A Generalization Of Global Topological Linearization Theorem

The classical conclusions of topological linearication of the differential equation x' -Ax + h(x) (none of the eigenvalues of A has zero real part) are given by Hartman and Grobman. But, their conclusions are only limited to the small neighborhood of origin. Later, Palmer shows that there is a homeomorphim H(R R) sending the solutions of the system x' = A(t)x + h(l,x) onto the solutions of its linear system x' - A(t)x if h(t,x) is bounded. That is global linearization.Professor Shi Jinlin omits the limitation that h(x) must be bounded and points out that x' - Ax + h(x) can be topologically linearlized when h(x) has a proper(where Re (A)<0 , Re (B) >0 , f(x) , (x) are unbounded) , while the firstequation x' = Ax + f(x) only includes x , but not includes y . This willundoubtedly confine the adaptation scope of the conclusion.What we discuss is the system Where x Rn,y Rm, A is a n n matrix, B is a m m matrix. f(x) ( Rn - Rn ), g(y) ( Rm - Rn ), (x) ( Rn - Rm ) and (y) ( Rm - Rm ) are continuous functions.This system allows the first equation also includes y and is more general. We points out that the system can be topologically linearlized under some conditions. Then, we can obtain more general conclusion.