Global attractivity results for nonlinear delay differential equations (Q1295887)

The authors consider the nonlinear and nonautonomous delay differential equation (1) \(x'(t)=-p(t)f(x(t-\tau))\), where \(\tau\) is a positive constant and \(p:[0,\infty)\to [0,\infty)\) and \(f:\mathbb{R}\to \mathbb{R}\) are continuous functions with \(xf(x)>0\) if \(x\neq 0\). The main result of the paper is the following: Let \(f\) be a nondecreasing function such that \(|f(x)|<|x|\) if \(x\neq 0\). Suppose that (2) \(\int_{t-\tau}^t p(s)ds\leq\frac{3}{2}\) for all large \(t>0\) and \(\int_0 ^{\infty}p(s)ds=\infty\). Then every solution to (1) tends to zero as \(t\to\infty\). A result which guarantees that the constant \(\frac{3}{2}\) in (2) is best possible is also given. The authors take notice of connections among their results and results by \textit{C. Qian} [J. Math. Anal. Appl. 197, No. 2, 529-547 (1996; Zbl 0851.34075)] and {J. W.-H. So} and \textit{J. S. Yu} [Proc. Am. Math. Soc. 123, No. 9, 2687-2694 (1995; Zbl 0844.34080)].