Abstract
In this typescript, we study system of nonlinear implicit coupled differential equations of arbitrary (non–integer) order having nonlocal boundary conditions on closed interval [0, 1] with Caputo fractional derivative. We establish sufficient conditions for the existence, at least one and a unique solution of the proposed coupled system with the help of Krasnoselskii’s fixed point theorem and Banach contraction principle. Moreover, we scrutinize the Hyers–Ulam stability for the considered problem. We present examples to illustrate our main results.
Acknowledgements
All authors would like to thank the referees for suggestions to make better this typescript in the present form.
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Author contributions: All authors unbiased contribution to this typescript.
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Research funding: There is no source of sponsor to bolster up this typescript financially.
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Conflict of interest statement: There is no competing interests concerning this analysis work.
References
[1] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus, Dordrecht, Springer, 2007.10.1007/978-1-4020-6042-7Search in Google Scholar
[2] K. B. Oldham, “Fractional differential equations in electrochemistry,” Adv. Eng. Software, vol. 41, pp. 9–12, 2010. https://doi.org/10.1016/j.advengsoft.2008.12.012.Search in Google Scholar
[3] B. M. Vintagre, I. Podlybni, A. Hernandez, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fract. Cal. Appl. Anal., vol. 3, no. 3, pp. 231–248, 2000.Search in Google Scholar
[4] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Beijing, Springer, Heidelberg, Higher Education Press, 2010.Search in Google Scholar
[5] F. A. Rihan, “Numerical modeling of fractional order biological systems,” Abstr. Appl. Anal., vol. 2013, pp. 1–11, 2013. https://doi.org/10.1155/2013/816803.Search in Google Scholar
[6] R. Hilfer, Application of Fractional Calculus in Physics, Singapore, World Scientific, 2000.10.1142/3779Search in Google Scholar
[7] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent,” Geophys. J. Roy. Astron. Soc., vol. 13, no. 5, pp. 529–539, 1967. https://doi.org/10.1111/j.1365-246x.1967.tb02303.x.Search in Google Scholar
[8] A. Khan, K. Shah, Y. Li, and T. S. Khan, “Ulam type stability for a coupled systems of boundary value problems of nonlinear fractional differential equations,” J. Funct. Spaces, vol. 2017, no. 8, pp. 1–8, 2017. https://doi.org/10.1155/2017/3046013.Search in Google Scholar
[9] V. Lakshmikantham, S. Leela, and J. Vasundhara, Theory of Fractional Systems, Cambridge, Cambridge Academic Publishers, 2009.Search in Google Scholar
[10] I. Podlubny, Fractional Differential Equation, Mathematics in Science and Engineering, New York, Academic Press, 1999.Search in Google Scholar
[11] U. Riaz and A. Zada, “Analysis of (α, β)–order coupled implicit Caputo fractional differential equations using topological degree method,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 22, nos. 7–8, pp. 897–915, 2020. https://doi.org/10.1515/ijnsns-2020-0082.Search in Google Scholar
[12] U. Riaz, A. Zada, Z. Ali, I. L. Popa, S. Rezapour, and S. Etemad, “On a Riemann–Liouville type implicit coupled system via generalized boundary conditions,” Mathematics, vol. 9, no. 1205, pp. 1–22, 2021. https://doi.org/10.3390/math9111205.Search in Google Scholar
[13] A. Zada, F. Khan, U. Riaz, and T. Li, “Hyers–Ulam stability of linear summation equations,” Punjab Univ. J. Math., vol. 49, no. 1, pp. 19–24, 2017.10.1186/s13662-017-1248-5Search in Google Scholar
