Journal of Integral Equations and Applications

Smooth solutions to mixed-order fractional differential systems with applications to stability analysis

Javier A. Gallegos, Norelys Aguila-Camacho, and Manuel A. Duarte-Mermoud

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Conditions for existence, uniqueness and smoothness of solutions for systems of fractional differential equations of Caputo and/or Riemann-Liouville type having all of them in general and not of the same derivation order are established in this paper. It includes mixed-order, multi-order or non-commensurate fractional systems. The smooth property is shown to be relevant for drawing consequences on the global behavior of solutions for such systems. In particular, we obtain sufficient conditions for global boundedness of solutions to mixed-order nonlinear systems and asymptotic stability of nonlinear fractional systems using backstepping control.

Article information

J. Integral Equations Applications, Volume 31, Number 1 (2019), 59-84.

First available in Project Euclid: 27 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A33: Fractional derivatives and integrals 34A30: Linear equations and systems, general 34D20: Stability

Fractional differential equations smoothness stability boundedness backstepping


Gallegos, Javier A.; Aguila-Camacho, Norelys; Duarte-Mermoud, Manuel A. Smooth solutions to mixed-order fractional differential systems with applications to stability analysis. J. Integral Equations Applications 31 (2019), no. 1, 59--84. doi:10.1216/JIE-2019-31-1-59.

Export citation


  • R. Agarwal, S. Hristova and O.D. Regan, Stability of solutions to impulsive Caputo fractional differential equations, Electr. J. Diff. Eqs. 58 (2016), 1–22.
  • N. Aguila-Camacho, M.A. Duarte-Mermoud and J.A. Gallegos, Lyapunov functions for fractional order systems, Comm. Nonlin. Sci. Num. Simul. 19 (2014), 2951–2957.
  • K.B. Ali, A. Ghanmi and K. Kefi, Existence of solutions for fractional differential equations with Dirichlet boundary conditions, Electr. J. Diff. Eqs. 116 (2016), 1–11.
  • A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations, Diff. Eqs. 46 (2010), 660–666.
  • A. Alsaedi, B. Ahmad and M. Kirane, Maximum principle for certain generalized time and space fractional diffusion equations, Quart. Appl. Math. 73 (2015), 163–175.
  • D. Baleanu and O. Mustaf, On the global existence of solutions to a class of fractional differential equations, Comp. Math. Appl. 59 (2010), 1835–1841.
  • D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional calculus: Models and numerical methods, World Scientific, Singapore, 2016.
  • N.D. Cong, T.S. Doan and H.T. Tuan, Asymptotic stability of linear fractional systems with constant coefficients and small time dependent perturbations, 2016, arXiv:1601.06538v1 [math.DS].
  • W. Deng, C. Li and Q. Guo, Analysis of fractional differential equations with multi-orders, Fractals 15 (2007), 1–10.
  • K. Diethelm, The analysis of fractional differential equations, An application-oriented exposition using differential operators of Caputo type, Lect. Notes Math 2004 (2010).
  • ––––, Smoothness properties of solutions of Caputo-type fractional differential equations, Fract. Calc. Appl. Anal. 10 (2007), 151–160.
  • ––––, A fractional calculus based model for the simulation of an outbreak of dengue fever, Nonlinear Dynam. 71 (2013), 613–619.
  • K. Diethelm, S. Siegmund and H.T. Tuan Asymptotic behavior of solutions of linear multi-order fractional differential equation systems, 2017, 1708.08131v1 [math.CA].
  • M.A. Duarte-Mermoud, N. Aguila-Camacho, J.A. Gallegos and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Nonlin. Anal. Th. Meth. Appl. 22 (2014), 650–659.
  • J.A. Gallegos and M.A. Duarte-Mermoud, Boundedness and convergence on fractional order systems, J. Comp. Appl. Math. 296 (2016), 815–826.
  • ––––, On the Lyapunov Theory for fractional order systems, Appl. Math. Comp. 287 (2016), 161–170.
  • ––––, Convergence of fractional adaptive systems using gradient approach, ISA Trans. 69 (2017), 31–42.
  • ––––, Robustness and convergence of fractional systems and their applications to adaptive systems, Fract. Calc. Appl. Anal. 20 (2017), 895–913.
  • J.A. Gallegos, M. A. Duarte-Mermoud, N. Aguila-Camacho and R. Castro-Linares, On fractional extensions of Barbalat lemma, Syst. Contr. Lett. 84 (2015), 7–12.
  • S. Heidarkhani, Y. Zhou, G. Caristi, G. Afrouzi and S. Moradi, Existence results for fractional differential systems through a local minimization principle, Comp. Math. Appl. (2016),.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Math. Stud. (2006).
  • Y. Li, Y. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comp. Math. Appl. 59 (2010), 1810–1821.
  • R.K. Miller and A. Feldstein, Smoothness of solutions of Volterra integral equations with weakly singular kernels, SIAM J. Math. Anal. 2 (1971), 242–258.
  • M. Ortigueira and F. Coito, System initial conditions vs derivative initial conditions, Computers Math. Appl. 59 (2010), 1782–1789.
  • E. Zeidler, Nonlinear functional analysis and its applications, I, Fixed-point theorems, Springer-Verlag New York, 1986.