STABILITY AND EXISTENCE OF SOLUTIONS FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS

Jun Qian; Youhui Su; Xiaoling Han; Yongzhen Yun

doi:10.11948/20220336

2023 Volume 13 Issue 4

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(4): 2026-2047

Next Article Previous Article

Jun Qian, Youhui Su, Xiaoling Han, Yongzhen Yun. STABILITY AND EXISTENCE OF SOLUTIONS FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 2026-2047. doi: 10.11948/20220336

Citation:

Jun Qian, Youhui Su, Xiaoling Han, Yongzhen Yun. STABILITY AND EXISTENCE OF SOLUTIONS FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 2026-2047. doi: 10.11948/20220336

STABILITY AND EXISTENCE OF SOLUTIONS FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS

1.
School of Mathematics and Statistics, Xuzhou University of Technology, 221018 Xuzhou, China
2.
College of science, Shenyang University of Technology, 110870 Shenyang, China
3.
College of Mathematics and Statistics, Northwest Normal University, 730070 Lanzhou, China

Author Bio: Email: qianjun01@inesa.com (J. Qian); Email: yunyz@xzit.edu.cn (Y. Yun)
Corresponding authors: Email: suyh02@163.com (Y. Su); Email: hanxiaoling9@163.com (X. Han)
Fund Project: The authors were supported by National Natural Science Foundation of China (Nos. 12161079, 12126427)

Abstract

Abstract

In this paper, we study a coupled system of Caputo type fractional differential equations with integral boundary conditions. By Leray-Schauder alternative theorem, the existence of solutions for the factional differential system are obtained. The Hyers-Ulam stability of solutions is discussed and sufficient conditions for the stability are developed. The main results are well illustrated with examples and numerical simulation graphs. The interesting point of this article is that it not only gives approximate graphs of solution by using the iterative methods, but also verifies the Hyers-Ulam stability of the coupling system by numerical simulation.
- Fractional differential equation /
- fixed point theorem /
- Hyers-Ulam stability /
- numerical simulation
MSC: 34A08, 34K37, 34B18

FullText(HTML)

