Oscillation of Solutions to Third-Order Nonlinear Neutral Dynamic Equations on Time Scales

In this paper, we are concerned with the oscillation of solutions to a class of third-order nonlinear neutral dynamic equations on time scales. New oscillation criteria are presented by employing the Riccati transformation and integral averaging technique. Two examples are shown to illustrate the conclusions.

Oscillation of nonlinear third-order differential equations with several sublinear neutral terms

A class of third order differential equations with several sublinear neutral terms of the type ( a ( t ) ( b ( t ) ( x ( t ) + ∑ j = 1 n p j ( t ) x α j ( τ j ( t ) ) ) ′ ) ′ ) ′ + ∑ i = 1 m f i ( t , x ( σ i ( t ) ) ) = 0 , t ≥ t 0 > 0 $$\begin{array}{} \displaystyle \bigg( a(t)\Big( b(t)\Big(x(t)+\sum\limits_{j=1}^{n}p_{j}(t)x^{\alpha _{j}}(\tau _{j}(t))\Big)'\Big)'\bigg)' +\sum\limits_{i=1}^{m}f_{i}(t,x(\sigma _{i}(t)))=0,\qquad t\geq t_{0} \gt 0 \end{array}$$ is considered. Some oscillation criteria are presented to improve and complement those in the literature. Two examples are established to illustrate the main results.

Some Oscillation Results for Even-Order Differential Equations with Neutral Term

The objective of this work is to study some new oscillation criteria for even-order differential equation with neutral term rxzn−1xγ′+qxyγζx=0. By using the Riccati substitution and comparison technique, several new oscillation criteria are obtained for the studied equation. Our results generalize and improve some known results in the literature. We offer some examples to illustrate the feasibility of our conditions.

Oscillation criteria of third-order neutral differential equations with damping and distributed deviating arguments

Our objective in this paper is to study the oscillatory and asymptotic behavior of the solutions of third-order neutral differential equations with damping and distributed deviating arguments. New oscillation criteria are established, which are based on a refinement generalized Riccati transformation. An important tool for this investigation is the integral averaging technique. Moreover, we provide an example to illustrate the main results.

Oscillation Criteria of Solutions of Third Order Neutral Integro-Differential Equations

Some necessary and sufficient conditions are obtained that guarantee the oscillation of all solutions of two types of neutral integro-differential equations of third order. The integral is used in the sense of Riemann-Stieltjes. Some examples were included to illustrate the obtained results

Delay Differential Equations of Fourth-Order: Oscillation and Asymptotic Properties of Solutions

In this work, by using the comparison method and Riccati transformation, we obtain some oscillation criteria of solutions of delay differential equations of fourth-order in canonical form. These criteria complement those results in the literature. We give two examples to illustrate the main results. Symmetry plays an essential role in determining the correct methods for solutions to differential equations.

On the Oscillation of Even-Order Nonlinear Differential Equations with Mixed Neutral Terms

The oscillation of even-order nonlinear differential equations (NLDiffEqs) with mixed nonlinear neutral terms (MNLNTs) is investigated in this work. New oscillation criteria are obtained which improve, extend, and simplify the existing ones in other previous works. Some examples are also given to illustrate the validity and potentiality of our results.