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Stability, fixed points and inverses of delays

Published online by Cambridge University Press:  12 July 2007

Leigh C. Becker
Christian Brothers University, 650 East Parkway South, Memphis, TN 38104, USA (
T. A. Burton
Northwest Research Institute, 732 Caroline Street, Port Angeles, WA 98362, USA (


The scalar equation with variable delay r(t) ≥ 0 is investigated, where tr(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

Research Article
Copyright © Royal Society of Edinburgh 2006

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