Oscillation of solutions to odd-order nonlinear neutral functional differential equations.

Li, Tongxing; Thandapani, Ethiraju

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In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the $n$ -th order delay differential equations ${(r (t) {[x^{(n - 1)} (t)]}^{γ})}^{'} = q (t) f (x (τ (t))) .$ Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases $\int^{\infty} r^{- 1 / γ} (t) t = \infty$ and $\int^{\infty} r^{- 1 / γ} (t) t < \infty$ are discussed. ...

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On the oscillation of third-order quasi-linear neutral functional differential equations

Ethiraju Thandapani, Tongxing Li (2011)

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The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation ${[a (t) {({[x (t) + p (t) x (δ (t))]}^{''})}^{α}]}^{'} + q (t) x^{α} (τ (t)) = 0, E$ where $α > 0$ , $0 \leq p (t) \leq p_{0} < \infty$ and $δ (t) \leq t$ . By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.