Nonlinear Coupled Fractional Order Systems with Integro-Multistrip-Multipoint Boundary Conditions

Article Sidebar

PDF
Cite as:
Bashir Ahmad, Ahmed Alsaedi, Sotiris K. Ntouyas, Nonlinear Coupled Fractional Order Systems with Integro-Multistrip-Multipoint Boundary Conditions, Int. J. Anal. Appl., 17 (6) (2019), 940-957.

Main Article Content

Bashir Ahmad, Ahmed Alsaedi, Sotiris K. Ntouyas

Abstract

We study the existence and uniqueness of solutions for a nonlinear system of coupled fractional differential equations equipped with nonlocal coupled integro-multistrip-multipoint boundary conditions. Our results are new in the sense that the given boundary conditions connect the values of the known functions over the given domain with the ones described on different sub-segments and different nonlocal positions within the given domain. We make use of Banach contraction mapping principle, Leray-Schauder alternative and Krasnoselskii fixed point theorem to prove the desired results for the problem at hand. An example illustrating the existence and uniqueness result is also presented.

Article Details

References

  1. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  2. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  3. K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993
  4. J. Sabatier, O.P. Agrawal, J.A.T. Machado (Eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
  5. S. Liang, J. Zhang, Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval, Math. Comput. Modelling 54 (2011) 1334-1346.
  6. C. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl. 61 (2011), 191-202.
  7. J.R. Wang, Y. Zhou, M. Feckan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl. 64 (2012), 3008-3020.
  8. B. Ahmad, On nonlocal boundary value problems for nonlinear integro-differential equations of arbitrary fractional order, Results Math. 63 (2013), 183-194.
  9. C. Zhai, L. Xu, Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2820-2827.
  10. J. Henderson, N. Kosmatov, Eigenvalue comparison for fractional boundary value problems with the Caputo derivative, Fract. Calc. Appl. Anal. 17 (2014), 872-880.
  11. Y. Ding, Z. Wei, J. Xu, D. O'Regan, Extremal solutions for nonlinear fractional boundary value problems with p-Laplacian, J. Comput. Appl. Math. 288 (2015), 151-158.
  12. H. Wang, Existence of solutions for fractional anti-periodic BVP, Results Math. 68 (2015), 227-245.
  13. B. Ahmad, S.K. Ntouyas, Some fractional-order one-dimensional semi-linear problems under nonlocal integral boundary conditions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 110 (2016), 159-172.
  14. M. Faieghi, S. Kuntanapreeda, H. Delavari, D. Baleanu, LMI-based stabilization of a class of fractional-order chaotic systems, Nonlinear Dynam. 72 (2013), 301-309.
  15. F. Zhang, G. Chen C. Li, J. Kurths, Chaos synchronization in fractional differential systems, Phil. Trans. R. Soc. A 371 (2013), 20120155.
  16. B. Ahmad, S.K. Ntouyas, A. Alsaedi, Fractional differential equations and inclusions with nonlocal generalized RiemannLiouville integral boundary conditions, Int. J. Anal. Appl. 13 (2017), 231-247.
  17. S. Asawasamrit, N. Phuangthong, S.K. Ntouyas, J. Tariboon, Nonlinear sequential Riemann-Liouville and Caputo fractional differential equations with nonlocal and integral boundary conditions, Int. J. Anal. Appl. 17 (2019), 47-63.
  18. C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl. 62 (2011), 1251-1268.
  19. S.K. Ntouyas, M. Obaid, A coupled system of fractional differential equations with nonlocal integral boundary conditions, Adv. Difference Equ. 2012 (2012), Art. ID 130.
  20. B. Ahmad, S.K. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput. 266 (2015), 615-622.
  21. B. Ahmad, S.K. Ntouyas, A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Solitons & Fractals 83 (2016), 234-241.
  22. A. Alsaedi, S.K. Ntouyas, D. Garout, B. Ahmad, Coupled fractional-order systems with nonlocal coupled integral and discrete boundary conditions, Bull. Malays. Math. Sci. Soc. 42 (2019), 241-266.
  23. A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2005.
  24. M.A. Krasnoselskii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk 10 (1955), 123-127.