Home Existence and uniqueness of solutions for coupled systems of Liouville-Caputo type fractional integrodifferential equations with Erdélyi-Kober integral conditions
Article
Licensed
Unlicensed Requires Authentication

Existence and uniqueness of solutions for coupled systems of Liouville-Caputo type fractional integrodifferential equations with Erdélyi-Kober integral conditions

  • Muthaiah Subramanian ORCID logo and Akbar Zada ORCID logo EMAIL logo
Published/Copyright: September 23, 2020

Abstract

In this paper, we examine a coupled system of fractional integrodifferential equations of Liouville-Caputo form with nonlinearities depending on the unknown functions, as well as their lower-order fractional derivatives and integrals supplemented with coupled nonlocal and Erdélyi-Kober fractional integral boundary conditions. We explain that the topic discussed in this context is new and gives more analysis into the research of coupled boundary value problems. We have two results: the first is the existence result of the given problem by using the Leray-Schauder alternative, whereas the second referring to the uniqueness result is derived by Banach’s fixed-point theorem. Sufficient examples were also supplemented to substantiate the proof, and some variations of the problem were discussed.

Mathematics Subject Classification (2010): 26A33; 34A08; 34A12; 34B15; 37C25

Corresponding author: Akbar Zada, Department of Mathematics, University of Peshawar, Khyber Pakhtunkhwa 25000, Pakistan, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] B. Henry and S. Wearne, “Existence of Turing instabilities in a two-species fractional reaction-diffusion system,” SIAM J. Appl. Math., vol. 62, pp. 870–887, 2002, https://doi.org/10.1137/s0036139900375227.Search in Google Scholar

[2] T. Matsuzaki and M. Nakagawa, “A chaos neuron model with fractional differential equation,” J. Phys. Soc. Jpn., vol. 72, pp. 2678–2684, 2003, https://doi.org/10.1143/jpsj.72.2678.Search in Google Scholar

[3] W. Glockle and T. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophys. J., vol. 68, pp. 46–53, 1995, https://doi.org/10.1016/s0006-3495(95)80157-8.Search in Google Scholar

[4] N. Heymans and J. C. Bauwens, “Fractal rheological models and fractional differential equations for viscoelastic behavior,” Rheol. Acta, vol. 33, pp. 210–219, 1994, https://doi.org/10.1007/bf00437306.Search in Google Scholar

[5] R. Herrmann, Fractional Calculus: An Introduction for Physicists, Singapore, World Scientific, 2011.10.1142/8072Search in Google Scholar

[6] R. L. Magin, Fractional Calculus in Bioengineering, Chicago, USA, Begell House Publishers, 2006.Search in Google Scholar

[7] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Singapore, World Scientific, 2010.10.1142/p614Search in Google Scholar

[8] A. Ali, M. Sarwar, M. B. Zada, and K. Shah, “Degree theory and existence of positive solutions to coupled system involving proportional delay with fractional integral boundary conditions,” Math. Meth. Appl. Sci., vol. 113, 2020, https://doi.org/10.1002/mma.6311.Search in Google Scholar

[9] A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,” Adv. Differ. Equ., vol. 15, no. 1, pp. 31–49, 2004, https://doi.org/10.1080/10652460310001600717.Search in Google Scholar

[10] J. Klafter, S. C. Lim, and R. Metzler, Fractional Dynamics: Recent Advances, Singapore, World Scientific, 2012.10.1142/8087Search in Google Scholar

[11] Y. Liu, “Solvability of anti-periodic BVPs for impulsive fractional differential systems involving Caputo and Riemann–Liouville fractional derivatives,” Int. J. Nonlin. Sci. Num. Simul., vol. 19, no. 2, pp. 125–152, 2018, https://doi.org/10.1515/ijnsns-2017-0009.Search in Google Scholar

[12] S. Muthaiah, M. Murugesan, and N. Thangaraj, “Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations,” Adv. Nonlinear Anal., vol. 3, no. 3, pp. 162–173, 2019, https://doi.org/10.31197/atnaa.579701.Search in Google Scholar

[13] S. Muthaiah and D. Baleanu, “Existence of solutions for nonlinear fractional differential equations and inclusions depending on lower-order fractional derivatives,” Axioms, vol. 9, p. 44, 2020, https://doi.org/10.3390/axioms9020044.Search in Google Scholar

[14] S. K. Ntouyas and S. Etemad, “On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions,” Appl. Math. Comput., vol. 266, no. 1, pp. 235–246, 2016.10.1016/j.amc.2015.05.036Search in Google Scholar

