Asymptotic theory for a class of nonautonomous delay differential equations

This paper deals with asymptotic behavior of solutions of the nonlinear nonautonomous delay differential equation x′(t) = -∝_{t - r(t)}^t f(t, x(s))dμ(t, s), (su*) where xf(t, x) ≥ 0, f(t, 0) = 0, t - r(t) nondecreasing, μ(t, s) is nondecreasing and of bounded variation. General sufficient conditions, which are easy to verify, are obtained for the solutions to be bounded and asymptotically stable (locally and globally). These results improve many existing ones principally by allowing: (i) r(t) to be unbounded, (ii) both discrete and distributed delays, and (iii) the equation to be strongly nonlinear and nonautonomous. Various examples are given in the form of corollaries with a highly flexible integrand.