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Existence of solutions for a nonlinear fractional p-Laplacian boundary value problem

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Abstract

The main objective of this paper is to prove the existence of solutions for a fractional p-Laplacian boundary value problem containing both left Riemann–Liouville and right Caputo fractional derivatives. The proofs are based on the upper and lower solutions method and Schauder’s fixed point theorem. The paper is ended by a numerical example.

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References

  1. Ahmad, B.: Nonlinear fractional differential equations with anti-periodic type fractional boundary conditions. Differ. Equ. Dyn. Syst. 21, 387–401 (2013)

    Article  MathSciNet  Google Scholar 

  2. Ahmad, B., Nieto, J.J.: The monotone iterative technique for three-point second-order integrodifferential boundary value problems with p-Laplacian. Bound. Value Problems 2007, Article ID 57481 (2007)

  3. Chai, G.: Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator, Bound. Value Problems 2012, Article ID 18 (2012)

  4. Chaudhary, R.: Monotone iterative technique for Sobolev type fractional integro-differential equations with fractional nonlocal conditions. Rendiconti del Circolo Matematico di Palermo (in press)

  5. Ding, X.-L., Nieto, J.J.: Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms. Fract. Calc. Appl. Anal. 21(2), 312–335 (2018)

    Article  MathSciNet  Google Scholar 

  6. Fazli, H., Nieto, J.J.: An investigation of fractional Bagley–Torvik equation. Open Math. 17(1), 499–512 (2019)

    Article  MathSciNet  Google Scholar 

  7. Ferrara, M., Bisci, G.Molica, Zhang, B.: Existence of weak solutions for non-local fractional problems via Morse theory. Discrete Contin. Dyn. Syst. Ser. B 19(8), 2483–2499 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Guezane-Lakoud, A., Rodríguez-López, R.: On a fractional boundary value problem in a weighted space. SeMA 75(3), 435–443 (2018)

    Article  MathSciNet  Google Scholar 

  9. Guezane-Lakoud, A., Khaldi, R., Kılıçman, A.: Existence of solutions for a mixed fractional boundary value problem. Adv. Differ. Equ. 2017, Article ID 164 (2017)

  10. Guezane-Lakoud, A., Khaldi, R., Torres, D.F.M.: On a fractional oscillator equation with natural boundary conditions. Progr. Fract. Differ. Appl. 3(3), 1–7 (2017)

    Google Scholar 

  11. Guezane-Lakoud, A., Khaldi, R., Boucenna, D., Nieto, Juan J.: On a multipoint fractional boundary value problem in a fractional Sobolev space. Differ. Equ. Dyn. Syst. (2018) (in press)

  12. Han, Z., Lu, H., Sun, S., Liu, J.: Existence on positive solutions for boundary value problems of nonlinear fractional differential equations with p-Laplacian. Adv. Differ. Equ. 2013, Article ID 30 (2013)

  13. 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. Electron. J. Differ. Equ. 2012, Article ID 213 (2012)

  14. Heidarkhani, S., Ferrara, M., Caristi, G., Salari, A.: Existence of three solutions for impulsive nonlinear fractional boundary value problems. Opusc. Math. 37(2), 281–301 (2017)

    Article  MathSciNet  Google Scholar 

  15. Kamenskii, M., Obukhovskii, V., Petrosyan, G., Yao, J.-C.: Existence and approximation of solutions to nonlocal boundary value problems for fractional differential inclusions. Fixed Point Theory Appl. https://doi.org/10.1186/s13663-018-0652-1

  16. Khaldi, R., Guezane-Lakoud, A.: Higher order fractional boundary value problems for mixed type derivatives. J. Nonlinear Funct. Anal. 2017, Article ID 30 (2017)

  17. Khaldi, R., Guezane-Lakoud, A.: Upper and lower solutions method for higher order boundary value problems. Progr. Fract. Differ. Appl. 1, 53–57 (2017)

    Article  Google Scholar 

  18. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies. Elsevier, Amsterdam (2006)

    Google Scholar 

  19. Leibenson, L.S.: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirg. SSR, Ser. Biol. Nauk 9, 7–10 (1983)

    MathSciNet  Google Scholar 

  20. Liu, X., Jia, M., Xiang, X.: On the solvability of a fractional differential equation model involving the p-Laplacian operator. Comput. Math. Appl. 64, 3267–32777 (2012)

    Article  MathSciNet  Google Scholar 

  21. Mérida, L.M., Vidal, R.E.: The obstacle problem for the infinity fractional Laplacian. Rendiconti del Circolo Matematico di Palermo 67(1), 7–15 (2018)

    Article  MathSciNet  Google Scholar 

  22. Noeiaghdam, Z., Allahviranloo, T., Nieto, Juan J.: q-Fractional differential equations with uncertainty. Soft Comput. 23(19), 9507–9524 (2019)

    Article  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equation. Academic Press, Sain Diego (1999)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  25. Shi, A., Zhang, S.: Upper and lower solutions method and a fractional differential equation boundary value problem. Electron. J. Qual. Theory Differ. Equ. 30, 1–13 (2009)

    Article  MathSciNet  Google Scholar 

  26. Wang, Y.: Existence and nonexistence of positive solutions for mixed fractional boundary value problem with parameter and p-Laplacian operator. J. Funct. Spaces 2018, Article ID 1462825 (2018)

  27. Xu, J., O’Regan, D.: Positive solutions for a fractional p-Laplacian boundary value problem. Facul. Sci. Math. Filom. 31(6), 1549–1558 (2017)

    MathSciNet  Google Scholar 

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Authors and Affiliations

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

    I. Merzoug, A. Guezane-Lakoud & R. Khaldi

  2. Laboratory of Numerical Analysis, Optimisation and Statistics, Badji Mokhtar - Annaba University, Annaba, Algeria

    I. Merzoug

Authors
  1. I. Merzoug
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  2. A. Guezane-Lakoud
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  3. R. Khaldi
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Correspondence to A. Guezane-Lakoud.

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Merzoug, I., Guezane-Lakoud, A. & Khaldi, R. Existence of solutions for a nonlinear fractional p-Laplacian boundary value problem. Rend. Circ. Mat. Palermo, II. Ser 69, 1099–1106 (2020). https://doi.org/10.1007/s12215-019-00459-4

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