Asymptotic relationship between solutions of two linear differential systems

summary:In this paper new generalized notions are defined: ${\bold\Psi}$-boundedness and ${\bold\Psi}$-asymptotic equivalence, where ${\bold\Psi}$ is a complex continuous nonsingular $n\times n$ matrix. The ${\bold\Psi}$-asymptotic equivalence of linear differential systems $ y'= A(t) y$ and $ x'= A(t) x+ B(t) x$ is proved when the fundamental matrix of $ y'= A(t) y$ is ${\bold\Psi}$-bounded