Stability with Initial Time Difference of Caputo Fractional Differential Equations by Lyapunov Functions

Abstract

The stability with initial data difference for nonlinear nonautonomous Caputo fractional differential equation is introduced. This type of stability generalizes the concept of stability in the literature and it enables us to compare the behavior of two solutions when both the initial times and initial values are different. Our theory is based on a new definition of the derivative of a Lyapunov like function along the given fractional equation. Comparison results for scalar fractional differential equations are presented and sufficient conditions for stability, uniform stability and asymptotic stability with initial time difference are obtained.