Skip to main content
SpringerLink
Log in

Analysis and Applications of Sequential Hybrid \(\psi \)-Hilfer Fractional Differential Equations and Inclusions in Banach Algebra

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

Abstract

This research inscription gets to grips with a specific kind of sequential hybrid fractional differential equation en-capsuling a collective fractional derivative known as the \(\psi \)-Hilfer type fractional operator. The existence of the solutions of the forehanded equations is tackled by using Dhage fixed point theorem on Banach algebras while their uniqueness is handled capitalizing on the Banach fixed point theorem. On the top of this, the stability within the scope of Ulam-Hyers of solutions to these systems are considered. Finally, pertinent examples with applications are presented to corroborate the reported results.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Availability of Data and Materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. 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 (2006)

    Google Scholar 

  2. Boutiara, A.: Multi-term fractional q-difference equations with q-integral boundary conditions via topological degree theory, Commun. Optim. Theory. 20(1): 1–16. (2021). https://doi.org/10.23952/cot.2021.1

  3. Boutiara, A., Benbachir, M., Guerbati, K.: Measure of noncompactness for nonlinear hilfer fractional differential equation in banach spaces. Ikonion J. Math. 1(2), 55–67 (2019)

    Google Scholar 

  4. Boutiara, A., Benbachir, M., Guerbati, K.: Caputo type fractional differential equation with nonlocal erdélyi-kober type integral boundary conditions in banach spaces. Surv. Math. Appl. 15, 399–418 (2020)

    MATH  Google Scholar 

  5. Boutiara, A., Etemad, S., Alzabut, J., Hussain, A., Subramanian, M., Rezapour, S.: On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria. Adv. Diff. Equat. 2021(1), 1–23 (2021)

    MATH  Google Scholar 

  6. Boutiara, A., Etemad, S., Hussain, A., Rezapour, S.: The generalized U-H and U-H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving \(\varphi \)-Caputo fractional operators. Adv. Diff. Equat. 95, 1–21 (2021). https://doi.org/10.1186/s13662-021-03253-8

    Article  MATH  Google Scholar 

  7. Boutiara, A., Matar, M.M., Kaabar, M.K., Martinez, F., Etemad, S., Rezapour, S.: Some qualitative analyses of neutral functional delay differential equation with generalized caputo operator. J. Funct. Spaces. 109, 1–3 (2021)

    MATH  Google Scholar 

  8. Boutiara, A.: Mixed fractional differential equation with nonlocal conditions in Banach spaces. J. Math. Model. 9(3), 451–463 (2021)

    MATH  Google Scholar 

  9. Selvam, A.G.M., Alzabut, J., Vignesh, D., Jonnalagadda, J.M., Abodayeh, K.: Existence and stability of nonolinear discrete fractional initial value problems with application to vibrating eardrum. Math. Biosci. Eng. 18(4), 3907–3921 (2021)

    Article  MATH  Google Scholar 

  10. Lasota, A., Opial, Z.: An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci. Set. Sci. Math. Astronom. Phy. 13, 781–786 (1965)

    MATH  Google Scholar 

  11. Dhage, B.C., Ntouyas, S.K.: Existence results for boundary value problems for fractional hybrid differential inclusions. Topol. Methods Nonlinear Analy. 44(1), 229–238 (2014)

    Article  MATH  Google Scholar 

  12. Dhage, B.C.: Existence results for neutral functional differential inclusions in Banach algebras. Nonlinear Anal. 64, 1290–1306 (2006)

    Article  MATH  Google Scholar 

  13. Dhage, B.C.: A fixed point theorem in Banach algebras involving three operators with applications. Kyungpook Math. J. 44, 145–155 (2004)

    MATH  Google Scholar 

  14. Dhage, B.C.: A nonlinear alternative in Banach algebras with applications to functional differential equations, Nonlinear. Funct. Anal. Appl. 8, 563–575 (2004)

