LYAPUNOV THEORY FOR FRACTIONAL DIFFERENTIAL EQUATIONS

Recently there has been a surge in the study of the theory of fractional differential equations. The existing results have been collected in the forthcoming monograph on that subject [1]. In this paper, we prove necessary comparison theorems utilizing Lyapunov-like functions, define stability concepts in terms of a norm and prove stability results parallel to Lyapunov’s results relative to stability and asymptotic stability. Since there are many stability concepts and corresponding results in the literature for ODEs, it is useful to work with stability concepts in terms of two measures, which includes several important stability notions as special cases [2, 3]. A few choices of the two measures are given to demonstrate the versatility of this approach. We extend this versatile approach to provide necessary conditions for stability criteria in terms of two measures for fractional differential equations. We use Caputo’s derivative of arbitrary order which is more suitable to discuss Lyapunov stability theory.