[14] A. Zada, U. Riaz, and F. U. Khan, “Hyers–Ulam stability of impulsive integral equations,” Boll. Unione Mat. Ital., vol. 12, no. 3, pp. 453–467, 2019. https://doi.org/10.1007/s40574-018-0180-2.Search in Google Scholar
[15] S. M. Ulam, A Collection of the Mathematical Problems, New York, Interscience, 1960.Search in Google Scholar
[16] D. H. Hyers, “On the stability of the linear functional equation,” Proc. Natl. Acad. Sci. U.S.A., vol. 27, no. 4, pp. 222–224, 1941. https://doi.org/10.1073/pnas.27.4.222.Search in Google Scholar PubMed PubMed Central
[17] M. Alam, A. Zada, and U. Riaz, “On a coupled impulsive fractional integrodifferential system with Hadamard derivatives,” Qual. Theory Dyn. Syst., vol. 12, no. 8, pp. 1–31, 2022. https://doi.org/10.1007/s12346-021-00535-0.Search in Google Scholar
[18] J. Hung, S. M. Jung, and Y. Li, “On the Hyers–Ulam stability of non–linear differential equations,” Bull. Korean Math. Soc., vol. 52, pp. 685–697, 2015.10.4134/BKMS.2015.52.2.685Search in Google Scholar
[19] D. H. Hyers, G. Isac, and T. Rassias, Stability of Functional Equations in Several Variables, Basel, Switzerland, Birkhäuser, 1998.10.1007/978-1-4612-1790-9Search in Google Scholar
[20] S. M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis, New York, Springer, 2011.10.1007/978-1-4419-9637-4Search in Google Scholar
[21] S. M. Jung, “Hyers–Ulam stability of linear differential equations of first order,” Appl. Math. Lett., vol. 19, pp. 854–858, 2006. https://doi.org/10.1016/j.aml.2005.11.004.Search in Google Scholar
[22] U. Riaz, A. Zada, Z. Ali, M. Ahmad, J. Xu, and Z. Fu, “Analysis of nonlinear coupled systems of impulsive fractional differential equations with Hadamard derivatives,” Math. Probl Eng., vol. 2019, pp. 1–20, 2019. https://doi.org/10.1155/2019/5093572.Search in Google Scholar
[23] U. Riaz, A. Zada, Z. Ali, Y. Cui, and J. Xu, “Analysis of coupled systems of implicit impulsive fractional differential equations involving Hadamard derivatives,” Adv. Differ. Equ., vol. 2019, no. 226, pp. 1–27, 2019. https://doi.org/10.1186/s13662-019-2163-8.Search in Google Scholar
[24] R. Rizwan, A. Zada, H. Waheed, and U. Riaz, “Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 24, no. 6, pp. 2405–2423, 2023. https://doi.org/10.1515/ijnsns-2020-0240.Search in Google Scholar
[25] A. Zada, M. Alam, and U. Riaz, “Analysis of q–fractional implicit boundary value problems having Stieltjes integral conditions,” Math. Methods Appl. Sci., vol. 2020, pp. 1–33, 2020. https://doi.org/10.1002/mma.7038.Search in Google Scholar
[26] Z. Ali, K. Shah, A. Zada, and P. Kumam, “Mathematical analysis of coupled systems with fractional order boundary conditions,” Fractals, vol. 28, no. 8, pp. 1–17, 2020. https://doi.org/10.1142/s0218348x20400125.Search in Google Scholar
[27] Z. Ali, A. Zada, and K. Shah, “On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations,” Bull. Malays. Math. Sci. Soc., vol. 4, pp. 2681–2699, 2019. https://doi.org/10.1007/s40840-018-0625-x.Search in Google Scholar
[28] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North–Holland Mathematics Studies, vol. 204, Amsterdam, Elsevier, 2006.Search in Google Scholar
[29] M. Altman, “A fixed point theorem for completely continuous operators in Banach spaces,” Bull. Acad. Polon. Sci., vol. 3, pp. 409–413, 1955.Search in Google Scholar
[30] C. Urs, “Coupled fixed point theorem and applications to periodic boundary value problem,” Miskolic Math. Notes, vol. 14, pp. 323–333, 2013. https://doi.org/10.18514/mmn.2013.598.Search in Google Scholar
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