[15] J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Netherlands, Springer, 2007.10.1007/978-1-4020-6042-7Search in Google Scholar

[16] B. Samet and H. Aydi, “Lyapunov-type inequalities for an anti-periodic fractional boundary value problem involving ψ-Caputo fractional derivative,” J. Inequal. Appl., vol. 2018, p. 286, 2018, https://doi.org/10.1186/s13660-018-1850-4.Search in Google Scholar PubMed PubMed Central

[17] B. Samet and H. Aydi, “On some inequalities involving Liouville-Caputo fractional derivatives and applications to special means of real numbers,” Mathematics, vol. 6, no. 10, pp. 1–9, 2018, https://doi.org/10.3390/math6100193.Search in Google Scholar

[18] Eiman, K. Shah, M. Sarwar, and D. Baleanu, “Study on Krasnoselskiis fixed point theorem for Caputo-Fabrizio fractional differential equations,” Adv. Differ. Equ., vol. 2020, p. 178, 2020, https://doi.org/10.1186/s13662-020-02624-x.Search in Google Scholar

[19] M. Sher, K. Shah, and J. Rassias, “On qualitative theory of fractional order delay evolution equation via the prior estimate method,” Math. Meth. Appl. Sci., vol. 112, 2020, https://doi.org/10.1002/mma.6390.Search in Google Scholar

[20] M. Subramanian and D. Baleanu, “Stability and existence analysis to a coupled system of Caputo type fractional differential equations with Erdelyi-Kober integral boundary conditions,” Appl. Math. Inf. Sci., vol. 14, no. 3, pp. 415–424, 2020.10.18576/amis/140307Search in Google Scholar

[21] M. Subramanian, A. R. V. Kumar, and T. N. Gopal, “Analysis of fractional boundary value problem with non-local integral strip boundary conditions,” Nonlinear Stud., vol. 26, no. 2, pp. 445–454, 2019.Search in Google Scholar

[22] M. Subramanian, A. R. V. Kumar, and T. N. Gopal, “Analysis of fractional boundary value problem with non local flux multi-point conditions on a Caputo fractional differential equation,” Stud. Univ. Babes-Bolyai Math., vol. 64, no. 4, pp. 511–527, 2019, https://doi.org/10.24193/subbmath.2019.4.06.Search in Google Scholar

[23] A. Zada and S. Ali, “Stability analysis of multi-point boundary value problem for sequential fractional differential equations with non-instantaneous impulses,” Int. J. Nonlin. Sci. Num. Simul., vol. 19, nos 7–8, pp. 763–774, 2018, https://doi.org/10.1515/ijnsns-2018-0040.Search in Google Scholar

[24] M. Javidi and B. Ahmad, “Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton–zooplankton system,” Ecol. Model., vol. 318, pp. 8–18, 2015, https://doi.org/10.1016/j.ecolmodel.2015.06.016.Search in Google Scholar

[25] K. Balachandran and J. Kokila, “Controllability of non-linear implicit fractional dynamical systems,” IMA J. Appl. Math., vol. 79, pp. 562–570, 2014, https://doi.org/10.1093/imamat/hxt003.Search in Google Scholar

[26] Y. Ding, Z. Wang, and H. Ye, “Optimal control of a fractional-order HIV-immune system with memory,” IEEE Trans. Control Syst. Technol., vol. 20, pp. 763–769, 2012, https://doi.org/10.1109/tcst.2011.2153203.Search in Google Scholar

[27] F. Zhang, G. Chen, C. Li, and J. Kurths, “Chaos synchronization in fractional differential systems,” Phil. Trans. R. Soc. A, vol. 371, p. 20120155, 2013, https://doi.org/10.1098/rsta.2012.0155.Search in Google Scholar PubMed

[28] 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. 42, no. 5, pp. 2681–2699, 2019, https://doi.org/10.1007/s40840-018-0625-x.Search in Google Scholar

[29] S. Ali, T. Abdeljawad, K. Shah, F. Jarad, and M. Arif, “Computation of iterative solutions along with stability analysis to a coupled system of fractional order differential equations,” Adv. Differ. Equ., vol. 2019, p. 215, 2019, https://doi.org/10.1186/s13662-019-2151-z.Search in Google Scholar

[30] P. Duraisamy and T. Nandha Gopal, “Existence and uniqueness of solutions for a coupled system of higher order fractional differential equations with integral boundary conditions,” Discontin. Nonlinearity Complex., vol. 7, no. 1, pp. 1–14, 2018, https://doi.org/10.5890/dnc.2018.03.001.Search in Google Scholar

