3 research outputs found

    Necessary and sufficient conditions for the oscillation of higher-order differential equations involving distributed delays

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    In this article, we establish necessary and sufficient conditions for the oscillation of both bounded and unbounded solutions of the differential equation \begin{equation} \bigg[x(t)+\int_{0}^{\lambda}p(t,v)x(\tau(t,v))\,\mathrm{d}v\bigg]^{(n)}+\int_{0}^{\lambda}q(t,v)x(\sigma(t,v))\,\mathrm{d}v=\varphi(t)\quad\text{for } t \geq t_{0},\notag \end{equation} where nNn\in\mathbb{N}, t0,λR+t_{0},\lambda\in\mathbb{R}^{+}, pC([t0,)×[0,λ]R)p\in C([t_{0},\infty)\times[0,\lambda] \mathbb{R}), qC([t0,)×[0,λ],R+)q\in C([t_{0},\infty)\times[0,\lambda],\mathbb{R}^{+}), τC([t0,)×[0λ],R)\tau\in C([t_{0},\infty)\times[0 \lambda],\mathbb{R}) with limtinfv[0,λ]τ(t,v)=\lim_{t\to\infty}\inf_{v\in[0,\lambda]}\tau(t,v)=\infty and supv[0,λ]τ(t,v)t\sup_{v\in[0,\lambda]}\tau(t,v)\leq t for all tt0t\geq t_{0}, σC([t0,)×[0,λ],R)\sigma\in C([t_{0},\infty)\times[0,\lambda],\mathbb{R}) with limtinfv[0,λ]σ(t,v)=\lim_{t\to\infty}\inf_{v\in[0,\lambda]}\sigma(t,v)=\infty, and φC([t0,),R)\varphi\in C([t_{0},\infty),\mathbb{R}). We also give illustrating examples to show the applicability of these results

    Oscillation of first-order neutral differential equations with distributed deviating arguments

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    WOS: 000253509100015In this paper we shall consider first-order neutral differential equations with distributed deviating arguments. Some new sufficient conditions are presented for the oscillation of all solutions. (c) 2007 Elsevier Ltd. All rights reserved
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