Asymptotic solutions of nondiagonal linear difference systems

Abstract

This paper, relying on dichotomic properties of the matrix difference system $ W(n+1)=A(n)W(n)A^{-1}(n)$, gives conditions under which a perturbed system $ y(n+1)=(A(n)+B(n))y(n)$, by means of a nonautonomous change of variables $ y(n)=S(n)x(n)$, can be reduced to the form $ x(n+1)=A(n)x(n)$. From this, a theory of asymptotic integration of the perturbed system follows, where the linear system $ x(n+1)=A(n)x(n)$ is nondiagonal. As a consequence of these results, we prove that the diagonal system $ x(n+1)=\Lambda(n)x(n)$ has a Levinson dichotomy iff system $ W(n+1)=\Lambda(n)W(n)\Lambda^{-1}(n)$ has an ordinary dichotomy.