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: dx→/dt=A(t)x→((t,x→)∈R+×Rn) with linear small perturbations. It is proved that, if A(t)A(t) is a upper-triangular real n by n   matrix-valued function on R+R+, continuous and uniformly bounded, and if there is a relatively dense sequence {Ti}0∞ in R+R+, say 0=T00ε>0, a constant δ>0δ>0, such that for every linear equations (B)(B), dz→/dt=B(t)z→((t,z→)∈R+×Rn) satisfying supt∈R+‖B(t)−A(t)‖<δsupt∈R+‖B(t)−A(t)‖<δ, where the real n by n   matrix-valued function B(t)B(t) is continuous with respect to t∈R+t∈R+, one haslim supt→+∞1tlog‖z→(t;z→0)‖<χ*+(A)+ε(∀z→0∈Rn), where z→(t)=z→(t;z→0), is the solution of Eq. (B)(B) with z→(0)=z→0. For the nonuniformly expanding case, there is a similar statement.