Jensen's Inequality and Liapunov's Direct Method

  • Leigh C. Becker Department of Mathematics. Christian Brothers University 650 E. Parkway South. Memphis, TN 38104-5581, USA.
  • T. A. Burton Northwest Research lnstitute 732 Caroline St. Port. Angeles, WA 98362, USA.
Keywords: Liapunov's direct method, asymptotic stability, Jensen's inequality


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'(txt). 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 xt bounded using Jensen's inequality. It is a basic introduction to stability and it provides a new avenue for stability investigations.

How to Cite
L. C. Becker and T. A. Burton, “Jensen’s Inequality and Liapunov’s Direct Method”, CUBO, vol. 6, no. 3, pp. 67–90, Oct. 2004.