On Nonoscillation of Mixed Advanced-Delay Differential Equations with Positive and Negative Coefficients

For a mixed (advanced--delay) differential equation with variable delays and coefficients $$\dot{x}(t) \pm a(t)x(g(t)) \mp b(t)x(h(t)) = 0, t\geq t_0$$ where $$a(t)\geq 0, b(t)\geq 0, g(t)\leq t, h(t)\geq t$$ explicit nonoscillation conditions are obtained.Comment: 17 pages; 2 figures; to appear in Computers & Mathematics with Application

Oscillatory and asymptotic behaviour of a neutral differential equation with oscillating coefficients

In this paper, we obtain sufficient conditions so that every solution of $$\big(y(t)- \sum_{i=1}^n p_i(t) y(\delta_i(t))\big)'+\sum_{i=1}^m q_i(t) y(\sigma_i(t)) = f(t)$$ oscillates or tends to zero as $t \to \infty$. Here the coefficients $p_i(t), q_i(t)$ and the forcing term $f(t)$ are allowed to oscillate; such oscillation condition in all coefficients is very rare in the literature. Furthermore, this paper provides an answer to the open problem 2.8.3 in [7, p. 57]. Suitable examples are included to illustrate our results

Oscillatory behaviour of a higher order nonlinear neutral delay type functional differential equation with oscillating coefficients

summary:In this paper we are concerned with the oscillation of solutions of a certain more general higher order nonlinear neutral type functional differential equation with oscillating coefficients. We obtain two sufficient criteria for oscillatory behaviour of its solutions

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

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 $n\in\mathbb{N}$, $t_{0},\lambda\in\mathbb{R}^{+}$, $p\in C([t_{0},\infty)\times[0,\lambda] \mathbb{R})$, $q\in C([t_{0},\infty)\times[0,\lambda],\mathbb{R}^{+})$, $\tau\in C([t_{0},\infty)\times[0 \lambda],\mathbb{R})$ with $\lim_{t\to\infty}\inf_{v\in[0,\lambda]}\tau(t,v)=\infty$ and $\sup_{v\in[0,\lambda]}\tau(t,v)\leq t$ for all $t\geq t_{0}$, $\sigma\in C([t_{0},\infty)\times[0,\lambda],\mathbb{R})$ with $\lim_{t\to\infty}\inf_{v\in[0,\lambda]}\sigma(t,v)=\infty$, and $\varphi\in C([t_{0},\infty),\mathbb{R})$. We also give illustrating examples to show the applicability of these results

Asymptotically polynomial solutions of difference equations of neutral type

Asymptotic properties of solutions of difference equation of the form $$\Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n$$ are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation

Oscillation criteria for a certain even order neutral difference equation with an oscillating coefficient

AbstractIn this paper we are concerned with the oscillation of solutions of a certain higher order linear neutral type difference equation with an oscillating coefficient. We obtain some sufficient criteria for oscillatory behaviour. In particular, the results are new even when n=2 and there are few results in the case of p is an oscillatory function

Classification of Solutions of Non-homogeneous Non-linear Second Order Neutral Delay Dynamic Equations with Positive and Negative Coefficients

In this paper we have studied the non-homogeneous non-linear second order neutral delay dynamic equations with positive and negative coefficients of the form classified all solutions of this type equations and obtained conditions for the existence or non-existence of solutions into four classes and these four classes are mutually disjoint. Examples are included to illustrate the validation of the main results