Stability of linear differential equations with periodic coefficients in Hilbert space
Gert Almkvist · Pacific Journal of Mathematics · 1966
In this paper we study the stability of the solutions of the differential equation (1) u'(t) = A(t) u(t) for ί ^ 0 in a separable Hubert space.It is assumed that A(f) is periodic with period one and satisfies the following symmetry condition: There exists a continuous constant invertible operator Q such that A(t)* = -Q A(t) Q-1 for all t £ 0 .We use a perturbation technique.Let A(t) = Ao(t)-\-B(f) where Ao(t) is compact and antihermitian for all t.We denote by Uo(t) the solution operator of u ι (t) = Ao(t)u(t).It is shown that (1) is stable if B(t) satisfies a certain smallness condition involving the distribution of the eigenvalues of Ϊ7o(l) and the action of B(t) on the eigenvectors of ϋi(l).The results can be applied to the second order equation y" + C(f)y = 0 where C(t) is selfadjoint for all t.