\(L_p\)-perturbations of invariant subbundles for linear systems
Let \(x= f_i(t)\), \(i= 1,\dots, n\), be linearly independent solutions to the differential equation \[ x'= A(t) x,\quad x\in\mathbb{R}^n,\quad t\geq 0,\tag{1} \] which have asymptotically separated direction \(\Phi_i(t)= f_i(t)|f_i(t)|^{-1}\): \[ \liminf_{t\to+\infty} |\Phi_j(t)- \Phi_it(t)|> 0,\quad j\neq k.\tag{2} \] Consider the perturbed system \[ y'= (A(t)+ R(t)) y\tag{3} \] with \(R(t)\) ``close to zero'' in some appropriate sense. In this paper, the authors investigate when for any ``small'' \(R(t)\) equation (3) has at least one family of solutions \(f_i(t, R)\) whose directions \(\Phi_i(t,R)= f_i(t, R)|f_i(t,R)|^{-1}\) are close to \(\Phi_i(t)\) for all \(t\geq 0\). In the case, when such a family exists, they propose to study the degree of proximity between the logarithms \(\lambda_i(t,R)= \ln|f_i(t, R)|\) and \(\lambda_i(t)= \ln|f_i(t)|\). The authors use Riccati equations and the ordinary and exponential dichotomies to get simple recurrent formula for the asymptotic integration of linear systems subjected to \(L_p\)-perturbations with arbitrary \(p\geq 1\). Moreover, they establish conditions which are necessary and sufficient for the persistence of one-dimensional invariant subbundles for the linear system under \(L_p\)-perturbations. In this way, they prove the sharp nature of the well known Levinson and Hartman-Wintner asymptotic theorems.
Levinson and Hartman-Wintner asymptotic theorems
scientific article; zbMATH DE number 1879630