On Periodic Solutions for Implicit Problem with Nonlinear Fractional Differential Equation Involving the Riesz-Caputo Fractional Derivative

Authors

  • Soufyane Bouriah
  • Abdelkrim Salim Hassiba Benbouali University of Chlef
  • Rosana Rodriguez-Lopez
  • Mouffak Benchohra

Keywords:

Coincidence degree theory, Riesz-Caputo fractional derivative, coupled system, uniquenes

Abstract

The main goal of this paper is to study the existence and uniqueness of periodic solutions for implicit problem with nonlinear fractional differential equation (NFDEs) involving the Riesz-Caputo fractional derivative. The proofs are based upon the coincidence degree theory of Mawhin. An example is constructed to authenticate and affirm the main findings.

References

[1] S. Abbas, B. Ahmad, M. Benchohra and A. Salim, Fractional Difference, Differential Equations and Inclusions: Analysis and Stability, Morgan Kaufmann, Cambridge, 2024. https://doi.org/10.1016/C2023-0-00030-9

[2] R. S. Adiguzel, U. Aksoy, E. Karapinar, I. M. Erhan, On The Solutions Of Fractional Differential Equations Via Geraghty Type Hybrid Contractions, Appl. Comput. Math. 20 (2021), 313-333.

[3] D. Baleanu, Z. B. G¨uven¸c, and J. A. T. Machado New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, 2010.

[4] F. Chen, D. Baleanu, and G. Wu, Existence results of fractional differential equations with Riesz-Caputo derivative, Eur. Phys. J. 226 (2017), 3411-3425.

[5] F. Chen, A. Chen, and X. Wu, Anti-periodic boundary value problems with Riesz-Caputo derivative, Adv. Difference Equ. 2019 (2019). https://doi.org/10.1186/s13662-019-2001-z

[6] M. Benchohra, S. Bouriah, A. Salim and Y. Zhou, Fractional Differential Equations: A Coincidence Degree Approach, De Gruyter, Berlin, Boston, 2024. https://doi.org/10.1515/9783111334387

[7] M. Benchohra, E. Karapınar, J. E. Lazreg and A. Salim, Advanced Topics in Fractional Differential Equations: A Fixed Point Approach, Springer, Cham, 2023. https://doi.org/10.1007/978-3-031-26928-8

[8] M. Benchohra, E. Karapınar, J. E. Lazreg and A. Salim, Fractional Differential Equations: New Advancements for Generalized Fractional Derivatives, Springer, Cham, 2023. https://doi.org/10.1007/978-3-031-34877-8

[9] J. Dehong, M.Yuan and G. Weigao, A Singular Fractional Differential Equation with Riesz-Caputo Derivative, Journal of Applied Analysis and Computation, Volume 14, Number 2,( April 2024), 642-656.

[10] R.E. Gaines and J. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., vol. 568, Springer-Verlag, Berlin, 1977.

[11] J. R. Graef, J. Henderson and A. Ouahab, Impulsive Differential Inclusions. A Fixed Point Approch, De Gruyter, Berlin/Boston, 2013.

[12] C. Y. Gu, G. C. Wu, Positive solutions of fractional differential equations with the Riesz space derivative. Appl. Math. Lett. 95 (2019), 59-64.

[13] E. Karapınar, R. Sevinik-Adıguzel, U. Aksoy, ˙I.M. Erhan, A new approach to the existence and uniqueness of solutions for a class of nonlinear q-fractional boundary value problems, Appl. Comput. Math. (2025), 235-249. https://doi.org/10.30546/1683-6154.24.2.2025.235

[14] A. A. Kilbas, Hari M. Srivastava and Juan J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

[15] S. Krim, A. Salim and M. Benchohra, Nonlinear contractions and Caputo tempered implicit fractional differential equations in b-metric spaces with infinite delay. Filomat. 37 (22) (2023), 7491-7503. https://doi.org/10.2298/FIL2322491K

[16] N. Laledj, A. Salim, J. E. Lazreg, S. Abbas, B. Ahmad and M. Benchohra, On implicit fractional q-difference equations: Analysis and stability. Math. Meth. Appl. Sci. 2 (2022), 1-23. https://doi.org/10.1002/mma.8417

[17] M. Li and Y. Wang, Existence and iteration of monotone positive solutions for fractional boundary value problems with Riesz-Caputo derivative. Engineering Letters 29 (2021), 1-5.

[18] J. Mawhin, NSFCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, 1979.

[19] A. Naas, M. Benbachir, M. S. Abdo and A. Boutiara, Analysis of a fractional boundary value problem involving Riesz- Caputo fractional derivative. ATNAA. 1 (2022), 14-27.

[20] D. O’Regan, Y.J. Chao and Y.Q. Chen, Topological Degree Theory and Application, Taylor and Francis Group, Boca Raton, London, New York, 2006. [21] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[22] A. Salim, S. Abbas, M. Benchohra and E. Karapinar, A Filippov’s theorem and topological structure of solution sets for fractional q-difference inclusions. Dynam. Syst. Appl. 31 (2022), 17-34. https://doi.org/10.46719/dsa202231.01.02

[23] A. Salim, S. Abbas, M. Benchohra and E. Karapinar, Global stability results for Volterra-Hadamard random partial fractional integral equations. Rend. Circ. Mat. Palermo (2). (2022), 1-13. https://doi.org/10.1007/s12215-022-00770-7

[24] A. Salim, M. Benchohra and J. E. Lazreg, Nonlocal k-generalized ψ-Hilfer impulsive initial value problem with retarded and advanced arguments, Appl. Anal. Optim. 6 (2022), 21-47.

[25] A. Salim, S. Bouriah, M. Benchohra, J. E. Lazreg and E. Karapınar, A study on k-generalized ψ-Hilfer fractional differential equations with periodic integral conditions. Math. Methods Appl. Sci. (2023), 1-18. https://doi.org/10.1002/mma.9056

[26] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Amsterdam, 1987, Engl. Trans. from the Russian.

[27] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

Downloads

Published

2026-03-31

Issue

Section

Articles