Advertisement

Existence and Ulam’s Stability for Conformable Fractional Differential Equations with Constant Coefficients

  • Mengmeng Li
  • JinRong WangEmail author
  • D. O’Regan
Article

Abstract

In this article, we develop a standard idea in seeking the solution of linear ODEs to derive the representation of solutions to conformable fractional linear differential equations with constant coefficients by adopting the variation of constants method. In addition, we present the existence of solutions to conformable fractional nonlinear differential equations with constant coefficients under mild conditions on the nonlinear term. Also, we transfer the concepts of Ulam’s stability for ODEs to this type of equation and give the Ulam–Hyers and Ulam–Hyers–Rassias stability results on finite time and infinite time intervals.

Keywords

Conformable fractional differential equations Representation of solutions Existence Ulam–Hyers and Ulam–Hyers–Rassias stability 

Mathematics Subject Classification

34A08 

Notes

Acknowledgements

The authors thank the referees for their careful reading and comments on the manuscript.

References

  1. 1.
    Diethelm, K.: The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  2. 2.
    Zhou, Y., Wang, J., Zhang, L.: Basic Theory of Fractional Differential Equations, 2nd edn. World Scientifc, Singapore (2016)CrossRefGoogle Scholar
  3. 3.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  4. 4.
    Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta. Appl. Math. 109, 973–1033 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Agarwal, R.P., Hristova, S., O’Regan, D.: A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Fract. Calc. Appl. Anal. 19, 290–318 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Wang, J., Fečkan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, 806–831 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wang, J., Fečkan, M., Zhou, Y.: Center stable manifold for planar fractional damped equations. Appl. Math. Comput. 296, 257–269 (2017)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Wang, J., Fečkan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Wang, J., Zhou, Y., Fečkan, M.: On recent developments in the theory of boundary value problems for impulsive fractional differential equations. Comput. Math. Appl. 64, 3008–3020 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Wang, J., Zhou, Y., Fečkan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wang, J., Li, X.: Ulam–Hyers stability of fractional Langevin equations. Appl. Math. Comput. 258, 72–83 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Wang, J., Li, X.: A uniformed method to Ulam–Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, J., Zhou, Y., Wei, W., Xu, H.: Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls. Comput. Math. Appl. 62, 1427–1441 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li, M., Wang, J.: Finite time stability of fractional delay differential equations. Appl. Math. Lett. 64, 170–176 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhou, Y., Peng, L.: On the time-fractional Navier–Stokes equations. Comput. Math. Appl. 73, 874–891 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhou, Y., Peng, L.: Weak solution of the time-fractional Navier–Stokes equations and optimal control. Comput. Math. Appl. 73, 1016–1027 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhou, Y., Zhang, L.: Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems. Comput. Math. Appl. 73, 1325–1345 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Khalil, R., Al, M., Horani, A., Yousef, M.Sababheh: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Abdeljawad, T., AL Horani, M., Khalil, R.: Conformable fractional semigroups of operators. J. Semigroup Theory Appl. 2015, Art. ID 7, 1–9 (2015)Google Scholar
  21. 21.
    Chung, W.: Fractional Newton mechanics with conformable fractional derivative. J. Comput. Appl. Math. 290, 150–158 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Syst. Appl. 10, 109–137 (2015)MathSciNetGoogle Scholar
  23. 23.
    Al-Rifae, M., Abdeljawad, T.: Fundamental results of conformable Sturm–Liouville eigenvalue problems. Complexity 2017, Art. ID 3720471, 1–7 (2017)Google Scholar
  24. 24.
    Horani, M.A.L., Hammad, M.A., Khalilb, R.: Variation of parameters for local fractional nonhomogenous linear–differential equations. J. Math. Comput. Sci. 16, 147–153 (2016)CrossRefGoogle Scholar
  25. 25.
    Abdeljawad, T., Alzabut, J., Jarad, F.: A generalized Lyapunov-type inequality in the frame of conformable derivatives. Adv. Differ. Equ. 321, 1–10 (2017)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Pospíšil, M., Pospíšilová Škripková, L.: Sturms theorems for conformable fractional differential equations. Math. Commun. 21, 273–281 (2016)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Hammad, M.A., Khalil, R.: Abel’s formula and Wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl. 13, 177–183 (2014)zbMATHGoogle Scholar
  28. 28.
    Zheng, A., Feng, Y., Wang, W.: The Hyers–Ulam stability of the conformable fractional differential equation. Math. Aeterna 5, 485–492 (2015)Google Scholar
  29. 29.
    Iyiola, O.S., Nwaeze, E.R.: Some new results on the new conformable fractional calculus with application using D’Alambert approach. Progr. Fract. Differ. Appl. 2, 1–7 (2016)CrossRefGoogle Scholar
  30. 30.
    Bayour, B., Torres, D.F.M.: Existence of solution to a local fractional nonlinear differential equation. J. Comput. Appl. Math. 312, 127–133 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Tariboon, J., Ntouyas, S.K.: Oscillation of impulsive conformable fractional differential equations. Open Math. 14, 497–508 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsGuizhou UniversityGuiyangPeople’s Republic of China
  2. 2.School of Mathematics, Statistics and Applied MathematicsNational University of IrelandGalwayIreland

Personalised recommendations