Asymptotic behavior at infinity between the solutions of two systems of ordinary differential equations (Q1108454)

Consider the systems of ordinary differential equations (1) \(\dot x=A(t)x+f_ 1(t,x)\), (2) \(\dot y=A(t)y=f_ 2(t,y)\) where \(x,y\in X=R^ n\) (or \(C^ n)\), A(t) is \(n\times n\), \(t\in J=[0,\infty)\), \(f_ 1(t,x)\) and \(f_ 2(t,y)\) defined on \(J\times X\). Equations (1) and (2) are asymptotically equivalent if for each bounded solution \(x=x(t)[y=y(t)]\) of (1) [(2)] there exists a bounded solution \(y=y(t)[x=x(t)]\) of (2) [(1)] such that [x(t)-y(t)]\(\to 0\) as \(t\to \infty\). Such a concept can be also formulated in the following way: To each solution \(x(t)\in L^{\infty}[y(t)\in L^{\infty}]\) of (1) [(2)] there corresponds a solution \(y(t)\in L^{\infty}[x(t)\in L^{\infty}]\) of (2) [(1)] such that \(x(t)-y(t)\in L_ 0^{\infty}\). The main purpose of this paper is to use suitable Banach spaces, stronger than L(J,X), to obtain more general results related to the problem under consideration. We plan to carry out some applications by exploiting the new ideas extending the definition of the asymptotic equivalence.