Continuity in weak topology: higher order linear systems of ODE

We will introduce a type of Fredholm operators which are shown to have a certain continuity in weak topologies. From this, we will prove that the fundamental matrix solutions of k-th, k ⩾ 2, order linear systems of ordinary differential equations are continuous in coefficient matrixes with weak topologies. Consequently, Floquet multipliers and Lyapunov exponents for periodic systems are continuous in weak topologies. Moreover, for the scalar Hill’s equations, Sturm-Liouville eigenvalues, periodic and anti-periodic eigenvalues, and rotation numbers are all continuous in potentials with weak topologies. These results will lead to many interesting variational problems. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10325102, 10531010), the National Basic Research Program of China (Grant No. 2006CB805903), and Teaching and Research Award Program for Outstanding Young Teachers, Ministry of Education of China (2001)