Existence of positive solutions to a coupled system with three­point boundary conditions via degree theory

PDF
  • S. Samina,
  • Kamal Shah,
  • Rahmat Ali Khan,

Published in Issue 2025-11-09

How to Cite

Existence of positive solutions to a coupled system with three­point boundary conditions via degree theory. (2025). Communications in Nonlinear Analysis, 3(1). https://oiccpress.com/cna/article/view/17895

PDF views: 288

Abstract

In this paper, we study the existence of solutions of nonlinear fractional hybrid differential equations. Byusing the topological degree theory, some results on the existence of solutions are obtained. The results aredemonstrated by a proper example.

Keywords

  • Coupled system,
  • fractional derivative,
  • green function,
  • growth condition,
  • topological degree theory,
  • fixed-point theorem
PDF

References

  1. [1] R. P. Agarwal, M. Belmekki and M. Benchohra, A survey on semilinear dierential equations and inclusionsinvolving Riemann-Liouville fractional derivative, Adv. Difference Equ., 2009 (2009), 47 pages.
  2. [2] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations withthree-point boundary conditions, Comput. Math. Appl., 58 (2009), 1838-1843.
  3. [3] B. Ahmad, J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractionaldifferential equations via Leray-Schauder degree theory, Topol. Methods Nonlinear Anal., 35 (2010), 295-304.
  4. [4] T. S. Aleroev, The Sturm-Liouville problem for a second-order ordinary differential equation with fractionalderivatives in the lower terms, Differentsial'nye Uravneniya, 18 (1982), 341-342.
  5. [5] Z. B. Bai, H. S. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation,J. Math. Anal. Appl., 311 (2005), 495-505.
  6. [6] D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer, NY, USA, (2012).[7] M. Benchohra, J. R. Graef and S. Hamani, Existence results for boundary value problems with nonlinear fractionaldifferential equations, Appl. Anal., 87 (2008), 851-863.
  7. [8] L. Cai, J. Wu, Analysis of an HIV/AIDS treatment model with a nonlinear incidence rate, Chaos Solitons Fractals,41 (2009), 175-182.
  8. [9] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, (1985).
  9. [10] K. Diethelm, The Analysis of Fractional Differential Equations, in Lecture Notes in Mathematics, Springer-Verlag,(2010).
  10. [11] V. Gaychuk, B. Datsko, V. Meleshko and D. Blackmore , Analysis of the solutions of coupled nonlinear fractionalreaction-diffusion equations, Chaos Solitons Fractals, 41 (2009), 1095-1104.
  11. [12] F. Isaia, On a nonlinear integral equation without compactness, Acta Math. Univ. Comenian., 75 (2006),233-240.
  12. [13] R. A. Khan, K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary valueproblems, Commun. Appl. Anal., 19 (2015), 515-526.
  13. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,Elsevier Science B.V., Amsterdam, (2006).
  14. [15] V. Lakshmikantham, S. Leela and J. Vasundhara Devi, Theory of Fractional Dynamic Systems, CambridgeScientic Publishers, (2009).
  15. [16] L. Lv, J. Wang and W. Wei, Existence and uniqueness results for fractional differential equations with boundaryvalue conditions, Opuscul. Math., 31 (2011), 629-643.
  16. [17] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley andSons, New York, USA, (1993).
  17. [18] M. W. Michalski, Derivatives of Noninteger Order and their Applications: Master dissertation Polish Acad. Sci.,(1993).
  18. [19] I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).
  19. [20] M. Rehman, R. A. Khan, Existence and uniqueness of solutions for multipoint boundary value problems forfractional differential equations, Appl. Math. Lett., 23 (2010), 1038-1044.
  20. [21] K. Shah, A. Ali and R. A. Khan, Degree theory and existence of positive solutions to coupled systems of multi-pointboundary value problems, Bound. Value Probl., 2016 (2016), 12 pages.
  21. [22] X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math.Lett., 22 (2009), 64-69.
  22. [23] K. Shah, R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractionalorder differential equations with anti-periodic boundary conditions, Dier. Equ. Appl. 7 (2015), 245-262.
  23. [24] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields andMedia, Springer, (2011).
  24. [25] J. Wang, Y. Zhou and W. Wei, Study in fractional differential equations by means of topological degree methods,Numer. Funct. Anal. Optim., 33 (2012), 216-238.
  25. [26] J. Wang, H. Xiang and Z. Liu , Positive solution to non zero boundary values problem for a coupled system ofnonlinear fractional differential equations, Int. J. Diff. Equ., 2010 (2010), 12 pages.
  26. [27] A. Yang, W. Ge , Positive solutions of multi-point boundary value problems of nonlinear fractional differentialequation at resonance, J. Korea Sco. Math.,16 (2009), 181-193.
  27. [28] W. Yang , Positive solution to non zero boundary values problem for a coupled system of nonlinear fractionaldifferential equations, Comput. Math. Appl., 63 (2012), 288-297.
  28. [29] W. Yang, Positive solution to non zero boundary values problem for a coupled system of nonlinear fractionaldifferential equations, Comput. Math. Appl., 63 (2012), 288-297.
  29. [30] S. Zhang, Positive solutions to singular boundary value problem for nonlinear fractional differential equation,Comput. Math. Appl., 59 (2010), 1300-1309.
  30. [31] S. Q. Zhang, Existence of solution for a boundary value problem of fractional order, Acta Math. Sci., 26B (2006),220-228.
  31. [32] S. Q. Zhang, Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron.J. Differential Equations, 36 (2006), 1-12.
  32. [33] E. Zeidler, Nonlinear Functional Analysis an Its Applications, Springer, NewYork (1986).