We are interested in comparing the oscillatory and asymptotic properties of the equations $L_n[x\left(t\right)-P\left(t\right)x\left(g\left(t\right)\right)]+\delta f(t,x\left(h\left(t\right)\right))=0$ with those of the equations $M_n[x\left(t\right)-P\left(t\right)x\left(g\left(t\right)\right)]+\delta Q\left(t\right)q\left(x\left(r\left(t\right)\right)\right)=0.$

In this paper the oscillatory and asymptotic properties of the solutions of the functional differential equation $L_{n}u\left(t\right)+p\left(t\right)f\left(u\left[g\left(t\right)\right]\right)=0$ are compared with those of the functional differential equation ${\alpha }_{n}u\left(t\right)+q\left(t\right)h\left(u\left[w\left(t\right)\right]\right)=0$.

In this paper we study asymptotic properties of the third order trinomial delay differential equation $$y^{\text{'}\text{'}\text{'}}\left(t\right)-p\left(t\right)y^{\text{'}}\left(t\right)+g\left(t\right)y\left(\tau \left(t\right)\right)=0$$ by transforming this equation to the binomial canonical equation. The results obtained essentially improve known results in the literature. On the other hand, the set of comparison principles obtained permits to extend immediately asymptotic criteria from ordinary to delay equations.

To explore the impacts of time delay on nonlinear dynamics of consensus models, we incorporate time-varying delay into a two-agent system to study its long-time behaviors. By the classical 3/2 stability theory, we establish a sufficient condition for the system to experience unconditional consensus. Numerical examples show the effectiveness of the proposed protocols and present possible Hopf bifurcations when the time delay changes.

Consider the delay differential equation $$\dot{x}\left(t\right)=g(x\left(t\right),x(t-r)),\left(1\right)$$ where $r>0$ is a constant and $g{ℝ}^2\to ℝ$ is Lipschitzian. It is shown that if $r$ is small, then the solutions of (1) have the same convergence properties as the solutions of the ordinary differential equation obtained from (1) by ignoring the delay.