Jensen's Inequality and Liapunov's Direct Method

Abstract

In 1940 Marachkoff introduced the annulus argument to prove the zero solution of (1) x'= f(t, x), f(t, 0) = 0, is asymptotically stable if f is bounded when x is bounded and if a positive definite Liapunov function for (1) exists with negative definite derivative. This paved the way for researchers seeking new asymptotic stability conditions for not only (1) but also for systems of functional differential equations x' = F'(t, x_t). However, Marachkoff's approach excludes unbounded F having features that actually promote asymptotic stability. This paper provides an alternative that does not require F be bounded for x_t bounded using Jensen's inequality. It is a basic introduction to stability and it provides a new avenue for stability investigations.