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

  • Saïd MazouziEmail author
  • Nasser-Eddine Tatar


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.


Exponential decay Memory term Fractional damping Multiplier technique 

AMS subject classifications

34D20 34K37 



The authors are thankful to the referees of the present paper for their careful reading and valuable comments. The second author is grateful to King Fahd University of Petroleum and Minerals for its continuous financial support through Project No. IN161010.


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© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Applied MathematicsBadji Mokhtar-Annaba UniversityAnnabaAlgeria
  2. 2.Department of Mathematical SciencesKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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