Asymptotic behavior of solutions of some differential equations (Q1076895)

The asymptotic behavior of solutions of functional differential equations \(\dot x(t)=f(t,x(\cdot)),\) where \(x(\cdot):[\alpha,t]\to R^ n\), \(\alpha\geq -\infty\), is a continuous function, under the assumption that there exists a non-negative continuous functional V(t,x(\(\cdot))\) such that \(\dot V_{(1)}(t,x(\cdot))\leq -p(t)W(x(t))\) for \(t\geq 0\), where W(x) is nonnegative and continuous on \(R^ n\) and \(p:[0,\infty)\to [0,\infty)\) satisfies \(\int_{I}p(t)dt=\infty\) for each set \(I=\cup^{\infty}_{k=1}[\alpha_ k,\beta_ k]\) such that \(\alpha_ k<\beta_ k<\alpha_{k+1}\), \(\beta_ k-\alpha_ k\geq \delta >0\).