 11, Number 4 (2007), 1221-1235.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500404815

Digital Object Identifier
doi:10.11650/twjm/1500404815

Mathematical Reviews number (MathSciNet)
MR2348564

Zentralblatt MATH identifier
1140.34416

#### Citation

Xu, Zhiting. OSCILLATION THEOREMS RELATED TO AVERAGING TECHNIQUE FOR SECOND ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS. Taiwanese J. Math. 11 (2007), no. 4, 1221--1235. doi:10.11650/twjm/1500404815. https://projecteuclid.org/euclid.twjm/1500404815