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Stability, fixed points and inverses of delays

Published online by Cambridge University Press:  12 July 2007

Leigh C. Becker  and
T. A. Burton
Leigh C. Becker
Affiliation:
Christian Brothers University, 650 East Parkway South, Memphis, TN 38104, USA ([email protected])
T. A. Burton
Affiliation:
Northwest Research Institute, 732 Caroline Street, Port Angeles, WA 98362, USA ([email protected])
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Abstract

The scalar equation 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 ψ a mapping P for (1) can be defined on a complete metric space Cψ 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 Cψ. Finally, we parlay the methods for (1) into results for

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 136 , Issue 2 , April 2006 , pp. 245 - 275
Copyright
Copyright © Royal Society of Edinburgh 2006

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