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Fractional differential equations and Lyapunov functionals. (English) Zbl 1226.34004

In this interesting paper, the author constructs several Lyapunov functionals for a scalar fractional differential equation and studies the qualitative behaviour of the solutions. It is shown that the kernel in the integral equation obtained from the fractional differential equation is convex with a singularity and is also completely monotone, as in the case of the resolvent kernel. When the kernel is not integrable, it is shown that the resolvent kernel is positive and integrable. These kernels give rise to essentially different types of Lyapunov functionals. It has to be noted that the Lyapunov functionals are explicitly given in terms of known functions.

MSC:

34A08 Fractional ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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