The linear homogeneous differential equation with variable delays $ẏ\left(t\right)={\sum }_{j=1}^n{\alpha }_j\left(t\right)[y\left(t\right)-y(t-{\tau }_j\left(t\right))]$ is considered, where ${\alpha }_j\in C(I,ℝ͞͞⁺)$, I = [t₀,∞), ℝ⁺ = (0,∞), ${\sum }_{j=1}^n{\alpha }_j\left(t\right)>0$ on I, ${\tau }_j\in C(I,ℝ⁺),$ the functions $t-{\tau }_j\left(t\right)$, j=1,...,n, are increasing and the delays ${\tau }_j$ are bounded. A criterion and some sufficient conditions for convergence of all solutions of this equation are proved. The related problem of nonconvergence is also discussed. Some comparisons to known results are given.

We show that if the Lyapunov exponents of a linear delay equation x′ = L(t)x t are limits, then the same happens with the exponential growth rates of the solutions to the equation x′ = L(t)x t + f(t, x t) for any sufficiently small perturbation f.

It is proved that under some conditions the set of solutions to initial value problem for second order functional differential system on an unbounded interval is a compact $R_{\delta }$-set and hence nonvoid, compact and connected set in a Fréchet space. The proof is based on a Kubáček’s theorem.

The neutral differential equation (1.1) $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}[x\left(t\right)+x(t-\tau )]+\sigma F(t,x\left(g\left(t\right)\right))=0,$$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma =\pm 1$, $F(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and is nondecreasing in $u\in (0,\infty )$, and $limg\left(t\right)=\infty$ as $t\to \infty$. It is shown that equation (1.1) has a solution $x\left(t\right)$ such that (1.2) $$\underset{t\to \infty }{lim}\frac{x\left(t\right)}{t^k}\text{exists}\text{and}\text{is}\text{a}\text{positive}\text{finite}\text{value}\text{if}\text{and}\text{only}\text{if}{\int }_{t_0}^{\infty }{t}^{n-k-1}F(t,c{\left[g\left(t\right)\right]}^k)\mathrm{d}t<\infty \text{for}\text{some}c>0.$$ Here, $k$ is an integer with $0\le k\le n-1$. To prove the existence of a solution $x\left(t\right)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used.

In this paper we have considered a nonlinear and nonautonomous stage-structured HIV/AIDS epidemic model with an imperfect HIV vaccine, varying total population size and distributed time delay to become infectious due to intracellular delay between initial infection of a cell by HIV and the release of new virions. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique....

The integrodifferential system with aftereffect (“heredity” or “prehistory”) dx/dt=Ax+-t R(t,s)x(s,)ds, is considered; here $\epsilon$ is a positive small parameter, $A$ is a constant $n\times n$ matrix, $R(t,s)$ is the kernel of this system exponentially decreasing in norm as $t\to \infty$. It is proved, if matrix $A$ and kernel $R(t,s)$ satisfy some restrictions and $\epsilon$ does not exceed some bound ${\epsilon }_{*}$, then the $n$-dimensional set of the so-called principal two-sided solutions $\tilde{x}(t,\epsilon )$ approximates in asymptotic sense the infinite-dimensional set of solutions...

We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions are written in terms of the parameters and are explicit for the case of asymptotic decay. For exponential decay, they are easily resolvable numerically. The analytical method is based on construction of a Lyapunov functional (asymptotic decay) and a forward-backward estimate for the square mean (exponential decay).