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

Existence of nonoscillatory solutions of higher-order neutral delay difference equations with variable coefficients

AbstractIn this paper, we consider the following higher-order neutral delay difference equations with positive and negative coefficients: Δm(xn + cxn−k) + pnxn−r − qnxn−l = 0, n≥n0, where c ϵ R, m ⩾ 1, k ⩾ 1, r, l ⩾ 0 are integers, and {pn}∞n=n0 and {qn}n=n0∞ are sequences of nonnegative real numbers. We obtain the global results (with respect to c) which are some sufficient conditions for the existences of nonoscillatory solutions

Existence of non-oscillatory solutions of a kind of first-order neutral differential equation

This paper deals with the existence of non-oscillatory solutions to a kind of first-order neutral equations having both delay and advance terms. The new results are established using the Banach contraction principle

Bounded solutions of kk-dimensional system of nonlinear difference equations of neutral type

The $k$-dimensional system of neutral type nonlinear difference equations with delays in the following form \begin{equation*} \begin{cases} \Delta \Big(x_i(n)+p_i(n)\,x_i(n-\tau_i)\Big)=a_i(n)\,f_i(x_{i+1}(n-\sigma_i))+g_i(n),\\ \Delta \Big(x_k(n)+p_k(n)\,x_k(n-\tau_k)\Big)=a_k(n)\,f_k(x_1(n-\sigma_k))+g_k(n), \end{cases} \end{equation*} where $i=1,\dots,k-1$, is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded solutions of the above system with various $(p_i(n))$, $i=1,\dots,k$, $k\geq 2$

Oscillatory and nonoscillatory properties of solutions of functional differential equations and difference equations

Oscillation and nonoscillation of solutions of functional differential equations and difference equations are analyzed qualitatively. A qualitative approach is usually concerned with the behavior of solutions of a given equation and does not seek explicit solutions. The dissertation is divided into five chapters. The first chapter is essentially introductory in nature. Its main purpose is to introduce certain well-known basic concepts and to present some result that are not as well-known. In chapter 2 and chapter 3 we present sufficient conditions for oscillation of solutions of neutral differential equations of the form [a(t)[x(t) + p(t)x([tau](t))] [superscript](n-1)] [superscript]\u27 + q(t)f(x([sigma](t))) = 0and [x(t) + p(t)x([tau](t))] [superscript](n) + q[subscript]1(t)f(x([sigma][subscript]1(t))) + q[subscript]2(t)f(x([sigma][subscript]2(t))) = h(t)respectively. Chapter 4 discusses the oscillation, nonoscillation, and the asymptotic behavior of solutions of higher order functional differential equations of the form (r[subscript]2(r[subscript]1 x[superscript]\u27(t))[superscript]\u27)[superscript]\u27 + q(t)f(x([sigma](t))) = h(t)and x[superscript](n)(t) + F(t,x([sigma][subscript]1(t)),...,x([sigma][subscript]m(t))) = h(t).Chapter 5 is devoted the study of oscillatory solutions of neutral type difference equations of the form [delta][a[subscript]n[delta][superscript]m-1(x[subscript]n + p[subscript]nx[subscript][tau][subscript]n)] + q[subscript]nf(x[subscript][sigma][subscript]n) = 0and that of asymptotic behavior for n → [infinity] of solutions of equations of the form [delta][superscript]mx[subscript]n + F(n, x[subscript][sigma][subscript]n) = h[subscript]n.The results obtained here are the discrete analogs of several of those in chapter 1 and chapter 4;A function x(t) : [a,[infinity]) → R is said to be oscillatory if it has a zero on [T,[infinity]) for every T ≥ a; otherwise it is called nonoscillatory. Similarly a sequence \x[subscript]n of real numbers is oscillatory if it is not eventually positive or eventually negative; otherwise it is nonoscillatory

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

EXISTENCE FOR NONOSCILLATORY SOLUTIONS OF FORCED HIGHER-ORDER NONLINEAR NEUTRAL DYNAMIC EQUATIONS

Abstract. In this paper, we first study the existence of nonoscillatory solutions of dy- on a time scale T. By using Krasnosel'skii's fixed point theorem and some new techniques, we obtain sufficient conditions for the existence of nonoscillatory solutions for general p i (t), f i (x) and q(t) which means that they are allowed oscillate. Then, we extend our results to equations of the form [x(t) + p(t)x(τ (t))] ∆ m + F (t, x(δ(t))) = q(t). We establish sufficient and necessary conditions for the existence of nonoscillatory solutions of this equation. Our results not only generalize and improve the known results stated for differential and difference equations using the time scale theory, but also improve some of the results for dynamic equations on time scales. Some examples are included to illustrate the results