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Oscillation theorems related to averaging technique for second order emden-fowler type neutral differential equations. (English) Zbl 1167.34028

Summary: Some oscillation theorems are established by the averaging techniques for the second order Emden-Fowler type neutral delay differential equation
\[ (r(t)x'(t))'+q_1(t)|y(t-\sigma_1)|^{\alpha-1} y(t-\sigma_1)+ q_2(t)|y(t-\sigma_2)|^{\beta-1} y(t-\sigma_2)=0,\quad t\geq t_0, \]
where \(x(t)=y(t)+p(t)y(t-\tau)\), \(\tau\), \(\sigma_1\) and \(\sigma_2\) are nonnegative constants, \(0<\alpha<1\), \(\beta>1\), and \(r\), \(p\), \(q_1,q_2\in C(t_0,\infty),\mathbb R)\). These theorems obtained here extend and improve some known results. In particular, two interesting examples that point out the applications of our results are also included.

MSC:

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
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