Optimal estimates for the uncoupling of differential equations (Q1915997)

The authors consider the autonomous differential equation (1) \(\dot z= Cz+ H(z)\), where \(C= A\times B\) is a bounded linear operator on a Banach space \(Z= X\times Y\) and \(H: Z\to Z\) is a Lipschitz continuous function with \(H(0)= 0\). It is assumed that a gap exists in the spectrum of \(C\). Optimal conditions on this spectral gap are obtained for the existence of a conjugacy between (1) and the system of uncoupled equations \((\dot x, \dot y)= (Ax, By)+ (F(x, \Phi(x)), G(\Psi(y), y))\), where \(z= (x, y)\), \(H(z)= (F(x, y), G(x, y))\), and \(\Phi: X\to Y\), \(\Psi: Y\to X\) are Lipschitz functions. Optimal conditions are also given for the regularity of the invariant manifolds graph \(\Phi\) and graph \(\Psi\), and for the regualarity of the leaves of two foliations of the phase space associated with the uncoupling.