Oscillation criteria for nonlinear delay differential equations of second order

We prove oscillation theorems for the nonlinear delay differential equation $\left( \left\vert y^{\prime }(t)\right\vert ^{\alpha -2}y^{\prime}(t)\right) ^{\prime }+q(t)\left\vert y(\tau (t))\right\vert ^{\beta-2}y(\tau (t))=0, t\geq t_{\ast }>0,$ where $\beta >1,$ $\alpha >1,$ $q(t)\geq 0$ and locally integrable on $[t_{\ast },\infty ),$ $\tau (t)$ is a continuous function satisfiying $0<\tau (t)\leq t$ and lim$_{t\rightarrow \infty }\tau (t)=\infty .$ The results obtained essentially improve the known results in the literature and can be applied to linear and half-linear delay type differential equations

OSCILLATION OF SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Abstract. We establish some new oscillation criteria for the second order neutral delay differential equation The obtained results supplement those of Dzurina and Stavroulakis, Sun and Meng, Xu and Meng, Baculíková and Lacková. We also make a slight improvement of one assumption in the paper of Xu and Meng

Nonoscillation of Second-Order Dynamic Equations with Several Delays

Existence of nonoscillatory solutions for the second-order dynamic equation (A0xΔ)Δ(t)+∑i∈[1,n]ℕAi(t)x(αi(t))=0 for t∈[t0,∞)T is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case A0(t)≡1 for t∈[t0,∞)R and for second-order nondelay difference equations (αi(t)=t+1 for t∈[t0,∞)N). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary A0 and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced

Sharp results for oscillation of second-order neutral delay differential equations

The aim of the present paper is to continue earlier works by the authors on the oscillation problem of second-order half-linear neutral delay differential equations. By revising the set method, we present new oscillation criteria which essentially improve a number of related ones from the literature. A couple of examples illustrate the value of the results obtained

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 Caused By Impulses

AbstractThe present paper is devoted to the investigation of the oscillation of a kind of very extensively studied second order nonlinear delay differential equations with impulses, some interesting results are obtained, which illustrate that impulses play a very important role in giving rise to the oscillations of equations

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