Liapunov Functionals, Fixed Points, and Stability by Krasnoselskii's Theorem

Abstract

This is a paper in a series of investigations into the use of fixed point theorems to prove stability. Here, we use a modification of a fixed point theorem of Krasnoselskii. The work concerns a scalar functional differential equation $x' =-a(t)x^3 + b(t)x^3(t-r(t))$ where $r(t)$ need be neither bounded nor differentiable, while a and b can be unbounded. Such problems have proved very challenging in the theory of Liapunov's direct method. We show that it fits very nicely into the framework of the modified Krasnoselskii theorem so that asymptotic stability is readily concluded.