Skip to main content
SpringerLink
Log in

On a Mixed Nonlinear Fractional Boundary Value Problem with a New Class of Closed Integral Boundary Conditions

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

Abstract

In this paper, we investigate the existence of solutions for a fractional integro-differential equation with mixed nonlinearities subject to a new class of nonlocal closed integral boundary conditions. The proposed problem contains a right Liouville–Caputo fractional derivative operator and mixed Riemann-Liouville integral operators. The standard fixed point theorems are applied to derive the desired results, which are well-illustrated with examples. Some interesting observations are presented.

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

Access this article

Log in via an institution

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. Feng, M., Zhang, X., Ge, W.: Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications. J. Appl. Math. Comput. 33, 137–153 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zheng, L., Zhang, X.: Modeling and Analysis of Modern Fluid Problems. Mathematics in Science and Engineering, Elsevier/Academic Press, London, UK (2017)

    MATH  Google Scholar 

  3. Nicoud, F., Schfonfeld, T.: Integral boundary conditions for unsteady biomedical CFD applications. Int. J. Numer. Meth. Fluids 40, 457–465 (2002)

    Article  MATH  Google Scholar 

  4. Čiegis, R., Bugajev, A.: Numerical approximation of one model of the bacterial self-organization. Nonlinear Anal. Model Control 17, 253–270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Yusufoglu, E., Turhan, I.: A mixed boundary value problem in orthotropic strip containing a crack. J. Franklin Inst. 349, 2750–2769 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Renterghem, T.V., Botteldooren, D., Verheyen, K.: Road traffic noise shielding by vegetation belts of limited depth. J. Sound Vib. 331, 2404–2425 (2012)

    Article  Google Scholar 

  7. Whyburn, W.M.: Differential equations with general boundary conditions. Bull. Am. Math. Soc. 48, 692–704 (1942)

    Article  MathSciNet  MATH  Google Scholar 

  8. Conti, R.: Recent trends in the theory of boundary value problems for ordinary differential equations. Boll. Un. Mat. Ital. 22, 135–178 (1967)

    MathSciNet  MATH  Google Scholar 

  9. Ahmad, B., Ntouyas, S.K.: Nonlocal Nonlinear Fractional-Order Boundary Value Problems. World Scientific, Singapore (2021)

    MATH  Google Scholar 

  10. Henderson, J., Luca, R., Tudorache, A.: On a system of fractional differential equations with coupled integral boundary conditions, Fract. Calc. Appl. Anal. 18, 361–386 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Agarwal, R., Hristova, S., O’Regan, D.: Integral presentations of the solution of a boundary value problem for impulsive fractional integro-differential equations with Riemann-Liouville derivatives. AIMS Math. 7, 2973–2988 (2022)

    Article  MathSciNet  Google Scholar 

  12. Peng, L., Zhou, Y.: The existence of mild and classical solutions for time fractional Fokker-Planck equations. Monatsh. Math. 199, 377–410 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kirane, M., Abdeljabbar, A.: Nonexistence of global solutions of systems of time fractional differential equations posed on the Heisenberg group. Math. Methods Appl. Sci. 45, 7336–7345 (2022)

    Article  MathSciNet  Google Scholar 

  14. Webb, J.R.L., Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions. NoDEA Nonlinear Diff. Eqn. Appl. 15, 45–67 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ahmad, B., Alruwaily, Y., Ntouyas, S.K., Alsaedi, A.: Existence and stability results for a fractional order differential equation with non-conjugate Riemann–Stieltjes integro-multipoint boundary conditions. Mathematics 7, 249 (2019)

    Article  Google Scholar 

  16. Lin, L., Liu, Y., Zhao, D.: Study on implicit-type fractional coupled system with integral boundary conditions. Mathematics 9(4), 300 (2021)

    Article  Google Scholar 

  17. Shivanian, E.: Error estimate and stability analysis on the study of a high-order nonlinear fractional differential equation with Caputo-derivative and integral boundary condition. Comput. Appl. Math. 41, no. 8, Paper No. 395 (2022)

  18. Agarwal, R.P., Assolami, A., Alsaedi, A., Ahmad, B.: Existence results and Ulam-Hyers stability for a fully coupled system of nonlinear sequential Hilfer fractional differential equations and integro-multistrip-multipoint boundary conditions. Qual. Theory Dyn. Syst. 21(125), 33 (2022)

    MathSciNet  MATH  Google Scholar 

  19. Wongcharoen, A., Ntouyas, S.K., Wongsantisuk, P., Tariboon, J.: Existence results for a nonlocal coupled system of sequential fractional differential equations involving \(\psi \)-Hilfer fractional derivatives. Adv. Math. Phys. 2021, 5554619 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kucche, K.D., Mali, A.D.: On the nonlinear \((k,\psi )\)-Hilfer fractional differential equations. Chaos Solitons Fractals 152, 111335 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bouacida, I., Kerboua, M., Segni, S.: Controllability results for Sobolev type \( \psi \)-Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evol. Equ. Control Theory 12(1), 213–229 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  22. Panneer, S.A., Govindaraj, V.: Reachability of fractional dynamical systems with multiple delays in control using \(\psi \)-Hilfer pseudo-fractional derivative. J. Math. Phys. 63(102706), 14 (2022)

