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Positive Solutions for Higher-Order Nonlinear Fractional Differential Equations

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Abstract

This paper investigates the existence of positive solutions for a class of higher-order nonlinear fractional differential equations with initial conditions given for ordinary as well as fractional derivatives of the unknown function. We assume that the nonlinear term f involves also derivatives of fractional order. The results are established by converting the problem into an equivalent integral equation and applying Guo–Krasnoselskii’s fixed-point theorem in cones.

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References

  1. Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed point theory and applications. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  2. Bai, Z.: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. TMA 72, 916–924 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Voß, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II: Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, pp. 217–224. Springer, Berlin (1999)

    Chapter  Google Scholar 

  4. Goodrich, C.S.: Existence of a positive solution to a class of fractional differential equations. Appl. Math. Lett. 23, 1050–1055 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Guezane-Lakoud, A., Bensebaa, S.: Solvability of a fractional boundary value problem with fractional derivative condition. Arab. J. Math. 3, 39–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guo, D.J., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics, Science and Engineering, vol. 5. Academic Press, Inc, San Diego (1988)

    Google Scholar 

  7. 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 

  8. Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. CISM Courses and Lectures, vol. 378, pp. 291–348. Springer, Vienna (1997)

    Chapter  Google Scholar 

  9. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  10. Ntouyas, S.K., Wang, G., Zhang, L.: Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments. Opusc. Math. 31, 433–442 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Podlubny, I.: Fractional Differential Equations Mathematics in Science and Engineering, vol. 198. Academic Press, Inc, San Diego (1999)

    Google Scholar 

  12. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon (1993)

    MATH  Google Scholar 

  13. Zhang, X., Liu, L., Wu, Y.: The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives. Appl. Math. Comput. 218, 8526–8536 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Zhang, X., Liu, L., Wu, Y.: Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives. Appl. Math. Comput. 219, 1420–1433 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a singular fractional differential system involving derivatives. Commun. Nonlinear Sci. Numer. Simul. 18, 1400–1409 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang, X., Liu, L., Wu, Y.: The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium. Appl. Math. Lett. 37, 26–33 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, X., Liu, L., Wu, Y., Lu, Y.: The iterative solutions of nonlinear fractional differential equations. Appl. Math. Comput. 219, 4680–4691 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Zhang, X., Liu, L., Wu, Y., Wiwatanapataphee, B.: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252–263 (2015)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to express our gratitude to the anonymous referees and the editor for their valuable comments and suggestions.

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Authors and Affiliations

  1. Laboratory of Applied Mathematics and Modeling, University of 8 May 1945 Guelma, P.O. Box 401, 24000, Guelma, Algeria

    Abdourazek Souahi & Amara Hitta

  2. Laboratory of Advanced Materials, University Badji Mokhtar-Annaba, P.O. Box 12, 23000, Annaba, Algeria

    Assia Guezane-Lakoud

Authors
  1. Abdourazek Souahi
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  2. Assia Guezane-Lakoud
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  3. Amara Hitta
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Correspondence to Abdourazek Souahi.

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Souahi, A., Guezane-Lakoud, A. & Hitta, A. Positive Solutions for Higher-Order Nonlinear Fractional Differential Equations. Vietnam J. Math. 45, 441–450 (2017). https://doi.org/10.1007/s10013-016-0218-7

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  • DOI: https://doi.org/10.1007/s10013-016-0218-7

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