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

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

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

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

Asymptotic and oscillatory behavior of higher order quasilinear delay differential equations

In the paper, we offer such generalization of a lemma due to Philos (and partially Staikos), that yields many applications in the oscillation theory. We present its disposal in the comparison theory and we establish new oscillation criteria for $n-$th order delay differential equation \begin{equation*} \left(r(t)\left[x'(t)\right]^{\gamma}\right)^{(n-1)}+q(t)x^{\gamma}(\tau(t))=0.\tag{$E$} \end{equation*} The presented technique essentially simplifies the examination of the higher order differential equations

On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations

By using comparison principles, we analyze the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Due to less restrictive assumptions on the coefficients of the equation and on the deviating argument, our criteria improve a number of related results reported in the literature.publishedVersio

Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given

Existence of Nonoscillatory Solutions of First Order Nonlinear Neutral Dierence Equations

In this paper, we discuss the existence of nonoscillatory solutions of first order nonlinear neutral difference equations of the form We use the Knaster-Tarski xed point theorem to obtain some sucient conditions for the existence of nonoscillatory solutions of above equations. Example are given to illustrate the main results

Oscillation criteria for second order superlinear neutral delay differential equations

New oscillation criteria for the second order nonlinear neutral delay differential equation $[y(t)+p(t)y(t-\tau )]^{^{\prime \prime}}+q(t)\,f(y(g(t)))=0$, $t\geq t_{0}$ are given. The relevance of our theorems becomes clear due to a carefully selected example

Asymptotic properties of solutions of second order quasilinear functional differential equations of neutral type

summary:This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasilinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as $t \to\infty$