Estimates for the Green functions of nonautonomous higher order differential equations

Gil’, Michael

doi:10.1090/S0002-9939-09-09829-3

Estimates for the Green functions of nonautonomous higher order differential equations
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Proc. Amer. Math. Soc. 137 (2009), 3045-3055 Request permission

Abstract:

We consider the equation \[ \sum _{k=0}^{n} a_k(t)x^{(n-k)}(t)=0\;\;(t > 0),\] where $a_0(t)\equiv 1;\;a_k(t)\;(k=1, ..., n)$ are bounded continuous functions. It is assumed that all the roots $r_k(t)\;\;(k=1, ..., n)$ of the polynomial $z^n+a_1(t)z^{n-1}+ ... +a_n(t)$ are real for all $t\geq 0$. Sharp estimates for the Green function to the Cauchy problem and their derivatives are derived.

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