Skip to main content
SpringerLink
Log in

Generalized Fractional Differential Systems with Stieltjes Boundary Conditions

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this work, we study the existence of solutions to a new class of boundary value problems consisting of a system of nonlinear differential equations with generalized fractional derivative operators of different orders and nonlocal boundary conditions containing Riemann-Stieltjes and generalized fractional integral operators. We emphasize that the nonlinearities in the given system are of general form as they depend on the unknown functions as well as their lower order generalized fractional derivatives. The uniqueness result for the given problem is proved by applying the Banach contraction mapping principle, while the existence of solutions for the given system is shown with the aid of Leray-Schauder alternative. Two concrete examples are given for illustrating the obtained results. The paper concludes with some interesting observations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbas, M.I., Fečkan, M.: Michal Feckan, Investigation of an implicit Hadamard fractional differential equation with Riemann-Stieltjes integral boundary condition. Math. Slovaca 72(4), 925–934 (2022)

    Article  MATH  Google Scholar 

  2. Ahmad, B., Alghanmi, M., Ntouyas, S.K., Alsaedi, A.: Fractional differential equations involving generalized derivative with Stieltjes and fractional integral boundary conditions. Appl. Math. Lett. 84, 111–117 (2018)

    Article  MATH  Google Scholar 

  3. Ahmad, B., Ntouyas, S.K.: Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Appl. Math. Comput. 266, 615–622 (2015)

    MATH  Google Scholar 

  4. Ahmad, B., Ntouyas, S., Alsaedi, A.: On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals 83, 234–241 (2016)

    Article  MATH  Google Scholar 

  5. Ahmad, B., Luca, R.: Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions. Frac. Calc. Appl. Anal. 21(2), 423–441 (2018)

    Article  MATH  Google Scholar 

  6. Ahmad, B., Alghanmi, M., Alsaedi, A.: Existence results for a nonlinear coupled system involving both Caputo and Riemann-Liouville generalized fractional derivatives and coupled integral boundary conditions. Rocky Mountain J. Math. 50(6), 1901–1922 (2020)

    Article  MATH  Google Scholar 

  7. Alruwaily, Y., Ahmad, B., Ntouyas, S.K., Alzaidi, A.S.M.: Existence Results for Coupled Nonlinear Sequential Fractional Differential Equations with Coupled Riemann-Stieltjes Integro-Multipoint Boundary Conditions. Fractal Fract. 6, 123 (2022). https://doi.org/10.3390/fractalfract6020123

    Article  Google Scholar 

  8. Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Chaos in a fractional order Duffing system, In:Proceedings of the 1997 European conference on circuit theory and design(ECCTD97), Budapest, Hungary, 30 August-3. Budapest: Hungary: Technical University of Budapest (1997) 1259-1262

  9. Asawasamrit, S., Thadang, Y., Ntouyas, S.K., Tariboon, J.: Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann-Stieltjes Fractional Integral Boundary Conditions. Axioms 10(3), 130 (2021)

    Article  Google Scholar 

  10. Belmor, S., Ravichandran, C., Jarad, F.: Nonlinear generalized fractional differential equations with generalized fractional integral conditions. J. Taibah University Sci 14(1), 114–123 (2020)

    Article  Google Scholar 

  11. Belmor, S., Jarad, F., Abdeljawad, T., Kilinç, G.: A study of boundary value problem for generalized fractional differential in- clusion via endpoint theory for weak contractions. Advances in Difference Equations 2020(1), 1–11 (2020)

    Article  MATH  Google Scholar 

  12. Belmor, S., Jarad, F., Abdeljawad, T.: On Caputo-, Hadamard type coupled systems of nonconvex fractional differential inclusions. Adv. Diff. Equ. 2021(1), 1–12 (2021)

    Article  MATH  Google Scholar 

  13. Belmor, S., Jarad, F., Abdeljawad, T., Alqudah, M.A.: On fractional differential inclusion problems involving fractional order derivative with respect to another function. Fractals 28(08), 2040002 (2020)

    Article  MATH  Google Scholar 

  14. Faieghi, M., Kuntanapreeda, S., Delavari, H., Baleanu, D.: LMI-based stabilization of a class of fractional-order chaotic systems. Nonlinear Dyn. 72, 301–309 (2013)

    Article  MATH  Google Scholar 

  15. Ge, Z.M., Jhuang, W.R.: Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor. Chaos Solitons Fractals 33, 270–289 (2007)

    Article  MATH  Google Scholar 

  16. Ge, Z.M., Ou, C.Y.: Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals 35, 705–717 (2008)

    Article  Google Scholar 

  17. Granas, A., Dugundji, J.: Fixed Point Theory. Springer-Verlag, New York (2003)

    Book  MATH  Google Scholar 

  18. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003)

    Article  Google Scholar 

  19. Hartley, T.T., Lorenzo, C.F., Killory, Q.H.: Chaos in a fractional order Chua’s system. IEEE Trans. CAS-I 42, 485–490 (1995)

