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Long time behavior for a fractional Picard problem in a Hilbert space. (English) Zbl 1428.34018

Summary: Of concern is a nonlinear second order initial value differential problem involving a convolution of a singular kernel with the derivative of the state. The problem describes the dynamics of a single-degree-of-freedom fractional oscillator. It is a generalization of the standard harmonic oscillator. The model also generalizes some well-known fractionally damped second order differential equations such as the Bagley-Torvik equation. Moreover, it extends models using exponential non-viscous damping to the more challenging singular case. We prove an exponential stability result of the equilibrium using the multiplier technique. A new energy functional, different from the classical one and different from the one obtained by the diffusive representation, is introduced.

MSC:

34A08 Fractional ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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