Nonlinear Dynamics

, Volume 71, Issue 4, pp 635–640

A uniqueness criterion for fractional differential equations with Caputo derivative

  • Dumitru Băleanu
  • Octavian G. Mustafa
  • Donal O’Regan
Original Paper

Abstract

We investigate the uniqueness of solutions to an initial value problem associated with a nonlinear fractional differential equation of order α∈(0,1). The differential operator is of Caputo type whereas the nonlinearity cannot be expressed as a Lipschitz function. Instead, the Riemann–Liouville derivative of this nonlinearity verifies a special inequality.

Keywords

Fractional differential equation Uniqueness of solution Caputo differential operator Riemann–Liouville derivative 

References

  1. 1.
    Agarwal, R.P., Lakshmikantham, V.: Uniqueness and nonuniqueness criteria for ordinary differential equations. World Scientific, Singapore (1993) MATHCrossRefGoogle Scholar
  2. 2.
    Băleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Boston (2012) MATHGoogle Scholar
  3. 3.
    Băleanu, D., Avkar, T.: Lagrangians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004) Google Scholar
  4. 4.
    Băleanu, D., Mustafa, O.G.: On the asymptotic integration of a class of sublinear fractional differential equations. J. Math. Phys. 50, 123520 (2009) MathSciNetCrossRefGoogle Scholar
  5. 5.
    Băleanu, D., Mustafa, O.G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59, 1835–1841 (2010) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Băleanu, D., Mustafa, O.G., O’Regan, D.: A Nagumo-like uniqueness theorem for fractional differential equations. J. Phys. A 44, 392003 (2011) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Băleanu, D., Agarwal, R.P., Mustafa, O.G., Coşulschi, M.: Asymptotic integration of some nonlinear differential equations with fractional time derivative. J. Phys. A 44, 055203 (2011) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bhalekar, S., Daftardar-Gejji, V., Băleanu, D., Magin, R.L.: Transient chaos in fractional Bloch equations. Comput. Math. Appl. (2012). doi:10.1016/j.camwa.2012.01.069 MATHGoogle Scholar
  9. 9.
    Delavari, H., Băleanu, D., Sadati, J.: Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn. 67(4), 2433–2439 (2012) MATHCrossRefGoogle Scholar
  10. 10.
    Agrawal, O.P., Defterli, O., Băleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16(13), 1967–1976 (2010) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964) MATHGoogle Scholar
  12. 12.
    Herzallah, M.A.E., Băleanu, D.: Fractional Euler–Lagrange equations revisited. Nonlinear Dyn. (2012). doi:10.1007/s11071-011-0319-5 Google Scholar
  13. 13.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland, New York (2006) MATHGoogle Scholar
  14. 14.
    Lovelady, D.L., Martin, R.H. Jr.: A global existence theorem for a nonautonomous differential equation in a Banach space. Proc. Am. Math. Soc. 35, 445–449 (1972) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lovelady, D.L.: A necessary and sufficient condition for exponentially bounded existence and uniqueness. Bull. Aust. Math. Soc. 8, 133–135 (1973) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    McShane, E.J.: Linear functionals on certain Banach spaces. Proc. Am. Math. Soc. 1, 402–408 (1950) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) MATHGoogle Scholar
  18. 18.
    Mustafa, O.G., O’Regan, D.: On the Nagumo uniqueness theorem. Nonlinear Anal. TMA 74, 6383–6386 (2011) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mustafa, O.G.: On the uniqueness of flow in a recent tsunami model. Appl. Anal. (2011). doi:10.1080/00036811.2011.569499 (on-line) Google Scholar
  20. 20.
    Mustafa, O.G.: A Nagumo-like uniqueness result for a second order ODE. Monatshefte Math. (2011). doi:10.1007/s00605-011-0324-2 (on-line) Google Scholar
  21. 21.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) MATHGoogle Scholar
  22. 22.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) MATHGoogle Scholar
  23. 23.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon & Breach, New York (1993) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Dumitru Băleanu
    • 1
    • 2
  • Octavian G. Mustafa
    • 3
  • Donal O’Regan
    • 4
  1. 1.Department of Mathematics & Computer ScienceÇankaya UniversityBalgatTurkey
  2. 2.Institute of Space SciencesMăgureleRomania
  3. 3.Faculty of Exact SciencesUniversity of CraiovaCraiovaRomania
  4. 4.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

Personalised recommendations