Abstract
One of the main properties studied in the qualitative theory of differential equations is the stability of solutions. The stability of fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. However, the Lyapunov approach to fractional differential equations causes many difficulties. In this paper a new definition (based on the Caputo fractional Dini derivative) for the derivative of Lyapunov functions to study a nonlinear Caputo fractional differential equation is introduced. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are given. Examples are presented to illustrate the theory.
Highlights
One of the main problems in the qualitative theory of differential equations is stability of solutions
The initial conditions of fractional differential equations with the Caputo derivative have a clear physical meaning, and, as a result, the Caputo derivative is usually used in real applications
We will define strict stability for fractional equations following the idea for ordinary differential equations
Summary
One of the main problems in the qualitative theory of differential equations is stability of solutions. In this paper the strict stability of nonlinear nonautonomous Caputo fractional differential equations is defined and studied using continuous Lyapunov functions. The Caputo fractional Dini derivative of a Lyapunov function is defined in an appropriate way Note that this type of derivative is introduced in [ ] and used to study the stability and asymptotic stability of Caputo fractional differential equations. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are obtained. Section presents basic definitions concerning strict stability and the new definition of the Caputo fractional Dini derivative of Lyapunov functions among the nonlinear fractional differential equations. ( ) The Riemann-Liouville (RL) fractional derivative of order q ∈ ( , ) of m(t) is given by (see, for example, Section . . . [ ], or [ ])
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