    MATH  Google Scholar 

  15. Dhage, B.C.: Basic results in the theory of hybrid differential equations with mixed perturbation of second type. Funct. Diff. Equ. 19, 87–106 (2012)

    MATH  Google Scholar 

  16. Thaiprayoon, C., Sudsutad, W., Alzabut, J., Etamed, S., Rezapour, S.: On the qualitative analysis of the fractional boundary value problem describing thermostat control model via \(\psi -\) Hilfer fractional operator. Adv. Diff. Equat. 2021, 201 (2021)

    Article  MATH  Google Scholar 

  17. Srivastava, H., Tomovski, Z.: Fractional calculus with an integral operator containing a generalized Mittag–Leffler function in the kernel. Appl. Math. and Comput. 211–(1), 198–210 (2009)

    MATH  Google Scholar 

  18. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  19. Alzabut, J., Selvam, A.G.M., El-Nabulsi, R.A., Vignesh, D., Samei, M.E.: Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions. Symmetry 13, 473 (2021)

    Article  Google Scholar 

  20. Alzabut, J., Selvam, A.G.M., Vignesh, D.: Yousef gholami, solvability and stability of nonlinear hybrid \(\Delta -\) difference equations of fractional order. Int. J. Nonlinear Sci. Numer. Simulat. (2021). https://doi.org/10.1515/ijnsns-2021-0005

    Article  Google Scholar 

  21. Aubin, J., Cellna, A.: Differential inclusions: set-valued maps and viability theory. Springer, Verlag (1984)

    Book  Google Scholar 

  22. Sousa, J.V.D.C., de Oliveira, E.C.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)

    Article  MATH  Google Scholar 

  23. Sousa, J.V.D.C., de Oliveira, E.C.: On the stability of a hyperbolic fractional partial differential equations. Differ. Equ. Dyn. Syst. (2019). https://doi.org/10.1007/s12591-019-00499-3

    Article  Google Scholar 

  24. Deimling, K.: Multi-valued differential equations. Walter de Gruyter, Berlin (1992)

    Book  MATH  Google Scholar 

  25. Abdo, M.S., Panchal, S.K., Saeed, A.M.: Fractional boundary value problem with \(\psi \)-Caputo fractional derivative. Proc. Indian Acad. Sci. Math. Sci. 129(5), 65 (2019)

    Article  MATH  Google Scholar 

  26. Ahmad, M., Zada, A., Alzabut, J.: Hyers-Ulam stability of a coupled system of fractional differential equations of Hilfer - Hadamard type. Demonstratio Mathematica 52(1), 283–295 (2019)

    Article  MATH  Google Scholar 

  27. Herzallah, M.A.E., Baleanu, D.: On fractional order hybrid differential equations. Abst. Appl. Analy. 2014, 389386 (2014)

    MATH  Google Scholar 

  28. Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)

    Article  MATH  Google Scholar 

  29. Almeida, R., Malinowska, A.B., Teresa, N., Monteiro, T.: Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math. Meth. Appl. Sci. 41, 336–352 (2018)

    Article  MATH  Google Scholar 

  30. Almeida, R.: Fractional differential equations with mixed boundary conditions. Bull. Malays. Math. Sci. Soc. 42, 1687–1697 (2019)

    Article  MATH  Google Scholar 

  31. Almeida, R.: Functional differential equations involving the \(\psi \)-Caputo fractional derivative. Fractal Fract. 4(2), 29 (2020)

    Article  Google Scholar 

  32. Enns, R.H., Mcguire, G.C.: Nonlinear physics with mathematica for scientists and engineers. Birkhauser Bost. 3, 7643–7659 (2001)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  34. Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  35. Chasreechai, S., Sitthiwirattham, T.: Existence results of initial value problems for hybrid fractional sum-difference equations. Discrete Dyn. Nature Soc. 2018, 5268528 (2018)

