Abstract
This paper deals with some existence and uniqueness results for a class of nonlinear fractional coupled systems with k-generalized \(\psi \)-Hilfer fractional differential equations and periodic conditions. The arguments are based on Mawhin’s coincidence degree theory. We demonstrate several results by changing the required conditions of the theorems. Furthermore, illustrative examples are presented to demonstrate the plausibility of our results.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.
References
Abbas, S., Benchohra, M., Lazreg, J.E., Nieto, J.J.: On a coupled system of Hilfer and Hilfer-Hadamard fractional differential equations in Banach spaces. J. Nonlinear Funct. Anal. 2018, 12 (2018)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer-Verlag, New York (2012)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2014)
Abdo, M.S., Abdeljawad, T., Kucche, K.D., et al.: On nonlinear pantograph fractional differential equations with Atangana-Baleanu-Caputo derivative. Adv. Differ. Equ. 2021, 65 (2021)
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci. 43, 1–12 (2020). https://doi.org/10.1002/mma.6652
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: On the solutions of fractional differential equations via Geraghty type hybrid contractions. Appl. Comput. Math. 20, 313–333 (2021)
Adiguzel, R.S., Aksoy, U., Karapinar, E., Erhan, I.M.: Uniqueness of solution for higher-order nonlinear fractional differential equations with multi-point and integral boundary conditions. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 115(3), 16 (2021)
Agrawal, O.P.: Some generalized fractional calculus operators and their applications in integral equations. Fract. Calc. Appl. Anal. 15(4), 700–711 (2012)
Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simulat. 44, 460–481 (2017)
Almeida, R.: Functional differential equations involving the \(\psi \)-Caputo fractional derivative. Fractal Fract. 4(29), 1–8 (2020)
Almeida, R., Malinowska, A.B., Odzijewicz, T.: On systems of fractional differential equations with the \(\psi \)-Caputo derivative and their applications. Math. Methods Appl. Sci. 42, 1–16 (2019)
Afshari, H., Kalantari, S., Karapinar, E.: Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ. 2015, 1–12 (2015)
Afshari, H., Karapinar, E.: A discussion on the existence of positive solutions of the boundary value problems via \(\psi \)-Hilfer fractional derivative on \(b\)-metric spaces. Adv. Differ. Equ. 2020, 1–11 (2020)
Balachandran, K., Kiruthika, S., Trujillo, J.J.: Existence of solutions of nonlinear fractional pantograph equations. Acta. Math. Sci. 33(3), 712–720 (2013)
Benchohra, M., Bouazzaoui, F., Karapinar, E., Salim, A.: Controllability of second order functional random differential equations with delay. Mathematics 10, 16 (2022). https://doi.org/10.3390/math10071120
Benchohra, M., Bouriah, S., Graef, J.R.: Nonlinear implicit differential equation of fractional order at resonance. Electron. J. Differ. Equ. 2016(324), 1–10 (2016)
Benkhettou, N., Aissani, K., Salim, A., Benchohra, M., Tunc, C.: Controllability of fractional integro-differential equations with infinite delay and non-instantaneous impulses. Appl. Anal. Optim. 6, 79–94 (2022)
Bouriah, S., Foukrach, D., Benchohra, M., Graef, J.: Existence and uniqueness of periodic solutions for some nonlinear fractional pantograph differential equations with \(\psi \)-Caputo derivative. Arab. J. Math. 10, 575–587 (2021). https://doi.org/10.1007/s40065-021-00343-z
Bouriah, S., Salim, A., Benchohra, M.: On nonlinear implicit neutral generalized Hilfer fractional differential equations with terminal conditions and delay. Topol. Algebra Appl. 10, 77–93 (2022). https://doi.org/10.1515/taa-2022-0115
Chu, Y.M., Awan, M.U., Talib, S., Noor, M.A., Noor, K.I.: Generalizations of Hermite-Hadamard like inequalities involving \(\chi _{{\kappa }}\)-Hilfer fractional integrals. Adv. Differ. Equ. 2020, 594 (2020)
Derbazi, C., Baitiche, Z.: Coupled systems of \(\psi \)-Caputo differential equations with initial conditions in Banach spaces. Mediter. J. Math. 17, 169 (2020)
Derbazi, C., Hammouche, H., Salim, A., Benchohra, M.: Measure of noncompactness and fractional hybrid differential equations with hybrid conditions. Differ. Equ. Appl. 14, 145–161 (2022). https://doi.org/10.7153/dea-2022-14-09
Diaz, R., Teruel, C.: \({q, k}\)-Generalized gamma and beta functions. J. Nonlinear Math. Phys. 12, 118–134 (2005)
