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 to third order neutral type difference equations with delay and advanced arguments

summary:In this paper, we present several sufficient conditions for the existence of nonoscillatory solutions to the following third order neutral type difference equation $$\Delta ^3(x_n+a_n x_{n-l} +b_n x_{n+m})+p_n x_{n-k} - q_n x_{n+r}=0,\quad n\geq n_0$$ via Banach contraction principle. Examples are provided to illustrate the main results. The results obtained in this paper extend and complement some of the existing 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

Differential/Difference Equations

The study of oscillatory phenomena is an important part of the theory of differential equations. Oscillations naturally occur in virtually every area of applied science including, e.g., mechanics, electrical, radio engineering, and vibrotechnics. This Special Issue includes 19 high-quality papers with original research results in theoretical research, and recent progress in the study of applied problems in science and technology. This Special Issue brought together mathematicians with physicists, engineers, as well as other scientists. Topics covered in this issue: Oscillation theory; Differential/difference equations; Partial differential equations; Dynamical systems; Fractional calculus; Delays; Mathematical modeling and oscillations

Oscillation and nonoscillation of third order functional differential equations

A qualitative approach is usually concerned with the behavior of solutions of a given differential equation and usually does not seek specific explicit solutions;This dissertation is the analysis of oscillation of third order linear homogeneous functional differential equations, and oscillation and nonoscillation of third order nonlinear nonhomogeneous functional differential equations. This is done mainly in Chapters II and III. Chapter IV deals with the analysis of solutions of neutral differential equations of third order and even order. In Chapter V we study the asymptotic nature of nth order delay differential equations;Oscillatory solution is the solution which has infinitely many zeros; otherwise, it is called nonoscillatory solution;The functional differential equations under consideration are:(UNFORMATTED TABLE OR EQUATION FOLLOWS) (b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + (q[subscript]1y)[superscript]\u27 + q[subscript]2y[superscript]\u27 = 0, &(b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + q[subscript]1y + q[subscript]2y(t - [tau]) = 0, &(b(ay[superscript]\u27)[superscript]\u27)[superscript]\u27 + qF(y(g(t))) = f(t), &(y(t) + p(t)y(t - [tau]))[superscript]\u27\u27\u27 + f(t, y(t), y(t - [sigma])) = 0, &(y(t) + p(t)y(t - [tau]))[superscript](n) + f(t, y(t), y(t - [sigma])) = 0, and &y[superscript](n) + p(t)f(t, y[tau], y[subscript]sp[sigma][subscript]1\u27,..., y[subscript]sp[sigma][subscript]n[subscript]1(n-1)) = F(t). (TABLE/EQUATION ENDS);The first and the second equations are considered in Chapter II, where we find sufficient conditions for oscillation. We study the third equation in Chapter III and conditions have been found to ensure the required criteria. In Chapter IV, we study the oscillation behavior of the fourth and the fifth equations. Finally, the last equation has been studied in Chapter V from the point of view of asymptotic nature of its nonoscillatory solutions

Stability, boundedness, oscillation and periodicity in functional differential equations.

Wudu Lu.Thesis (Ph.D.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 114-116).Abstract --- p.iiIntroduction --- p.ivChapter 1 --- The Fundamental Theory of NFDEs with Infinite Delay --- p.1Chapter 1.1 --- Introduction --- p.1Chapter 1.2 --- Phase spaces and NFDEs with infinite delay --- p.2Chapter 1.3 --- Local theory --- p.4Chapter 2 --- Periodicity and Bp -Boundedness in Neutral Systems of Non- linear D-operator with Infinite Delay --- p.12Chapter 2.1 --- Introduction --- p.12Chapter 2.2 --- Preliminaries --- p.15Chapter 2.3 --- Existence of periodic solutions --- p.22Chapter 2.4 --- BP-U.B and Bp -U.U.B of solutions --- p.29Chapter 2.5 --- Applications --- p.42Chapter 3 --- Stability in Neutral Differential Equations of Nonlinear D- operator with Infinite Delay --- p.47Chapter 3.1 --- Introduction --- p.47Chapter 3.2 --- Preliminaries --- p.49Chapter 3.3 --- Uniformly Asymptotic Stability --- p.57Chapter 3.4 --- Applications --- p.74Chapter 4 --- Nonoscillation and Oscillation of First Order Linear Neutral Equations --- p.79Chapter 4.1 --- Introduction --- p.79Chapter 4.2 --- Existence of Nonoscillatory Solutions --- p.80Chapter 4.3 --- Oscillation --- p.90Chapter 5 --- Nonoscillation and Oscillation of First Order Nonlinear Neu- tral Equations --- p.94Chapter 5.1 --- Introduction --- p.94Chapter 5.2 --- Existence of Nonoscillatory Solutions --- p.95Chapter 5.3 --- Oscillation --- p.102Bibliography --- p.108List of Author's Publications --- p.11

Volterra difference equations

This dissertation consists of five papers in which discrete Volterra equations of different types and orders are studied and results regarding the behavior of their solutions are established. The first paper presents some fundamental results about subexponential sequences. It also illustrates the subexponential solutions of scalar linear Volterra sum-difference equations are asymptotically stable. The exact value of the rate of convergence of asymptotically stable solutions is found by determining the asymptotic behavior of the transient renewal equations. The study of subexponential solutions is also continued in the second and third articles. The second paper investigates the same equation using the same process as considered in the first paper. The discussion focuses on a positive lower bound of the rate of convergence of the asymptotically stable solutions. The third paper addresses the rate of convergence of the solutions of scalar linear Volterra sum-difference equations with delay. The result is proved by developing the rate of convergence of transient renewal delay difference equations. The fourth paper discusses the existence of bounded solutions on an unbounded domain of more general nonlinear Volterra sum-difference equations using the Schaefer fixed point theorem and the Lyapunov direct method. The fifth paper examines the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations and establishes some new criteria based on so-called time scales, which unifies and extends both discrete and continuous mathematical analysis. Beside these five research papers that focus on discrete Volterra equations, this dissertation also contains an introduction, a section on difference calculus, a section on time scales calculus, and a conclusion. --Abstract, page v