    Article  Google Scholar 

  20. Henderson, J., Luca, R., Tudorache, A.: On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 18(2), 361–386 (2015). https://doi.org/10.1515/fca-2015-0024

    Article  MATH  Google Scholar 

  21. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

  22. Jiang, J., Liu, L., Wu, Y.: Positive solutions to singular fractional differential system with coupled boundary conditions. Comm. Nonlinear Sc. Num. Sim. 18(11), 3061–3074 (2013)

    Article  MATH  Google Scholar 

  23. Jiao, Z., Chen, Y.Q., Podlubny, I.: Distributed-order Dyn. Syst. Springer, New York (2012)

    Book  Google Scholar 

  24. Katugampola, U.N.: New approach to a generalized fractional integral. Appl. Math. Comput. 218, 860–865 (2015)

    MATH  Google Scholar 

  25. Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl. 6, 1–15 (2014)

    MATH  Google Scholar 

  26. Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: Methods, results and problems I. Appl. Anal. 78, 153–192 (2001)

    Article  MATH  Google Scholar 

  27. Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: Methods, results and problems II. Appl. Anal. 81, 435–493 (2002)

    Article  MATH  Google Scholar 

  28. Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–11399 (1999)

    Article  Google Scholar 

  29. Lu, H., Sun, S., Yang, D., Teng, H.: Theory of fractional hybrid differential equations with linear perturbations of second type. Bound. Value Probl. 2013, 23 (2013)

    Article  MATH  Google Scholar 

  30. Lupinska, B., Odzijewicz, T.: A Lyapunov-type inequality with the Katugampola fractional derivative. Math. Methods Appl. Sci. 41, 8985–8996 (2018)

    Article  MATH  Google Scholar 

  31. Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MATH  Google Scholar 

  32. Ostoja-Starzewski, M.: Towards thermoelasticity of fractal media. J. Therm. Stress 30, 889–896 (2007)

    Article  MATH  Google Scholar 

  33. Povstenko, Y.Z.: Fract thermoelast. Springer, New York (2015)

    Book  Google Scholar 

  34. Redhwan, S.S., Shaikh, S.L., Abdo, M.S.: Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type. AIMS Math. 5(4), 3714–3730 (2020)

    Article  MATH  Google Scholar 

  35. Redhwan, S.S., Shaikh, S.L., Abdo, M.S., Shatanawi, W., Abodayeh, K., Almalahi, M.A., Aljaaidi, T.: Investigating a generalized Hilfer-type fractional differential equation with two-point and integral boundary conditions. AIMS Math. 7(2), 1856–1872 (2021)

    Article  Google Scholar 

  36. S.S. Redhwan, S.L. Shaikh, M.S. Abdo, Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations, arXiv:1911.08884 13 (2019)

  37. Redhwan, S.S., Shaikh, S.L., Abdo, M.S.: Caputo-Katugampola type implicit fractional differential equation with two-point anti-periodic boundary conditions. Results Nonlinear Anal. 5(1), 12–28 (2022)

    Article  Google Scholar 

  38. Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today. 55, 48–54 (2002)

    Article  Google Scholar 

  39. Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett. 22, 64–69 (2009)

    Article  MATH  Google Scholar 

  40. Sabatier, J., Agarwal, O.P., Ttenreiro Machado, J.A.: Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering, New York, Springer (2007)

    Book  Google Scholar 

  41. Ttenreiro Machado, J.A.: Discrete time fractional-order controllers. Frac. Cal. App. Anal. 4, 47–66 (2001)

    MATH  Google Scholar 

  42. Waheed, H., Zada, A., Rizwan, R., Popa, I.L.: Hyers-Ulam Stability for a Coupled System of Fractional Differential Equation With \(p\)-Laplacian Operator Having Integral Boundary Conditions. Qual. Theory Dyn. Syst. 21(3), Paper No. 92 (2022)

    Article  MATH  Google Scholar 

  43. Zada, A., Alam, M., Riaz, U.: Analysis of \(q\)-fractional implicit boundary value problems having stieltjes integral conditions. Math. Meth. Appl. Sci. 44(6), 4381–4413 (2020)

    Article  MATH  Google Scholar 

  44. Zhang, F., Chen, G., Li, C., Kurths, J.: Chaos synchronization in fractional differential systems. Phil. Trans. R. Soc. A 371, 20120155, 26 pages (2013)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Razi University, 67149, Kermanshah, Iran

    Nemat Nyamoradi

  2. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Bashir Ahmad

Authors
  1. Nemat Nyamoradi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bashir Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nemat Nyamoradi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nyamoradi, N., Ahmad, B. Generalized Fractional Differential Systems with Stieltjes Boundary Conditions. Qual. Theory Dyn. Syst. 22, 6 (2023). https://doi.org/10.1007/s12346-022-00703-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00703-w

Keywords

Search

Navigation