On the asymptotic stability of the solutions of functional differential equations with infinite delay (Q1320280)

Let \(W: R_ +\to R_ +\) be continuous, strictly increasing with \(W(0)= 0\), and let \(g: R_ -\to R_ +\) be integrable. The author gives sufficient conditions for uniform asymptotic stability of the zero solution using a Lyapunov functional \(V\) whose derivative is negative definite along any solution \(x\) in one of the following two senses: \[ V'(t,x_ t)\leq -\eta(t)W\left(\int^ 0_{-\infty} | x(t+s)| g(s)ds\right),\tag{1} \] \[ V'(t,x_ t)\leq -\eta(t)W(| x(t)|),\tag{2} \] where \(\eta\) is a suitable nonnegative scalar function.