We establish conditions for the linear differential equation y(t) + p(t)y(g(t)) = 0 to have property A. Explicit sufficient conditions for the oscillation of the the equation is obtained while dealing with the property A of the equations. A comparison theorem is obtained for the oscillation of the equation with the oscillation of a third order ordinary differential equation.

In this article, we analyze the first order linear delay differential equation
 \begin{equation*}
 x^{\prime }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\geq t_{0},
 \end{equation*}
 where $p,$ $\tau \in C\left( [t_{0},\infty ),\mathbb{R}^{+}\right) $ and $%
 (t)\leq t,\ \lim_{t\rightarrow \infty }\tau (t)=\infty $. Under assumption that (t)$ is not necessarily monoton...

We discuss the solution of linear second order differential-algebraic equations (DAEs) with variable coefficients. Since index reduction and order reduction for higher order, higher index differential-algebraic systems do not commute, appropriate index reduction methods for higher order DAEs are required. We present an index reduction method based on derivative arrays that allows to determine a...

In this paper, we consider the cost of null controllability for a large class of linear equations of parabolic or dispersive type in one space dimension in small time. By extending the work of Tenenbaum and Tucsnak in New blow-up rates for fast controls of Schrödinger and heat equations, we are able to give precise upper bounds on the time-dependance of the cost of fast controls when the time o...