The paper investigates the singular initial problem[4pt] ${\left(p\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\left(t\right)\right)=0,u\left(0\right)=u_0,u^{\text{'}}\left(0\right)=0$[4pt] on the half-line $[0,\infty )$. Here $u_0\in [L_0,L]$, where $L_0$, $0$ and $L$ are zeros of $f$, which is locally Lipschitz continuous on $ℝ$. Function $p$ is continuous on $[0,\infty )$, has a positive continuous derivative on $(0,\infty )$ and $p\left(0\right)=0$. Function $q$ is continuous on $[0,\infty )$ and positive on $(0,\infty )$. For specific values $u_0$ we prove the existence and uniqueness of damped solutions of this problem. With additional conditions for $f$, $p$ and $q$ it is shown that the problem has for each specified $u_0$ a unique...

In this paper we examine some features of the global dynamics of the four-dimensional system created by Lou, Ruggeri and Ma in 2007 which describes the behavior of the AIDS-related cancer dynamic model in vivo. We give upper and lower ultimate bounds for concentrations of cell populations and the free HIV-1 involved in this model. We show for this dynamics that there is a positively invariant polytope and we find a few surfaces containing omega-limit sets for positive half trajectories in the positive...

The unstable properties of the linear nonautonomous delay system $x^{\text{'}}\left(t\right)=A\left(t\right)x\left(t\right)+B\left(t\right)x(t-r\left(t\right))$, with nonconstant delay $r\left(t\right)$, are studied. It is assumed that the linear system $y^{\text{'}}\left(t\right)=(A\left(t\right)+B\left(t\right))y\left(t\right)$ is unstable, the instability being characterized by a nonstable manifold defined from a dichotomy to this linear system. The delay $r\left(t\right)$ is assumed to be continuous and bounded. Two kinds of results are given, those concerning conditions that do not include the properties of the delay function $r\left(t\right)$ and the results depending on the asymptotic properties of the...

In this paper we have considered completely the equation $$y^{\text{'}\text{'}\text{'}}+a\left(t\right)y^{\text{'}\text{'}}+b\left(t\right)y^{\text{'}}+c\left(t\right)y=0,(*)$$ where $a\in C^2([\sigma ,\infty ),R)$, $b\in C^1([\sigma ,\infty ),R)$, $c\in C\left(\right[\sigma ,\infty ),R)$ and $\sigma \in R$ such that $a\left(t\right)\le 0$, $b\left(t\right)\le 0$ and $c\left(t\right)\le 0$. It has been shown that the set of all oscillatory solutions of (*) forms a two-dimensional subspace of the solution space of (*) provided that (*) has an oscillatory solution. This answers a question raised by S. Ahmad and A.  C. Lazer earlier.

Conditions are given for a class of nonlinear ordinary differential equations $x^{\text{'}\text{'}}+a\left(t\right)w\left(x\right)=0$, $t\ge t_0\ge 1$, which includes the linear equation to possess solutions $x\left(t\right)$ with prescribed oblique asymptote that have an oscillatory pseudo-wronskian $x^{\text{'}}\left(t\right)-\frac{x\left(t\right)}{t}$.

Our aim in this paper is to present sufficient conditions under which all solutions of (1.1) tend to zero as $t\to \infty$.