References

[1]	R. P. Agarwal and S. Heikkila, On the solvability of first-order discontinuous scalar initial and boundary value problems, Anziam. J., 2003, 44(4), 513-538. doi: 10.1017/S1446181100012906 CrossRef Google Scholar
[2]	B. Ahmad, S. K. Ntouyas and A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos. Soliton. Fract., 2016, 83, 234-241. doi: 10.1016/j.chaos.2015.12.014 CrossRef Google Scholar
[3]	A. Alkhazzan, P. Jiang, D. Baleanu, et al, Stability and existence results for a class of nonlinear fractional differential equations with singularity, Math. Method. Appl. Sci., 2018, 41(18), 9321-9334. doi: 10.1002/mma.5263 CrossRef Google Scholar
[4]	M. T. Ashordiya, On boundary value problems for systems of linear generalized ordinary differential equations with singularities, Different. Equ., 2006, 42(3), 307-319. doi: 10.1134/S0012266106030013 CrossRef Google Scholar
[5]	A. Assanova, An integral-boundary value problem for a partial differential equation of second order, Turk. J. Math., 2019, 43(4), 1967-1978. doi: 10.3906/mat-1903-111 CrossRef Google Scholar
[6]	S. R. Baslandze and I. T. Kiguradze, On the unique solvability of a periodic boundary value problem for third-order linear differential equations, Different. Equ., 2006, 42(2), 165-171. doi: 10.1134/S0012266106020029 CrossRef Google Scholar
[7]	Y. Cui, W. Ma, et. al, New uniqueness results for boundary value problem of fractional differential equation, Nonlinear. Anal-Model., 2018, 23(1), 31-39. Google Scholar
[8]	Q. Dang and Q. Dang, Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem, Numer. Algorithms, 2020, 85(3), 887-907. doi: 10.1007/s11075-019-00842-3 CrossRef Google Scholar
[9]	A. Domoshnitsky, R. Hakl and B. Puza, Multi-point boundary value problems for linear functional-differential equations, Georgian. Math. J., 2017, 24(2), 193-206. doi: 10.1515/gmj-2016-0076 CrossRef Google Scholar
[10]	D. Dzhumabaev, E. Bakirova and S. Mynbayeva, A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation, Math. Method. Appl. Sci., 2020, 43(4), 1788-1802. doi: 10.1002/mma.6003 CrossRef Google Scholar
[11]	M. Faieghi, S. Kuntanapreeda, H. Delavari, et. al, LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear. Dynam., 2013, 72(1), 301-309. Google Scholar
[12]	Z. Ge and C. Ou, Chaos synchronization of fractional order modified duffing systems with parameters excited by a chaotic signal, Chaos. Soliton. Fract., 2008, 35(4), 705-717. doi: 10.1016/j.chaos.2006.05.101 CrossRef Google Scholar
[13]	J. R. Graef, L. Kong and B. Yang, Positive solutions for a fractional boundary value problem, Appl. Math. Lett., 2016, 56, 49-55. doi: 10.1016/j.aml.2015.12.006 CrossRef Google Scholar
[14]	A. Granas and J. Dugundji, Fixed point theory, Springer, New York, 2003. Google Scholar
[15]	D. H. Hyers, On the stability of the linear functional equation, P. Natl. Acad. Sci. USA., 1941, 27(4), 222-224. doi: 10.1073/pnas.27.4.222 CrossRef Google Scholar
[16]	Z. Jiao, Y. Chen and I. Podlubny, Distributed-order dynamic systems, Springer, New York, 2012. Google Scholar
[17]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. Google Scholar
[18]	X. Liu and M. Jia, Existence of solutions for the integral boundary value problems of fractional order impulsive differential equations, Math. Method. Appl. Sci., 2016, 39(3), 475-487. doi: 10.1002/mma.3495 CrossRef Google Scholar
[19]	N. I. Mahmudov and A. Al-Khateeb, Stability, existence and uniqueness of boundary value problems for a coupled system of fractional differential equations, Mathematics-Basel., 2019, 7(4), 354-366. Google Scholar
[20]	J. Nan, W. Hu, Y. Su, et. al, Stability and existence of solutions for fractional differential system with p-Laplacian operator on star graphs, Dynam. Syst. Appl., 2022, 31(3), 133-150. Google Scholar
[21]	R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Status. Solidi. B., 1986, 133(1), 425-430. doi: 10.1002/pssb.2221330150 CrossRef Google Scholar
[22]	I. Podlubny, Fractional differential equations, Academic. Press, San Diego, 1999. Google Scholar
[23]	I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math., 2010, 26, 103-107. Google Scholar
[24]	S. Rezapour, B. Tellab, C. T. Deressa, et. al, HU-type stability and numerical solutions for a nonlinear model of the coupled systems of Navier BVPs via the generalized differential transform method, Fractal. Fract., 2021, 5(4), 166-192. doi: 10.3390/fractalfract5040166 CrossRef Google Scholar
[25]	A. Sun, Y. Su and J. Sun, Existence of solutions to a class of fractional differential equations, J. Non. Modeling Anal., 2022, 4(3), 409-442. Google Scholar
[26]	A. Sun, Y. Su, Q. Yuan, et al, Existence of solutions to fractional differential equations with fractional-order derivative terms, J. Appl. Anal. Comput., 2021, 11(1), 486-520. Google Scholar
[27]	W. Sun, Y. Su and X. Han, Existence of solutions for a coupled system of Caputo-Hadamard fractional differential equations with $p$-Laplacian operator, J. Appl. Anal. Comput., 2022, 12(5), 1885-1900. Google Scholar
[28]	W. Sun, Y. Su, A. Sun and Q. Zhu, Existence and simulation of positive solutions for $m$-point fractional differential equations with derivative terms, Open. Math., 2021, 19, 1820-1846. doi: 10.1515/math-2021-0131 CrossRef Google Scholar
[29]	Y. Wang and S. Sun, Solvability to infinite-point boundary value problems for singular fractional differential equations on the half-line, J. Appl. Math. Comput., 2018, 57(1), 359-373. Google Scholar
[30]	H. Wang and R. Rodriguez-Lopez, On the existence of solutions to boundary value problems for interval-valued differential equations under gH-differentiability, Inform. Sciences., 2021, 553, 225-246. doi: 10.1016/j.ins.2020.10.052 CrossRef Google Scholar
[31]	Y. Xi, X. Zhang and Y. Su, Uniqueness results of fuzzy fractional differential equations under Krasnoselskii-Krein conditions, Dynam. Syst. Appl., 2021, 30(12), 1792-1801. Google Scholar
[32]	C. Yuan, X. Wen and D. Jiang, Existence and uniqueness of positive solution for nonlinear singular 2mth-order continuous and discrete Lidstone boundary value problems, Acta. Math. Sci., 2011, 31(1), 281-291. doi: 10.1016/S0252-9602(11)60228-2 CrossRef Google Scholar
[33]	X. Zhang, Y. Su and Y. Kong, Positive solutions for boundary value problem of fractional differential equation in Banach spaces, Dynam. Syst. Appl., 2021, 30(9), 1449-1462. Google Scholar
[34]	F. Zhang, Z. Ma and J. Yan, Functional boundary value problem for first order impulsive differential equations at variable times, Indian. J. Pure. Appl. Math., 2003, 34(5), 733-742. Google Scholar
[35]	X. Zhou and C. Xu, Well-posedness of a kind of nonlinear coupled system of fractional differential equations, Sci. China. Math., 2016, 59(6), 1209-1220. doi: 10.1007/s11425-015-5113-2 CrossRef Google Scholar