Long time behavior for a fractional Picard problem in a Hilbert space

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.