, Volume 26, Issue 7, pp 1287-1298
Date: 15 Jun 2010

Continuity in weak topology: First order linear systems of ODE

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Abstract

In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L p topologies, but also with respect to the weak topologies of the L p spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.

The second author is supported by National Basic Research Program of China (Grant No. 2006CB805903) and National Natural Science Foundation of China (Grant Nos. 10325102 and 10531010)