Vol. 111, No. 1, 1984

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ISSN: 0030-8730
Asymptotic behavior of a perturbed neutral functional-differential equation related to the solution of the unperturbed linear system

A. F. Izé and A. Ventura

Vol. 111 (1984), No. 1, 57–91
Abstract

In this paper we consider the problem of the relative asymptotic equivalence of the solutions of the systems

d-Dy = L (y )
dt  t     t
(1)

and

-d[Dx  − G(t,x )] = L(x )+ f(t,x ),
dt    t      t       t       t
(2)

where (1) is a linear system of neutral functional differential equations. The main theorem gives conditions under which the following result is verified. Given a solution yt of (1) there exists a solution xt of (2) such that

   ∥xt −-yt∥
lim   ∥yt∥   = 0.
(*)

The converse of this result, namely given a solution xt of (2) there is a solution yt of (1) such that (*) is satisfied is partially proved. A counterexample is given to show that the converse result is not true in general.

Mathematical Subject Classification 2000
Primary: 34K25
Milestones
Received: 2 April 1981
Revised: 14 June 1982
Published: 1 March 1984
Authors
A. F. Izé
A. Ventura