The paper discusses the asymptotic properties of solutions of the scalar functional differential equation $y^{\text{'}}\left(x\right)=ay\left(\tau \left(x\right)\right)+by\left(x\right),x\in [x_0,\infty ]$. Asymptotic formulas are given in terms of solutions of the appropriate scalar functional nondifferential equation.

We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated.

Our aim in this paper is to obtain sufficient conditions under which for every $\xi \in R^n$ there exists a solution $x$ of the functional differential equation $\dot{x}\left(t\right)={\int }_c^t\left[d_{s}Q(t,s)\right]f(t,x\left(s\right)),t\in [t_0,T]$ such that $li{m}_{t\to T-}x\left(t\right)=\xi$.

In this paper we consider the third-order nonlinear delay differential equation (*) $${\left(a\left(t\right){\left(x^{\text{'}\text{'}}\left(t\right)\right)}^{\gamma }\right)}^{\text{'}}+q\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)=0,t\ge t_0,$$ where $a\left(t\right)$, $q\left(t\right)$ are positive functions, $\gamma >0$ is a quotient of odd positive integers and the delay function $\tau \left(t\right)\le t$ satisfies ${lim}_{t\to infty}\tau \left(t\right)=infty$. We establish some sufficient conditions which ensure that (*) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria....

The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation $${\left[a\left(t\right){\left({[x\left(t\right)+p\left(t\right)x\left(\delta \left(t\right)\right)]}^{\text{'}\text{'}}\right)}^{\alpha }\right]}^{\text{'}}+q\left(t\right)x^{\alpha }\left(\tau \left(t\right)\right)=0,E$$ where $\alpha >0$, $0\le p\left(t\right)\le p_0<\infty$ and $\delta \left(t\right)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

We obtain some sufficient conditions for the existence of the solutions and the asymptotic behavior of both linear and nonlinear system of differential equations with continuous coefficients and piecewise constant argument.

Systems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.

The existence, uniqueness and asymptotic stability of weak solutions of functional-differential abstract nonlocal Cauchy problems in a Banach space are studied. Methods of m-accretive operators and the Banach contraction theorem are applied.