On the stability for the delay-differential equation \(\dot x(t)=- a(t)f(x(t-r(t)))\) (Q1822262)

Suppose that the zero solution of \(x'(t)=-a(t)f\{x(t-r(t))\}\) is unique and that \(\int^{t}_{t-r(t)}a(s)ds\to 0\) as \(t\to \infty\). Then the solution \(x=0\) is uniformly stable. If, in addition, \(\int^{\infty}_{0}a(s)ds=\infty,\) the solution \(x=0\) is asymptotically stable. The hypotheses are less restrictive than those of \textit{K. L. Cooke} [Bull. Am. Math. Soc. 72, 285-288 (1966; Zbl 0151.104)], \textit{T. A. Burton} and \textit{J. R. Haddock} [J. Math. Anal. Appl. 54, 37-48 (1976; Zbl 0344.34065)], and the author [ibid. (to appear)], but the conclusions are somewhat weaker.