    MATH  Google Scholar 

  36. Samko, S.G., Kilbas, A.A.: Marichev: fractional integrals and derivatives. Theory Appl. Gordon Breach 1, 109 (1993)

    Google Scholar 

  37. Sitho, S., Ntouyas, S.K., Tariboon, J.: Existence results for hybrid fractional integro-differential equations. Bound. Value Probl. 2015(1), 113 (2015)

    Article  MATH  Google Scholar 

  38. Bashiri, T., Vaezpour, S.M., Park, C.: Existence results for fractional hybrid differential systems in Banach algebras. Adv. Diff. Equat. 2016(1), 1–13 (2016)

    MATH  Google Scholar 

  39. Shatanawi, W., Boutiara, A., Abdo, M.S., Jeelani, M.B., Abodayeh, K.: Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative. Adv. Diff. Equat. 1, 1–19 (2021)

    MATH  Google Scholar 

  40. Singla, K., Gupta, R.K.: On invariant analysis of some time fractional nonlinear systems of partial differential equations. I J. Math.l Phys. 57, 101504 (2016)

    Article  MATH  Google Scholar 

  41. Grahovac, N.M., Zigic, M.M.: Modelling of the hamstring muscle group by use of fractional derivatives. Comput. Math. Appl. 59, 1695–1700 (2010)

    Article  MATH  Google Scholar 

  42. Sousa, J. Vanterler da C., Vellappandi, M., Govindaraj, V., Frederico, Gastão S. F.: Reachability of fractional dynamical systems using \(\psi \)-Hilfer pseudo-fractional derivative. J. Math. Phys. 62: 082703 (2021)

  43. El-Nabulsi, R.A.: Glaeske-Kilbas-Saigo fractional integration and fractional Dixmier trace. Acta Math. Vietnam. 37(2), 149–160 (2012)

    MATH  Google Scholar 

  44. Balachandran, K., Govindaraj, V., Rivero, M., Trujillo, J.J.: Controllability of fractional damped dynamical systems. Appl. Math. Computat. 257, 66–73 (2015)

    Article  MATH  Google Scholar 

  45. El-Nabulsi, R.A.: Nonlocal-in-time kinetic energy in nonconservative fractional systems, disordered dynamics, jerk and snap and oscillatory motions in the rotating fluid tube. Int. J. Non-Linear Mech. 93, 65–81 (2017)

    Article  Google Scholar 

  46. Borisut, P., Kumam, P., Ahmed, I., Jirakitpuwapat, W.: Existence and uniqueness for \(\psi \)-Hilfer fractional differential equation with nonlocal multi-point condition. Math. Methods Appl. Sci. 44(3), 2506–2520 (2021)

    Article  MATH  Google Scholar 

  47. El-Nabulsi, R.A.: On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics. Proceed. Royal Soc. A 476, 20190729 (2020)

    Article  MATH  Google Scholar 

  48. Moshrefi-Torbati, M., Hammond, J.K.: Physical and geometrical interpretation of fractional operators. J. Franklin Instit. 335B(6), 1077–1086 (1998)

    Article  MATH  Google Scholar 

  49. El-Nabulsi, R.A.: Path integral formulation of fractionally perturbed Lagrangian oscillators on fractal. J. Statist. Phys. 172(6), 1617–1640 (2018)

    Article  MATH  Google Scholar 

  50. Balachandran, K., Govindaraj, V., Rodríguez-Germa, L., Trujillo, J.J.: Controllability results for nonlinear fractional-order dynamical systems. J. Opt. Theory Appl 156, 33–44 (2013)

    Article  MATH  Google Scholar 

  51. Balachandran, K., Govindaraj, V., Rodríguez-Germa, L., Trujillo, J.J.: Controllability of nonlinear higher order fractional dynamical systems. Nonlinear Dyn. 71, 605–612 (2013)

