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.