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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An improved estimate in the method of freezing


Author: Robert E. Vinograd
Journal: Proc. Amer. Math. Soc. 89 (1983), 125-129
MSC: Primary 34C11; Secondary 34D05
DOI: https://doi.org/10.1090/S0002-9939-1983-0706524-1
MathSciNet review: 706524
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Abstract: Let $ \dot x = A(t)x$ and $ {\lambda _k}(t)$ be the eigenvalues of the matrix $ A(t)$. The main result of the Method of Freezing [1] states that if $ \left\Vert {A(t)} \right\Vert \leqslant a$. Re $ {\lambda _k}(t) \leqslant {\lambda _0}$ and $ \left\Vert {A(t) - A(s)} \right\Vert \leqslant \delta \left\vert {t - s} \right\vert$, then

$\displaystyle \left\Vert {x(t)} \right\Vert \leqslant \left\Vert {x({t_0})} \ri... ...}\exp ({\lambda _0} + 2a{\lambda _\delta })(t - {t_0})\quad (t \geqslant {t_0})$

for all solutions of the system, where

$\displaystyle {\lambda _\delta } = {({C_n} \cdot \delta /4{a^2})^{1/(n + 1)}}$

.

The previous best known value. $ {C_n} = n(n + 1)/2$, is reduced to the substantially smaller value $ 2{n^n}{e^{ - n}}/(n - 1)! < \sqrt {2n/\pi } $.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0706524-1
Article copyright: © Copyright 1983 American Mathematical Society