On Determining Phase Spaces for Functional Differential Equations

Funkcialaj Ekvacioj 31, (1988) 331-347

On Determining Phase Spaces for Functional Differential Equations By F. V. ATKINSON and J. R. HADDOCK* (University of Toronto, Canada and Memphis State University, U. S. A.)

1. Introduction In recent years, several investigators have studied properties (or axioms) of admissible spaces (solution spaces; phase spaces) for functional differential equations (FDEs) with infinite delay. The basic idea often has been to assign conditions to admissible spaces in such a way as to generate, among other things, standard existence-uniqueness-continuous dependence-continuation-orbit precompactness results (see, for example, [8]). -issible The purpose of this paper is to demonstrate that a certain class of spaces?C spaces?frequently arise in a natural way for many types of Volterra integrodifferential equations. Since definiotions of admissible spaces have been given in a variety of forms and in numerous papers, we do not list the properties here. Instead, we refer to [4], [5] or [6] for the definition we are using and for various discussions and additional references regarding these spaces. However, spaces. in Section 2, we give a detailed presentation of the aforementioned In particular, we list the properties needed for admissibility. In Sections 3 and 4 spaces, can be space, or a family of we provide results that indicate how a generated from a given Volterra integrodifferential equation. Also, fundamental results in differential equations will be discussed in connection with a given equation and these corresponding spaces. Before pursuing the matters mentioned in the above paragraphs, we briefly , $-¥infty0$

$g$

$0]$

Theorem 3.1.

Suppose

(3.7)

for some

$¥mathrm{C}$

is an

$n¥times n$

matrix

of continuous functions

and

$¥int_{-¥infty}^{0}|¥mathrm{C}(-s)|ds0$ ,

$g$

:

$M_{K}(t, ¥phi):=¥int_{-K}^{0_{1}}Q(t, t+s, ¥phi(s))|ds$

is continuous in

Proof, of

Let

$¥delta_{0}>0(¥delta_{0}¥leq 1)$

$t(t¥geq 0)$

$a>0$ ,

uniformly for

$¥phi$

in

$¥mathrm{C}_{g}$

with

$|¥phi|_{g}¥leq¥alpha$

.

be given. We wish to prove the existence such that $|t-t_{0}|0$

and

$t_{0}¥geq 0$

$|¥int_{-K}^{0}|Q(t, t+s, ¥phi(s))|ds-¥int_{-K}^{0}|Q(t_{0}, t_{0}+s, ¥phi(s))|ds|0$

$t_{0}¥geq 0$

$¥epsilon_{n}=1/n^{2}$

$¥delta_{0}>0$

.

For the given such that

$(¥delta_{0}¥leq 1)$

345

Phase Spaces for FDE $|¥int_{-¥infty}^{0}Q(t, t+s, ¥phi(s))ds-¥int_{-¥infty}^{0}Q(t_{0}, t_{0}+s, ¥phi(s))ds|0$ such that $¥phi$

$|¥phi|_{g}¥leq¥alpha$

$¥mathrm{C}_{g}$

$¥int_{-¥infty}^{-K}|Q(t, t+s, ¥phi(s))|ds0$

and such that

$t¥geq 0$

$(¥delta_{0}¥leq 1)$

$|¥phi|_{g}¥leq¥alpha$

as

$ L¥rightarrow¥infty$

For this $K$ , Lemma 4.2 guarantees the existence and imply

.

$t¥geq 0$

$|t-t_{0}|