Some properties of majorants of Lyapunov exponents for systems with unbounded coefficients

Abstract

A number of well-known results on majorants of Lyapunov exponents of linear differential systems with bounded coefficients on the half-line (such as their simultaneous attainability in the class of infinitesimal perturbations, belonging to the second Baire class in the compact-open topology, etc.) are generalized to systems with unbounded coefficients.For n ∈ N, consider the set M n of linear differential systemṡwith continuous (not necessarily bounded) operator functions A : R + → End R n (which are identified with the corresponding systems, and we write A ∈ M n ). We equip the set M n with the structure of a linear space over R with the natural operations of multiplication by a number and addition and with the uniform topology (the topology of uniform convergence of the coefficients of the systems on the half-line) defined by the metric (A, B) = sup t∈R + min{|A(t) − B(t)|, 1}, where we have set |Y | = sup{|Y x| : |x| = 1}, Y ∈ End R n , and |x| = x 2 1 + · · · + x 2 n , x = (x 1 , . . . , x n ) T . This topological space is denoted by M n U . By M n we denote the subset of M n that consists of all systems with bounded coefficients on the half-line. Definition 1. The Lyapunov exponents of system (1) are the numbers [1]where G i (R n ) is the set of i-dimensional subspaces of the space R n , the mapping Y | L is the restriction of the mapping Y to the set L, and X A (· , ·) is the Cauchy operator of system (1).In our notation, unlike [1], the Lyapunov exponents are numbered in nondescending order. Since we do not assume that the coefficients of the systems are bounded on the half-line, it follows that their Lyapunov exponents are points of the extended numerical line R ≡ R {−∞, ∞}. (As usual, we assume that −∞ < r < ∞ for each r ∈ R and −∞ < ∞.) If all Lyapunov exponents of system (1) are finite, then they coincide with the numbers defined in [2, p. 63].1279