Electron. J. Diff. Eqns., Vol. 2008(2008), No. 97, pp. 1-7.

Positivity of the Green functions for higher order ordinary differential equations

Michael I. Gil'

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.

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Michael I. Gil'
Department of Mathematics
Ben Gurion University of the Negev
P.0. Box 653, Beer-Sheva 84105, Israel
Email: gilmi@cs.bgu.ac.il

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