    MathSciNet  MATH  Google Scholar 

  23. Yang, Q., Bai, C., Yang, D.: Finite-time stability of nonlinear stochastic \(\psi \)-Hilfer fractional systems with time delay. AIMS Math. 7, 18837–18852 (2022)

    Article  MathSciNet  Google Scholar 

  24. Salim, A., Benchohra, M., Graef, J.R., Lazreg, J.E.: Initial value problem for hybrid \(\psi \)-Hilfer fractional implicit differential equations. J. Fixed Point Theory Appl. 24(7), 14 (2022)

    MathSciNet  MATH  Google Scholar 

  25. Ntouyas, S.K., Ahmad, B., Nuchpong, C., Tariboon, J.: On \((k,\psi )\)-Hilfer fractional differential equations and inclusions with mixed \((k,\psi )\)-derivative and integral boundary conditions. Axioms 11, 403 (2022)

    Article  Google Scholar 

  26. Ntouyas, S.K., Ahmad, B., Tariboon, J., Alhodaly, M.S.: Nonlocal integro-multi-point \((k,\psi )\)-Hilfer type fractional boundary value problems. Mathematics 10, 2357 (2022)

    Article  Google Scholar 

  27. Ahmad, B., Alnahdi, M., Ntouyas, S.K.: Existence results for a differential equation involving the right Caputo fractional derivative and mixed nonlinearities with nonlocal closed boundary conditions. Fractal Fract 7, 129 (2023)

    Article  Google Scholar 

  28. Ivashkevich, E.V.: Boundary height correlations in a two-dimensional Abelian sandpile. J. Phys. A Math. Gen. 27, 3643 (1994)

    Article  MathSciNet  Google Scholar 

  29. Piroux, G., Ruelle, P.: Boundary height fields in the Abelian sandpile model. J. Phys. A Math. Gen. 38, 1451 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Azimi-Tafreshi, N., Dashti-Naserabadi, H., Moghimi-Araghi, S., Ruelle, P.: The Abelian sandpile model on the honeycomb lattice. J. Stat. Mech. Theory Exp. 2010, P02004 (2010)

    Article  Google Scholar 

  31. Donatelli, M., Serra-Capizzano, S.: Antireflective boundary conditions for deblurring problems, J. Electr. Comput. Eng. 2010 , Article ID 241467, 18 (2010)

  32. Li, X., Robertsson, J., Curtis, A., van Manen1, D.: Internal absorbing boundary conditions for closed-aperture wavefield decomposition in solid media with unknown interiors. J. Acoust. Soc. Am. 152, 313–329 (2022)

  33. Mohammadimehr, M., Okhravi, S.V., Alavi, S.M.A.: Free vibration analysis of magneto-electro-elastic cylindrical composite panel reinforced by various distributions of CNTs with considering open and closed circuits boundary conditions based on FSDT. J. Vib. Control 24, 1551–1569 (2018)

    Article  MathSciNet  Google Scholar 

  34. Ahmad, B., Nieto, J.J., Pimentel, J.: Some boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62, 1238–1250 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Setukha, A.V.: On the three-dimensional Neumann boundary value problem with a generalized boundary condition in a domain with a smooth closed boundary (Russian). Differ. Equ. 41, 1237–1252 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, G., Ahmad, B., Zhang, L.: Existence results for nonlinear fractional differential equations with closed boundary conditions and impulses. Adv. Difference Equ. 2012(169), 13 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Ergoren, H., Kilicman, A.: Some existence results for impulsive nonlinear fractional differential equations with closed boundary conditions. Abstr. Appl. Anal. Art. ID 387629, 15 pp (2012)

  38. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V, Amsterdam, The Netherlands (2006)

    Google Scholar 

  39. Krasnosel’skiĭ, M.A.: Two remarks on the method of successive approximations. Uspekhi Mat. Nauk. 10, 123–127 (1955)

    MathSciNet  Google Scholar 

  40. Granas, A., Dugundji, J.: Fixed Point Theory, Springer, New York, NY, USA (2005)

Download references

Acknowledgements

This research work was funded by Institutional Fund Projects under Grant No. (IFPIP: 1249-130-1443). The authors gratefully acknowledge technical and financial support provided by the Ministry of Education and King Abdulaziz University, DSR, Jeddah, Saudi Arabia. The authors also thank the reviewers for their constructive remarks on their work.

Author information

Authors and Affiliations

  1. 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, Manal Alnahdi & Ahmed Alsaedi

  2. Department of Mathematics, University of Ioannina, 451 10, Ioannina, Greece

    Sotiris K. Ntouyas

Authors
  1. Bashir Ahmad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manal Alnahdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sotiris K. Ntouyas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ahmed Alsaedi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All the authors contributed equally to the preparation of the manuscript.

Corresponding author

Correspondence to Bashir Ahmad.

Ethics declarations

Conflict of nterest

The authors declare no competing interests.

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

Ahmad, B., Alnahdi, M., Ntouyas, S.K. et al. On a Mixed Nonlinear Fractional Boundary Value Problem with a New Class of Closed Integral Boundary Conditions. Qual. Theory Dyn. Syst. 22, 96 (2023). https://doi.org/10.1007/s12346-023-00781-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00781-4

Keywords

Mathematics Subject Classification

Search

Navigation