Science in China Series A: Mathematics Jun., 2008, Vol. 51, No. 6, 1036–1058 www.scichina.com math.scichina.com www.springerlink.com Continuity in weak topology: higher order linear systems of ODE ZHANG MeiRong Department of Mathematical Sciences, Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China (email: mzhang@math.tsinghua.edu.cn) Abstract We will introduce a type of Fredholm operators which are shown to have a certain con- tinuity 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. Keywords: continuity, weak topology, Fredholm operator, linear system, Hill’s equation, eigen- value, rotation number, Lyapunov exponent MSC(2000): 34A30, 34L40, 37E45, 34D08, 58C07, 46B50 1 Introduction Given an a(t) ∈ L 1 (S 2π )= L 1 (S 2π , R), called a potential, where S 2π = R/2πZ. The Hill’s equation x ′′ + a(t)x =0,x ∈ R, defines the rotation number ρ = ρ(a) ∈ [0, ∞). With a choice of the Sturm-Liouville boundary conditions or the periodic, or the anti-periodic boundary conditions, equation x ′′ +(λ + a(t))x = 0 defines a sequence of (real) eigenvalues {λ m (a)} m0 , where, for the Dirichlet boundary condition, λ 0 (a) is understood voidly. It will be proved in this paper that as functionals of a ∈ L 1 (S 2π ), all of these important quantities have a very strong continuity in potentials a(t). That is, (L 1 (S 2π ),w) → R, a → ρ(a), (L 1 (S 2π ),w) → R, a → λ m (a), are continuous, where w indicates the topology of weak convergence in L 1 (S 2π ). See Section 4 and Section 5 respectively. In order to prove these results, we will consider in Section 3 general second-order linear systems of ordinary differential equations (ODE) x ′′ + a(t)x =0, x ∈ R d , Received July 16, 2007; accepted September 17, 2007 DOI: 10.1007/s11425-008-0011-5 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)