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Differential Equations and Dynamical Systems

, Volume 27, Issue 4, pp 539–552 | Cite as

Existence of Positive Solutions for System of p-Laplacian Fractional Order Boundary Value Problems

  • Abdullah Y. Al-HossainEmail author
Original Research

Abstract

The purpose of this paper is to establish some results on the existence of positive solutions for a system of p-Laplacian fractional order boundary value problem. The main tool is a fixed point theorem of the cone expansion and compression of functional type due to Avery, Henderson and O’Regan. Some examples are also presented to illustrate the availability of the main results.

Keywords

Fractional differential equation p-Laplacian operator Positive solution Green’s functions Cone 

Mathematics Subject Classifications

34A34 34B10 34B15 34B18 34B27 

Notes

Acknowledgments

The author thank the referees for their valuable suggestions and comments.

References

  1. 1.
    Avery, R.I., Henderson, J., O’Regan, D.: Functional compression expansion fixed point theorem. Electron. J. Differ. Equ. 2008(22), 1–12 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anderson, D.R., Avery, R.I.: Fixed point theorem of cone expansion and compression of functional type. J. Differ. Equ. Appl. 8, 1073–1083 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Positive Solutions of Differential. Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  4. 4.
    Al-Hossain, A.Y.: Eigenvalues for iterative systems of nonlinear Caputo fractional order three point boundary value problems. J. Appl. Math. Comput. (2015). doi: 10.1007/s12190-015-0935-1 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chai, G.: Positive solutions for boundary value problem of fractional differential equation with \(p\)-Laplacian operator. Bound. Value Probl. 2012, 20 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Delbosco, D.: Fractional calculus and function spaces. J. Fract. Calc. 6, 45–53 (1994)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Deimling, K.: Nonlinear Functional Analysis. Springer, New York (1985)CrossRefGoogle Scholar
  8. 8.
    Han, Z., Lu, H., Sun, S., Yang, D.: Positive solutions to boundary value problems of \(p\)-Laplacian fractional differential equations with a parameter in the boundary conditions. Electron. J. Differ. Equ. 2012(213), 1–14 (2012)Google Scholar
  9. 9.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B. V., Amsterdam (2006)Google Scholar
  10. 10.
    Krasnosel’skii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964)zbMATHGoogle Scholar
  11. 11.
    Krasnosel’skii, M.A., Zabreiko, P.P.: Geometrical Methods of Nonlinear Analysis. Springer, New York (1984)CrossRefGoogle Scholar
  12. 12.
    Lakshmikanthan, V., Devi, J.: Theory of fractional differential equations in a Banach space. Eur. J. Pure Appl. Math. 1(14), 38–45 (2008)MathSciNetGoogle Scholar
  13. 13.
    Lakshmikanthan, V.: Theory of fractional differential equations. Nonlinear Anal. TMA 69, 3337–3343 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lu, H., Han, Z., Sun, S.: Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with \(p\)-Laplacian. Bound. Value Probl. 2014, 26 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  16. 16.
    Nageswararao, S.: Existence of positive solutions for Riemann-Liouville fractional order three-point boundary value problem. Asian-Eur. J. Math. 8(4), 15 (2015). doi: 10.1142/S1793557115500576 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nageswararao, S.: Multiple positive solutions for a system of Riemann-Liouville fractional order two-point boundary value problems. Panamer. Math. J. 25(1), 66–81 (2015)MathSciNetGoogle Scholar
  18. 18.
    Nageswararao, S.: Existence and multiplicity for a system of fractional higher-order two-point boundary value problem. J. Appl. Math. Comput. 51, 93–107 (2016). doi: 10.1007/s12190-015-0893-7 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nageswararao, S.: Solvability of second order delta-nabla \(p\)-Laplacian \(m\)-point eigenvalue problem on time scales. TWMS J. App. Eng. Math. 5(1), 98–109 (2015)MathSciNetGoogle Scholar
  20. 20.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  21. 21.
    Sun, J., Zhang, G.: A generalization of the cone expansion and compression fixed point theorem and applications. Nonlinear Anal. 67, 579–586 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sun, Y., Zhang, X.: Existence and nonexistence of positive solutions for fractional order two point boundary value problems. Adv. Difference Equ. 53, 1–11 (2014)MathSciNetGoogle Scholar
  23. 23.
    Yang, C., Yan, J.: Positive solutions for third-order Sturm-Liouville boundary value problems with \(p\)-Laplacian. Comput. Math. Appl. 59(6), 2059–2066 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhao, Y., Sun, S., Han, Z., Zhang, M.: Positive solutions for boundary value problems of nonlinear fractional differential equations. Appl. Math. Comput. 217(16), 6950–6958 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceJazan UniversityJazanKingdom of Saudi Arabia

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