The purpose of this paper is to study the asymptotic properties of nonoscillatory solutions of the third order nonlinear functional dynamic equation ${\left[p\left(t\right){\left[{\left(r\left(t\right)x^{\Delta }\left(t\right)\right)}^{\Delta }\right]}^{\gamma }\right]}^{\Delta }+q\left(t\right)f\left(x\left(\tau \left(t\right)\right)\right)=0$, t ≥ t₀, on a time scale , where γ > 0 is a quotient of odd positive integers, and p, q, r and τ are positive right-dense continuous functions defined on . We classify the nonoscillatory solutions into certain classes $C_i$, i = 0,1,2,3, according to the sign of the Δ-quasi-derivatives and obtain sufficient conditions in order that $C_i=\varnothing$. Also, we establish...

The aim of this paper is to study asymptotic properties of the solutions of the third order delay differential equation $${\left(\frac{1}{r\left(t\right)}{y}^{\text{'}}\left(t\right)\right)}^{\text{'}\text{'}}-p\left(t\right)y^{\text{'}}\left(t\right)+g\left(t\right)y\left(\tau \left(t\right)\right)=0.*$$ Using suitable comparison theorem we study properties of Eq. () with help of the oscillation of the second order differential equation.

In this paperwe study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term,we define a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered.

In this paper, we prove and discuss averaging results for ordinary differential equations perturbed by a small parameter. The conditions we assume on the right-hand sides of the equations under which our averaging results are stated are more general than those considered in the literature. Indeed, often it is assumed that the right-hand sides of the equations are uniformly bounded and a Lipschitz condition is imposed on them. Sometimes this last condition is relaxed to the uniform continuity in...

This contribution is devoted to the problem of asymptotic behaviour of solutions of scalar linear differential equation with variable bounded delay of the form $$\dot{x}\left(t\right)=-c\left(t\right)x(t-\tau \left(t\right)){(}^{*})$$ with positive function $c\left(t\right).$ Results concerning the structure of its solutions are obtained with the aid of properties of solutions of auxiliary homogeneous equation $$\dot{y}\left(t\right)=\beta \left(t\right)[y\left(t\right)-y(t-\tau \left(t\right))]$$ where the function $\beta \left(t\right)$ is positive. A result concerning the behaviour of solutions of Eq. (*) in critical case is given and, moreover, an analogy with behaviour of solutions of...

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.