Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 97, pp. 1-7.
Title: Positivity of the Green functions for higher order
ordinary differential equations
Author: Michael I. Gil' (Ben Gurion Univ. of the Negev, Israel)
Abstract:
We consider the equation
$$
\sum_{k=0}^n a_k(t)x^{(n-k)}(t)=0,\quad t\geq 0,
$$
where $a_0(t)\equiv 1$, $a_k(t)$ ($k=1, \dots, n$)
are real bounded functions.
Assuming that all the roots of the polynomial
$z^n+a_1(t)z^{n-1}+ \dots +a_n(t)$ ($t\geq 0$) are real,
we derive positivity conditions for the Green function for the
Cauchy problem. We also establish
a lower estimate for the Green function
and a comparison theorem for solutions.
Submitted May 8, 2008. Published July 25, 2008.
Math Subject Classifications: 34C10, 34A40.
Key Words: Linear ODE; Green function; fundamental solution;
positivity; comparison of solutions.