Lyapunov functions and boundedness for differential and delay equations (Q1107707)

In the study of ordinary and functional differential equations by means of Lyapunov's direct method, it frequently happens that the Lyapunov function fails to be radially unbounded and the derivative fails to be strictly negative definite at the same time. The classic example is given by the system \(\{x'=y\), \(y'=-q(x,y)y-g(x)\}\) in which q(x,y)\(\geq 0\), \(xg(x)>0\) if \(x\neq 0\). The Lyapunov function \(V=y^ 2+2\int^{x}_{0}g(s)ds\) yields \(V'=-2q(x,y)y^ 2.\) Thus, if \(\int^{x}_{0}g(s)ds\) is bounded, then the described situation obtains along the x-axis. In this paper it is shown how to convert the problem into a differential inequality yielding bounded solutions. Several examples are given for both ordinary and functional differential equations.