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A perturbation theorem for exponential dichotomies

Published online by Cambridge University Press:  14 November 2011

Kenneth J. Palmer
Kenneth J. Palmer
Affiliation:
Department of Mathematics and Computer Science, University of Miami, Coral Gables, FL 33124, U.S.A.
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Synopsis

Suppose the linear equation x' = A(t)x has an exponential dichotomy and suppose B(t) is close to A(t) in the following sense: on any interval of length 2T, B(t) is close to some translate A(t + τ) of A(t) (actually the conditions in the paper are slightly weaker than this). Then if T is sufficiently large, the equation y' = B(t)y also has an exponential dichotomy. This generalises the usual roughness theorem for exponential dichotomies.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 1-2 , 1987 , pp. 25 - 37
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Coppel, W. A.. Dichotomies in Stability Theory, Lecture Notes in Mathematics 629 (Berlin: Springer-Verlag, 1978).Google Scholar
2Johnson, R. A.. Remarks on linear differential systems with measurable coefficients (preprint).Google Scholar
3Massera, J. L. and Schäffer, J. J.. Linear Differential Equations and Function Spaces (New York: Academic Press, 1966).Google Scholar
4Sacker, R. J. and Sell, G. R.. Existence of dichotomies and invariant splittings for linear differential systems I. J. Differential Equations 15 (1974), 429–458.CrossRefGoogle Scholar
5Sacker, R. J. and Sell, G. R., A spectral theory for linear differential systems. J. Differential Equations 27 (1978), 320–358.CrossRefGoogle Scholar
6Sell, G. R.. Topological Dynamics and Ordinary Differential Equations (New York: Van Nostrand Reinhold, 1971).Google Scholar