Density of images of semigroup operators for linear neutral functional differential equations

Abstract

In this paper we consider properties of a system described by a linear functional differential equation of a neutral type. By T(t) we mean the operator: an initial state --+ the state at t. The state space is the Sobolev space WI’, hence T(t) maps W,P into itself, but other spaces may also be used. We mention, for example, that Hale [7] has considered the space of continuous functions C and Manitius [l] the space LTP x L2 (retarded case only). The main problem we consider here is to find conditions for density of the image of the operator T(t) (in the considered space). This problem was solved by Jakubczyk [2] for a simple neutral system described by a difference-differential equation. His paper was an inspiration for the author to solve this problem in a general case. The method used in our paper is based on properties of convolution operators and a ring structure of such operators. These ideas may be found in the papers of Jakubczyk [2] and that of Kamen [3]. In Section 3 we prove that a convolution operator in the space LP([O, T]; Iwn), T < +co, has a dense image if and only if a determinant of a kernel matrix of this operator over the ring K (defined in Section 2) is not a zero-divisor in this ring. This fact is then used to derive a criterion for density of im T(t). In the proof of the main result we use some representation of the operator T(h). A similar representation was used by Manitius in [l]. Density of the image of T(t) means that the system described by the functional differential equation is, in some sense, nondegenerate in the state space. In a neutral case the image of T(t) may cover the whole state space. The problem considered in this paper is closely related to the problem of completness of eigenfunctions associated with the system. Completness means that the space spanned by generalized eigenfunctions is dense in the state space. From papers of Henry [8] and Manitius [l], it follows that the above problems are equivalent (in the spaces C and [w” x L”). Thus our main result on density of im T(t) gives