[31] S. Saha Ray, “On the soliton solution and Jacobi Doubly periodic solution of the fractional coupled Schrödinger-KdV equation by a novel approach,” Int. J. Nonlin. Sci. Num. Simul., vol. 16, no. 2, pp. 79–95, 2015, https://doi.org/10.1515/ijnsns-2014-0050.Search in Google Scholar

[32] Samina, K. Shah, R. A. Khan, and D. Baleanu, “Study of implicit type coupled system of non-integer order differential equations with antiperiodic boundary conditions,” Math. Methods Appl. Sci., vol. 42, no. 6, pp. 1–10, 2019, https://doi.org/10.1002/mma.5496.Search in Google Scholar

[33] K. Shah, H. Khalil, and R. A. Khan, “Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations,” Chaos. Solitons. Fractals., vol. 77, pp. 240–246, 2015, https://doi.org/10.1016/j.chaos.2015.06.008.Search in Google Scholar

[34] H. H. Alsulami, S. K. Ntouyas, R. P. Agarwal, B. Ahmad, and A. Alsaedi, “A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions,” Adv. Differ. Equ., vol. 2017, p. 68, 2017, https://doi.org/10.1186/s13661-017-0801-1.Search in Google Scholar

[35] B. Ahmad, S. K. Ntouyas, and A. Alsaedi, “On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions,” Chaos. Solitons. Fractals., vol. 83, pp. 234–241, 2016, https://doi.org/10.1016/j.chaos.2015.12.014.Search in Google Scholar

[36] A. Alsaedi, S. K. Ntouyas, D. Garout, and B. Ahmad, “Coupled fractional-order systems with nonlocal coupled integral and discrete boundary conditions,” Bull. Malays. Math. Sci. Soc., vol. 42, no. 2, pp. 241–266, 2017, https://doi.org/10.1007/s40840-017-0480-1.Search in Google Scholar

[37] R. P. Agarwal, B. Ahmad, D. Garout, and A. Alsaedi, “Existence results for coupled nonlinear fractional differential equations equipped with nonlocal coupled flux and multi-point boundary conditions,” Chaos. Solitons. Fractals., vol. 102, pp. 1–13, 2017, https://doi.org/10.1016/j.chaos.2017.03.025.Search in Google Scholar

[38] M. Subramanian, A. R. V. Kumar, and T. N. Gopal, “Influence of coupled nonlocal slit-strip conditions involving Caputo derivative in fractional boundary value problem,” Discontin. Nonlinearity Complex., vol. 8, no. 4, pp. 429–445, 2019, https://doi.org/10.5890/dnc.2019.06.006.Search in Google Scholar

[39] B. Ahmad, J. J. Nieto, A. Alsaedi, and M. H. Aqlan, “A coupled system of Caputo-type sequential fractional differential equations with coupled (periodic/anti-periodic type) boundary conditions,” Mediterr. J. Math., vol. 14, no. 227, pp. 1–15, 2017, https://doi.org/10.1007/s00009-017-1027-2.Search in Google Scholar

[40] M. Subramanian, A. R. V. Kumar, and T. N. Gopal, “A strategic view on the consequences of classical integral sub-strips and coupled nonlocal multi-point boundary conditions on a combined Caputo fractional differential equation,” Proc. Jangjeon Math. Soc., vol. 22, no. 3, pp. 437–453, 2019.Search in Google Scholar

[41] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Boston, Elsevier, 2006.Search in Google Scholar

[42] I. Podlubny, Fractional Differential Equations, San Diego-Boston-New York-London-Tokyo-Toronto, Academic Press, 1999.Search in Google Scholar

[43] H. Kober, “On fractional integrals and derivatives,” Q J Math., vol. 11, no. 1, pp. 193–211, 1940 https://doi.org/10.1093/qmath/os-11.1.193.Search in Google Scholar

[44] Y. Zhou, J. Wang, and L. Zhang, Basic Theory of Fractional Differential Equations, Singapore, World Scientific, 2016.10.1142/10238Search in Google Scholar

[45] A. Granas and J. Dugundji, Fixed Point Theory, New York, Springer, 2003.10.1007/978-0-387-21593-8Search in Google Scholar

Received: 2019-12-10
Accepted: 2020-07-18
Published Online: 2020-09-23
Published in Print: 2021-08-26

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 27.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2019-0299/html
Scroll to top button