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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
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 } $.

References [Enhancements On Off] (What's this?)

  • [1] B. F. Bylov, D. M. Grobman, V. V. Nemyckiĭ and R. E. Vinograd, The theory of Lyapunov exponents, "Nauka", Moscow, 1966, pp. 130-138. (Russian) MR 0206415 (34:6234)
  • [2] W. A. Coppel, Dichotomies in stability theory, Springer-Verlag, Berlin and New York, 1978, p. 4. MR 0481196 (58:1332)
  • [3] Yu. L. Daleckiĭ and M. G. Kreĭn, Stability of solutions of differential equations in Banach space, Transl. Math. Mono., vol. 43, Amer. Math. Soc., Providence, R. I., 1974, p. 57. MR 0352639 (50:5126)

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