Skip to main content

Stability, fixed points and inverses of delays

The full text article is not available for purchase.

The publisher only permits individual articles to be downloaded by subscribers.

Abstract:

The scalar equation

x′(t) = −∫tt−r(t)a(t, s)g(x(s))ds (1)

with variable delay r(t) ≥ 0 is investigated, where t−r(t) is increasing and xg (x) > 0 (x ≠ 0) in a neighbourhood of x = 0. We find conditions for r, a and g so that for a given continuous initial function  a mapping P for (1) can be defined on a complete metric space C and in which P has a unique fixed point. The end result is not only conditions for the existence and uniqueness of solutions of (1) but also for the stability of the zero solution. We also find conditions ensuring that the zero solution is asymptotically stable by changing to an exponentially weighted metric on a closed subset of C. Finally, we parlay the methods for (1) into results for

x′(t) = −∫tt−r(t)a(t, s)g(s, x(s))ds (2)

and

x′(t) = −a(t)g(x(t−r(t))). (3)

Document Type: Research Article

Publication date: May 1, 2006

rse/proca/2006/00000136/00000002/art00002
dcterms_title,dcterms_description,pub_keyword
6
5
20
40
5

Access Key

Free Content
Free content
New Content
New content
Open Access Content
Open access content
Subscribed Content
Subscribed content
Free Trial Content
Free trial content
Cookie Policy
X
Cookie Policy
Ingenta Connect website makes use of cookies so as to keep track of data that you have filled in. I am Happy with this Find out more