    Article  MATH  Google Scholar 

  52. El-Nabulsi, R.A.: Non-standard fractional Lagrangians. Nonlinear Dyn. 74(1), 381–394 (2013)

    Article  MATH  Google Scholar 

  53. El-Nabulsi, R.A.: Finite two-point space without quantization on noncommutative space from a generalized fractional integral operator. Comp. Analy. Operat. Theory 12(7), 1609–1616 (2018)

    Article  Google Scholar 

  54. Babakhani, A., Yadollahzadeh, M., Neamaty, A.: Some properties of pseudo-fractional operators. J Pseudo-Diff. Operat Appl. 9(3), 677–700 (2018)

    Article  MATH  Google Scholar 

  55. El-Nabulsi, R.A.: Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent. Computat. Appl. Math. 33(1), 163–179 (2014)

    Article  MATH  Google Scholar 

  56. Caponetto, R., Dongola, G., Fortuna, L., Petras, I.: Fractional Order Systems: Modeling and Control Applications. World Scientific, Singapore (2010)

    Book  Google Scholar 

  57. Tabouche, N., Berhail, A., Matar, M.M., Alzabut, J., Selvam, A.G.M., Vignesh, D.: Existence and stability analysis of solution for Mathieu fractional differential equations with applications on some physical phenomena. Iranian J. Sci. Technol. Trans. A Sci. 45, 973–982 (2021)

    Article  Google Scholar 

  58. Shammakh, W., Selvam, A. George Maria., Vignesh, D., Alzabut, J.: A study of generalized hybrid discrete pantograph equation via Hilfer fractional operator. Fractal Fract. 6, 152 (2022)

    Article  Google Scholar 

  59. Alzabut, J., Selvam, A. George., Maria, Vignesh, Mohammadi, H., Rezapour, S.: On chaos of discrete time fractional order host-immune-tumor cells interaction model. J. Appl. Math. Comput. 1, 109 (2022)

    MATH  Google Scholar 

  60. Selvam, A.G.M., Baleanu, D., Alzabut, J., Vignesh, D., Abbas, S.: On Hyers-Ulam Mittag-Leffler stability of discrete fractional Duffing equation with application on inverted pendulum. Adv. Diff. Equat. 2020, 456 (2020)

    Article  MATH  Google Scholar 

  61. Jajarmi, Amin, Baleanu, Dumitru, Sadat Sajjadi, Samaneh, Nieto, Juan J.: Analysis and some applications of a regularized \(\psi \)–Hilfer fractional derivative. J. Computat. Appl. Math. 415, 114476 (2022)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

J. Alzabut would like to thank Prince Sultan University and Ostim Technical University for supporting this research.

Funding

Funding was provided by Prince Sultan University Prof. Jehad Alzabut (Grant No. 1).

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Sciences appliquées, Université de Ghardaia, Gharadaia, Algerie

    A. Boutiara

  2. Department of Mathematics and Sciences, Prince Sultan University, Riyadh, 11586, Saudi Arabia

    J. Alzabut

  3. Department of Industrial Engineering, Ostim Technical University, Ankara, 06374, Turkey

    J. Alzabut

  4. Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur, 635 601, Tamil Nadu, India

    A. G. M. Selvam

  5. Department of Mathematics, School of Advanced Sciences, Kalasalingam Academy of Research and Education, Krishnankoil, Srivilliputhur, Tamil Nadu, 626 126, India

    D. Vignesh

Authors
  1. A. Boutiara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Alzabut
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. G. M. Selvam
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. Vignesh
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Corresponding author

Correspondence to J. Alzabut.

Ethics declarations

Conflict of interest

The authors declare that they have 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

Boutiara, A., Alzabut, J., Selvam, A.G.M. et al. Analysis and Applications of Sequential Hybrid \(\psi \)-Hilfer Fractional Differential Equations and Inclusions in Banach Algebra. Qual. Theory Dyn. Syst. 22, 12 (2023). https://doi.org/10.1007/s12346-022-00710-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-022-00710-x

Keywords

Mathematics Subject Classification

Search

Navigation