Stability, fixed points and inverses of delays

Abstract

The scalar equation x′(t) = - ∫t-r(t)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 ψ a mapping P for (1) can be defined on a complete metric space C ψ 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 CTJ,. Finally, we parlay the methods for (1) into results for x′(t) = - ∫t-r(t)t a(t,s)g(s,x(s)ds (2) and x′(t) = -a(t)g(x(t - r(t))). (3) © 2006 The Royal Society of Edinburgh.