Exponential stability of nonautonomous linear differential equations with linear perturbations by Liao methods

In this paper, the author considers, by Liao methods, the stability of Lyapunov exponents of a nonautonomous linear differential equations: with linear small perturbations. It is proved that, if A(t) is a upper-triangular real n by n matrix-valued function on , continuous and uniformly bounded, and if there is a relatively dense sequence in , say 0=T00, a constant δ>0, such that for every linear equations (B), satisfying , where the real n by n matrix-valued function B(t) is continuous with respect to , one has where , is the solution of Eq. (B) with . For the nonuniformly expanding case, there is a similar statement.