New Criteria for Sharp Oscillation of Second-OrderNeutral Delay Differential Equations

In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based on a priori bound for the nonoscillatory solution, and deriving new results which are optimal in a nonneutral case. It is shown that the newly obtained results significantly improve a large number of existing ones.

On Sharp Oscillation Criteria for General Third-Order Delay Differential Equations

In this paper, effective oscillation criteria for third-order delay differential equations of the form, r2r1y′′′(t)+q(t)y(τ(t))=0 ensuring that any nonoscillatory solution tends to zero asymptotically, are established. The results become sharp when applied to a Euler-type delay differential equation and, to the best of our knowledge, improve all existing results from the literature. Examples are provided to illustrate the importance of the main results.

Oscillation of Half-Linear Differential Equations with Delay

We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

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.

Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [r(t)(x(t)+P(t)x(t-τ))(n-1)]′+∑i=1mQi(t)fi(x(t-σi))=0, t≥t0, where m≥1,n≥2 are integers, τ>0, σi≥0, r,P,Qi∈C([t0,∞),R), fi∈C(R,R) (i=1,2,…,m), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general Qi(t) (i=1,2,…,m) which means that we allow oscillatory Qi(t) (i=1,2,…,m). In particular, our results improve essentially and extend some known results in the recent references.

Oscillation of Nonlinear Differential Equations with Advanced Arguments

This paper is concerned with the oscillation of all solutions of the n-th order delay differential equation . The necessary and sufficient conditions for oscillatory solutions are obtained and other conditions for nonoscillatory solution to converge to zero are established.

On the Nonoscillation of Second-Order Neutral Delay Differential Equation with Forcing Term

This paper is concerned with nonoscillation of second-order neutral delay differential equation with forcing term. By using contraction mapping principle, some sufficient conditions for the existence of nonoscillatory solution are established. The criteria obtained in this paper complement and extend several known results in the literature. Some examples illustrating our main results are given.