Fixed points, stability, and harmless perturbations
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Much has been written about systems in which each constant is a solution and each solution approaches a constant. It is a small step to conjecture that functions promoting such behavior constitute harmless perturbations of stable equations. That idea leads to a new way of avoiding delay terms in a functional-differential equation. In this paper we use fixed point theory to show that such a conjecture is valid for a set of classical equations. 1. Introduction There is a large literature concerning equations typified by
x (t) = g x(t) − g x(t − L)
(1.1)
(as well as distributed delays) where g is an arbitrary Lipschitz function and L is a positive constant. Under suitable conditions, three dominant properties emerge. (i) Every constant function is a solution. (ii) Every solution approaches a constant. (iii) The differential equation has a first integral. It is but a small step, then, to conjecture that such a pair of terms as those appearing in the right-hand side of (1.1) constitute a harmless perturbation of a stable equation. While this can be helpful in a given equation, there is a very important additional application. For if we have a difficult stability problem of the form
x (t) = −g x(t − L) ,
(1.2)
then we can study
x (t) = −g x(t − L) + g x(t) − g x(t) ,
Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 35–46 DOI: 10.1155/FPTA.2005.35
(1.3)
36
Fixed points, stability, and harmless perturbations
having the aforementioned harmless perturbation so that we need only to study
x (t) = −g x(t) .
(1.4)
The idea of ignoring the delay is ancient when the delay is small or when there is a considerable monotonicity; neither will be present in this discussion. The thesis of this paper is that the conjecture is substantially correct and the solution is applied in a uniformly simple way using fixed point theory regardless of whether the delay is constant, variable, pointwise, distributed, finite, or infinite. 2. The conjecture Cooke and Yorke [10] introduced a population model of the form of (1.1), where g(x(t)) is the birth rate and g(x(t − L)) is the death rate. They also introduced other models with distributed delays and they posed a number of questions. The unusual aspect of their study centered on the fact that g is an arbitrary Lipschitz function, laying to rest the controversy over just what the growth properties should be in a given population. That paper generated a host of studies which continue to this day, as may be seen in [1, 2, 3, 13, 14, 15, 18, 19, 21], to mention just a few. Most of the subsequent studies asked that g should be monotone in some sense. Recently we noted [8] that every question raised in the Cooke-Yorke paper can be answered with two applications of the contraction mapping principle. This paper begins a study of the conjecture that the right-hand side of (1.1) is a harmless perturbation. Thus, we list a number of classical delay equations of both first and second order to test the conjectu
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