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

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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.

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References

  1. Diethelm, K.: The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics. Springer, New York (2010)

    Book  MATH  Google Scholar 

  2. Zhou, Y., Wang, J., Zhang, L.: Basic Theory of Fractional Differential Equations, 2nd edn. World Scientifc, Singapore (2016)

    Book  Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, J., Fečkan, M., Zhou, Y.: A survey on impulsive fractional differential equations. Fract. Calc. Appl. Anal. 19, 806–831 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang, J., Fečkan, M., Zhou, Y.: Center stable manifold for planar fractional damped equations. Appl. Math. Comput. 296, 257–269 (2017)

    MathSciNet  MATH  Google Scholar 

  8. Wang, J., Fečkan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, J., Zhou, Y., Fečkan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang, J., Li, X.: Ulam–Hyers stability of fractional Langevin equations. Appl. Math. Comput. 258, 72–83 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Wang, J., Li, X.: A uniformed method to Ulam–Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, M., Wang, J.: Finite time stability of fractional delay differential equations. Appl. Math. Lett. 64, 170–176 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zhou, Y., Peng, L.: On the time-fractional Navier–Stokes equations. Comput. Math. Appl. 73, 874–891 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhou, Y., Peng, L.: Weak solution of the time-fractional Navier–Stokes equations and optimal control. Comput. Math. Appl. 73, 1016–1027 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhou, Y., Zhang, L.: Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems. Comput. Math. Appl. 73, 1325–1345 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khalil, R., Al, M., Horani, A., Yousef, M.Sababheh: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Abdeljawad, T., AL Horani, M., Khalil, R.: Conformable fractional semigroups of operators. J. Semigroup Theory Appl. 2015, Art. ID 7, 1–9 (2015)

  21. Chung, W.: Fractional Newton mechanics with conformable fractional derivative. J. Comput. Appl. Math. 290, 150–158 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Anderson, D.R., Ulness, D.J.: Newly defined conformable derivatives. Adv. Dyn. Syst. Appl. 10, 109–137 (2015)

    MathSciNet  Google Scholar 

  23. Al-Rifae, M., Abdeljawad, T.: Fundamental results of conformable Sturm–Liouville eigenvalue problems. Complexity 2017, Art. ID 3720471, 1–7 (2017)

  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)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  26. Pospíšil, M., Pospíšilová Škripková, L.: Sturms theorems for conformable fractional differential equations. Math. Commun. 21, 273–281 (2016)

    MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tariboon, J., Ntouyas, S.K.: Oscillation of impulsive conformable fractional differential equations. Open Math. 14, 497–508 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, 550025, Guizhou, People’s Republic of China

    Mengmeng Li & JinRong Wang

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    D. O’Regan

Authors
  1. Mengmeng Li
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  2. JinRong Wang
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  3. D. O’Regan
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Corresponding author

Correspondence to JinRong Wang.

Additional information

Communicated by Rosihan M. Ali.

This work is supported by National Natural Science Foundation of China (Grant Number 11661016), Training Object of High Level and Innovative Talents of Guizhou Province (Grant Number (2016)4006), and Unite Foundation of Guizhou Province (Grant Number [2015]7640).

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Li, M., Wang, J. & O’Regan, D. Existence and Ulam’s Stability for Conformable Fractional Differential Equations with Constant Coefficients. Bull. Malays. Math. Sci. Soc. 42, 1791–1812 (2019). https://doi.org/10.1007/s40840-017-0576-7

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  • DOI: https://doi.org/10.1007/s40840-017-0576-7

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