The asymptotic bounds of solutions of linear delay systems
The author investigates asymptotic bounds of all solutions \(y\) to vector linear delay differential equations \[ y'(x)= A(x)y \bigl(\tau(x)\bigr) +B (x)y(x), x\in I= [x_0, \infty), \] in terms of solutions to the following scalar functional equations: \[ \bigl\| A(x) \bigr\| \varphi\bigl( \tau(x)\bigr) +b\psi (x)=0,\;x\in I, \quad \varphi \bigr( \tau(x) \bigr)= \varphi(x)-1,\;x\in I. \] More precisely, it is shown that \(| y(x) |=O (\psi(x) L^{\varphi(x)})\) as \(x\to \infty\), where \(L\) is a real positive constant. Here \(A\) and \(B\) are real function \(n \times n\) matrices which are continuous on the interval \(I\) and \(\tau: I\to\mathbb{R}\) is a continuous increasing function such that \(\tau(x) <x\) for \(x\in I\) and \(\tau(x) \to \infty\) as \(x\to \infty\). Some other results in this direction are derived and a few examples illustrating the results are presented.
vector linear delay differential equations
scientific article; zbMATH DE number 1218227