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On a Multipoint Fractional Boundary Value Problem in a Fractional Sobolev Space

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Abstract

In this paper, we study the existence of positive solutions in a Sobolev space for a Reimann Liouville fractional boundary value problem. The main tools are the lower and upper solutions method and Schauder fixed point theorem. A numerical example is given to illustrate the obtained results.

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References

  1. Agarwal, R.P., Benchohra, M., Hamani, S., Pinelas, S.: Upper and lower solutions method for impulsive differential equations involving the Caputo fractional derivative. Mem. Differ. Equ. Math. Phys. 53, 1–12 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Arshad, S., Lupulescu, V., O’Regan, D.: Lp-solutions of fractional integral equations. Fract. Calc. Appl. Anal. 17, 259–276 (2014)

    Article  MathSciNet  Google Scholar 

  3. Barton, T.A., Zhang, B.: \(L_{p}\)-solutions of fractional differential equations. Nonlinear Stud. 19(2), 161–177 (2012)

    MathSciNet  Google Scholar 

  4. Benchohra, M., Hamani, S., Ntouyas, Sotiris, K.: Boundary value problems for differential equations with fractional order. Surv. Math. Appl. 3, 1–12 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Bergounioux, M., Leaci, A., Nardi, G., Tomarelli, F.: Fractional Sobolev spaces and functions of bounded variation of one variable. Fract. Calc. Appl. Anal. 20(4), 936–962 (2017)

    Article  MathSciNet  Google Scholar 

  6. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York Dordrecht Heidelberg London (2010)

    Google Scholar 

  7. Burton, T.A., Zhang, B.: Lp-solutions of fractional differential equations. Nonlinear Stud. 19(2), 161–177 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Boucenna, D., Guezane-Lakoud, A., Nieto, Juan J., Khaldi, R.: On a multipoint fractional boundary value problem with integral conditions. 2017, Article ID 53, 1–13 (2017)

  9. Delbosco, D., Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation. J. Math. Anal. Appl. 204, 609–625 (1996)

    Article  MathSciNet  Google Scholar 

  10. Diethelm, K.: The Analysis of Fractional Differential Equations. An application-oriented exposition using differential operators of Caputo type, Springer, Heidelberg (2010)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Guezane-Lakoud, A., Khaldi, R., Torres, D.F.M.: On a fractional oscillator equation with natural boundary conditions. Prog. Fract. Differ. Appl. 3(3), 191–197 (2017)

    Article  Google Scholar 

  13. Guezane-Lakoud, A., Khaldi, R.: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 75, 2692–2700 (2012)

    Article  MathSciNet  Google Scholar 

  14. 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  Google Scholar 

  15. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  Google Scholar 

  16. Jiang, D., Yang, Y., Chu, J.J., O’Regan, D.: The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order. Nonlinear Anal. Theory Methods Appl. 67, 2815–2828 (2007)

    Article  MathSciNet  Google Scholar 

  17. Karoui, A., BenAouicha, H., Jawahdou, A.: Existence and numerical solutions of nonlinear quadratic integral equations defined on unbounded intervals. Numer. Funct. Anal. Opt. 31(6), 691–714 (2010)

    Article  MathSciNet  Google Scholar 

  18. Karoui, A., Jawahdou, A.: Existence and approximate \(L_{p}\) and continuous solutions of nonlinear integral equations of the Hammerstein and Volterra types. Appl. Math. Comput. 216(7), 2077–2091 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Khaldi, R., Guezane-Lakoud, A.: Upper and lower solutions method for higher order boundary value problems. Progr. Fract. Differ. Appl. 3(1), 53–57 (2017)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  21. Nieto, Juan J., Rodrıguez-Lopez, R.: Monotone method for first-order functional differential equations. Comput. Math. Appl. 52, 471–484 (2006)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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Acknowledgements

The authors are very grateful to the anonymous referees for their valuable comments and suggestions that improved this paper.

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Correspondence to A. Guezane-Lakoud.

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Guezane-Lakoud, A., Khaldi, R., Boucenna, D. et al. On a Multipoint Fractional Boundary Value Problem in a Fractional Sobolev Space. Differ Equ Dyn Syst 30, 659–673 (2022). https://doi.org/10.1007/s12591-018-0431-9

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