Gaines, R.E., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math, vol. 568. Springer-Verlag, Berlin (1977)
Heris, A., Salim, A., Benchohra, M., Karapinar, E.: Fractional partial random differential equations with infinite delay. Results Phys. (2022). https://doi.org/10.1016/j.rinp.2022.105557
Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific Publishing Company, Singapore (2011)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier, Amsterdam (2006)
Kucche, K.D., Mali, A.D.: On the nonlinear \((k,\psi )\)-Hilfer fractional differential equations. Chaos Solitons Fractals 2021, 14 (2021)
Laledj, N., Salim, A., Lazreg, J.E., Abbas, S., Ahmad, B., Benchohra, M.: On implicit fractional \(q\)-difference equations: analysis and stability. Math. Methods Appl. Sci. 45, 1–23 (2022). https://doi.org/10.1002/mma.8417
Mawhin, J.: NSFCBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI (1979)
Mubeen, S., Habibullah, G.M.: \(k\)-fractional integrals and application. Int. J. Contemp. Math. Sci. 7, 89–94 (2012)
O’Regan, D., Chao, Y.J., Chen, Y.Q.: Topological Degree Theory and Application. Taylor and Francis Group, Boca Raton, London, New York (2006)
Rahimkhani, P., Ordokhani, Y., Babolian, E.: Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J. Comput. Appl. Math. 309, 493–510 (2017)
Rashid, S., Aslam Noor, M., Inayat Noor, K., Chu, Y.M.: Ostrowski type inequalities in the sense of generalized \({\cal{K} }\)-fractional integral operator for exponentially convex functions. AIMS Math. 5, 2629–2645 (2020)
Saeed, U., Rehman, M.: Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. 2014, 8 (2014)
Salim, A., Abbas, S., Benchohra, M., Karapinar, E.: A Filippov’s theorem and topological structure of solution sets for fractional q-difference inclusions. Dyn. Syst. Appl. 31, 17–34 (2022). https://doi.org/10.46719/dsa202231.01.02
Salim, A., Abbas, S., Benchohra, M., Karapinar, E.: Global stability results for Volterra-Hadamard random partial fractional integral equations. Rend. Circ. Mat. Palermo 2, 1–13 (2022). https://doi.org/10.1007/s12215-022-00770-7
Salim, A., Ahmad, B., Benchohra, M., Lazreg, J.E.: Boundary value problem for hybrid generalized Hilfer fractional differential equations. Differ. Equ. Appl. 14, 379–391 (2022). https://doi.org/10.7153/dea-2022-14-27
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, 14 (2022). https://doi.org/10.1007/s11784-021-00920-x
Salim, A., Benchohra, M., Lazreg, J.E., Henderson, J.: On \(k\)-generalized \(\psi \)-Hilfer boundary value problems with retardation and anticipation. Adv. Theory Nonlinear Anal. Appl. 6, 173–190 (2022). https://doi.org/10.31197/atnaa.973992
Salim, A., Benchohra, M., Lazreg, J.E., Karapınar, E.: On \(k\)-generalized \(\psi \)-Hilfer impulsive boundary value problem with retarded and advanced arguments. J. Math. Ext. 15, 39 (2021)
Salim, A., Lazreg, J.E., Ahmad, B., Benchohra, M., Nieto, J.J.: A study on \(k\)-generalized \(\psi \)-Hilfer derivative operator. Vietnam J. Math. (2022). https://doi.org/10.1007/s10013-022-00561-8
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon (1993)
Shah, K., Vivek, D., Kanagarajan, K.: Dynamics and stability of \(\psi \)-fractional Pantograph equations with boundary conditions. Bol. Soc. Parana. Mat. 39(5), 43–55 (2021)
Sousa, J.V.C., Capelas de Oliveira, E.: A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator. Differ. Equ. Appl. 11, 87–106 (2019)
Sousa, J.V.C., Capelas de Oliveira, E.: Fractional order pseudo-parabolic partial differential equation: Ulam-Hyers stability. Bull. Braz. Math. Soc. 50, 481–496 (2019)
Sousa, J.V.C., Capelas de Oliveira, E.: On the \(\psi \)-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)
Funding
Not available.
Author information
Authors and Affiliations
Contributions
The study was carried out in collaboration of all authors. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
It is declared that authors has no competing interests.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
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.
About this article
Cite this article
Salim, A., Benchohra, M. & Lazreg, J.E. On Implicit k-Generalized \(\psi \)-Hilfer Fractional Differential Coupled Systems with Periodic Conditions. Qual. Theory Dyn. Syst. 22, 75 (2023). https://doi.org/10.1007/s12346-023-00776-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-023-00776-1