THEOREMS OF KIGURADZE-TYPE AND BELOHOREC-TYPE REVISITED ON TIME SCALES

This article concerns the oscillation of second-order nonlinear dynamic equations. By using generalized Riccati transformations, Kiguradzetype and Belohorec-type oscillation theorems are obtained on an arbitrary time scale. Our results cover those for differential equations and difference equations, and provide new oscillation criteria for irregular time scales. Some examples are given to illustrate our results

OSCILLATION of SECOND-ORDER HALF-LINEAR NEUTRAL NONCANONICAL DYNAMIC EQUATIONS

In This Paper, We Shall Establish Some New Criteria for the Oscillation of Certain Second-Order Noncanonical Dynamic Equations with a Sublinear Neutral Term. This Task is Accomplished by Reducing the Involved Nonlinear Dynamic Equation to a Second-Order Linear Dynamic Inequality. We Also Establish Some New Oscillation Theorems Involving Certain Integral Conditions. Three Examples, Illustrating Our Results, Are Presented. Our Results Generalize Results for Corresponding Differential and Difference Equations

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument

The chapter is devoted to study the oscillation of all solutions to second‐order nonlinear neutral damped differential equations with delay argument. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and integral averaging techniques

Oscillation Criteria for Fourth Order Nonlinear Positive Delay Differential Equations with a Middle Term

In this article, we establish some new criteria for the oscillation of fourth order nonlinear delay differential equations of the form (Equation presented) provided that the second order equation (Equation presented) is nonoscillatiory or oscillatory. This equation with g(t) = t is considered in [8] and some oscillation criteria for this equation via certain energy functions are established. Here, we continue the study on the oscillatory behavior of this equation via some inequalities

Asymptotic properties of solutions of certain third-order dynamic equations

AbstractIn this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation (r2(t)((r1(t)xΔ(t))Δ)γ)Δ+q(t)f(x(t))=0 on time scale T, where γ≥1 is a ratio of odd positive integers. Our results are essentially new even for third-order differential and difference equations, i.e., when T=R and T=N. Two examples of dynamic equations on different time scales are given to show the applications of our main results

Integral averaging technique for the interval oscillation criteria of certain second-order nonlinear differential equations

AbstractWe present new interval oscillation criteria related to integral averaging technique for certain classes of second-order nonlinear differential equations which are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [t0,∞), rather than on the whole half-line. They generalize and improve some known results. Examples are also given to illustrate the importance of our results

Oscillation Theorems for Second-Order Quasilinear Neutral Functional Differential Equations

New oscillation criteria are established for the second-order nonlinear neutral functional differential equations of the form (r(t)|z′(t)|α−1z′(t))’+f(t,x[σ(t)])=0, t≥t0, where z(t)=x(t)+p(t)x(τ(t)), p∈C1([t0,∞),[0,∞)), and α≥1. Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results

Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping

Some new oscillation criteria for a general class of second-order differential equations with nonlinear damping are shown. Except some general structural assumptions on the coefficients and nonlinear terms, we additionally assume only one sufficient condition (of Fite-Wintner-Leighton type). It is different compared to many early published papers which use rather complex sufficient conditions. Our method contains three items: classic Riccati transformations, a pointwise comparison principle, and a blow-up principle for sub- and supersolutions of a class of the generalized Riccati differential equations associated to any nonoscillatory solution of the main equation

Oscillation criteria for nonlinear second-order differential equations with damping

Some new oscillation criteria are given for general nonlinear second order ordinary differential equations with damping of the form x′′ + p(t)x′ + q(t) f (x) = 0, where f is with or without monotonicity. Our results generalize and extend some earlier results of Deng.Наведено деякі нові осцнляційні критерії для загальних нелінійних звичайних диференціальних рівнянь другого порядку із затуханням вигляду x" + p(t)x' + q(t)f(x) = 0, де функція f або монотонна, або немонотонна. Наведені результати узагальнюють та розширюють деякі результати, отримані раніше Денгом

Oscillation results for second-order nonlinear neutral differential equations

Published version of an article in the journal: Advances in Difference Equations. Also available from the publisher at: http://dx.doi.org/10.1186/1687-1847-2013-336 Open AccessWe obtain several oscillation criteria for a class of second-order nonlinear neutral differential equations. New theorems extend a number of related results reported in the literature and can be used in cases where known theorems fail to apply. Two illustrative examples are provided