Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Stability, fixed points and inverses of delays

Leigh C. Beckera1 and T. A. Burtona2

a1 Christian Brothers University, 650 East Parkway South, Memphis, TN 38104, USA (

a2 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


(Received November 23 2004)

(Accepted June 21 2005)