Stability by Fixed Point Methods
by
T. A. Burton
Northwest Research Institute

For more than 100 years Liapunov's direct method has been the primary tool for determining stability in ordinary and functional differential equations. But there are persistent difficulties with the method. Three of the simplest and best known difficulties include:

1. For x¢=F(t, x), unboundedness of F as t ® ¥ prevents us from effectively using negative definiteness of the derivative of the Liapunov function unless the Liapunov function is positive definite and decrescent. For functional differential equations the difficulty persists even if the Liapunov functional is positive definite and decrescent.

2. For x¢=a(t)x(t)+b(t)x(t-r) we generally require pointwise comparison of a(t) and b(t) for stability instead of averaging conditions.

3. For x¢=a(t)x(t)+b(t)x(t-r(t)) the delay r(t) is frequently an integral part of the Liapunov functional and lack of smoothness and/or boundedness of r(t) causes difficulties of two types with the derivative of the Liapunov functional.

Some years ago we began a study of how fixed point theory might be used to circumvent these kinds of difficulties. This talk will explore some of the most recent results of those investigations. It will focus mainly on examples and comparisons between Liapunov's direct method and fixed point methods.

Date received: January 8, 2003