Subspaces of Stable and Unstable Solutions of a Functional Differential Equation in a Fading Memory Space: The Critical Case

Abstract

We study the asymptotic behavior of the linear infinite delay, autonomous system of functional differential equations \[( * )\qquad \begin{array}{*{20}c}x'(t) + \mu *x(t) = f(t)\quad (t \geqq 0), \\ x(t) = \phi (t)\quad (t \leqq 0). \\\end{array} \] Here $\mu $ is an n-dimensional matrix-valued measure supported on $[0,\infty )$, finite with respect to a weight function, and f,$\phi $ and x are $C^n $-valued continuous or locally integrable functions bounded with respect to a fading memory norm. We find conditions that ensure that the state space of $(*)$ can be written as a direct sum of a stable subspace, which is characterized by the fact that solutions are small at infinity, a finite dimensional central-stable subspace in which solutions are neither small nor large at infinity, and a finite dimensional exponentially unstable subspace consisting of exponentially growing solutions. We give estimates for the rate of decay at infinity of solutions belonging to the stable subspace. Our results extend earlier work of Staffans [10], [11] since we analyze the critical case in which the components of the solutions are not exponentially separated, as well as the noncritical case.