In this paper, we investigated $ n $-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in $ 3 $-dimensional space at the end of the paper.
Citation: Salma Noor, Aman Ullah, Anwar Ali, Saud Fahad Aldosary. Analysis of a hybrid fractional coupled system of differential equations in $ n $-dimensional space with linear perturbation and nonlinear boundary conditions[J]. AIMS Mathematics, 2024, 9(6): 16234-16249. doi: 10.3934/math.2024785
In this paper, we investigated $ n $-dimensional fractional hybrid differential equations (FHDEs) with nonlinear boundary conditions in a nonlinear coupled system. For this purpose, we used Dhage's fixed point theory, and applied the Krasnoselskii-type coupled fixed point theorem to construct existence conditions of the solution of the FHDEs. To illustrated this idea, suitable examples are presented in $ 3 $-dimensional space at the end of the paper.
[1] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integral and derivatives: Theory and applications, Philadelphia: Gordon and Breach Science Publishers, 1993. |
[2] | J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027 |
[3] | R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A, 278 (2000), 107–125. https://doi.org/10.1016/S0378-4371(99)00503-8 doi: 10.1016/S0378-4371(99)00503-8 |
[4] | B. Ahmed, S. K. Ntouyas, A. Alsaedi, New existence results for nonlinear fractionaldifferential equations with three-point integral boundary conditions, Adv. Differ. Equ., 2011 (2011), 107384. https://doi.org/10.1155/2011/107384 doi: 10.1155/2011/107384 |
[5] | C. F. Li, X. N. Luo, Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59 (2010), 1363–1375. https://doi.org/10.1016/j.camwa.2009.06.029 doi: 10.1016/j.camwa.2009.06.029 |
[6] | R. P. Agarwal, M. Belmekki, M. Benchohra, A survey on semilinear differential equations and inclusions involving Riemann-Liouville fractional derivative, Adv. Differ. Equ., 2009 (2009), 981728. https://doi.org/10.1155/2009/981728 doi: 10.1155/2009/981728 |
[7] | K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence results for fractional impulsive integrodifferential equations in Banach spaces, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1970–1977. http://dx.doi.org/10.1016/j.cnsns.2010.08.005 doi: 10.1016/j.cnsns.2010.08.005 |
[8] | B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl., 58 (2009), 1838–1843. https://doi.org/10.1016/j.camwa.2009.07.091 doi: 10.1016/j.camwa.2009.07.091 |
[9] | V. Gafychuk, B. Datsko, V. Meleshko, Mathematical modeling of time fractional reaction-difusion systems, J. Comput. Appl. Math., 220 (2008), 215–225. https://doi.org/10.1016/j.cam.2007.08.011 doi: 10.1016/j.cam.2007.08.011 |
[10] | X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64–69. https://doi.org/10.1016/j.aml.2008.03.001 doi: 10.1016/j.aml.2008.03.001 |
[11] | C. Zhai, R. Jiang, Unique solutions for a new coupled system of fractional differential equations, Adv. Differ. Equ., 2018 (2018), 1. https://doi.org/10.1186/s13662-017-1452-3 doi: 10.1186/s13662-017-1452-3 |
[12] | A. Ali, M. Sarwar, M. B. Zada, K. Shah, Degree theory and existence of positive solutions to coupled system involving proportional delay with fractional integral boundary conditions, Math. Meth. Appl. Sci., 2020, 1–13. https://doi.org/10.1002/mma.6311 doi: 10.1002/mma.6311 |
[13] | A. Ali, M. Sarwar, M. B. Zada, T. Abdeljawad, Existence and uniqueness of solutions for coupled system of fractional differential equations involving proportional delay by means of topological degree theory, Adv. Differ. Equ., 2020 (2020), 470. https://doi.org/10.1186/s13662-020-02918-0 doi: 10.1186/s13662-020-02918-0 |
[14] | D. Baleanu, H. Khan, H. Jafari, R. A. Khan, M. Alipour, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Differ. Equ., 2015 (2015), 318. https://doi.org/10.1186/s13662-015-0651-z doi: 10.1186/s13662-015-0651-z |
[15] | T. Bashiri, S. M. Vaezpour, C. Park, A coupled fixed point theorem and application to fractional hybrid differential problems, Fixed Point Theory Appl., 2016 (2016), 23. https://doi.org/10.1186/s13663-016-0511-x doi: 10.1186/s13663-016-0511-x |
[16] | B. Ahmad, S. K. Ntouyas, A. Alsaedi, Existence results for a system of coupled hybrid fractional differential equations, Sci. World J., 2014 (2014), 426438. https://doi.org/10.1155/2014/426438 doi: 10.1155/2014/426438 |
[17] | W. Kumam, M. B. Zada, K. Shah, R. A. Khan, Investigating a coupled hybrid system of nonlinear fractional differential equations, Discrete Dyn. Nat. Soc., 2018 (2018), 5937572. https://doi.org/10.1155/2018/5937572 doi: 10.1155/2018/5937572 |
[18] | A. Ali, M. Sarwar, K. Shah, T. Abdeljawad, Study of coupled system of fractional hybrid differential equations via prior estimate method, Fractals, 30 (2022), 2240213. https://doi.org/10.1142/S0218348X22402137 doi: 10.1142/S0218348X22402137 |
[19] | H. Akhadkulov, F. Alsharari, T. Y. Ying, Applications of Krasnoselskii-Dhage type fixed-point theorems to fractional hybrid differential equations, Tamkang J. Math. 52 (2021), 281–292. https://doi.org/10.5556/j.tkjm.52.2021.3330 doi: 10.5556/j.tkjm.52.2021.3330 |
[20] | B. C. Dhage, S. B. Dhage, J. R. Graef, Dhage iteration method for initial value problems for nonlinear first order hybrid integrodifferential equations, J. Fixed Point Theory Appl., 18 (2016), 309–326. https://doi.org/10.1007/s11784-015-0279-3 doi: 10.1007/s11784-015-0279-3 |