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On a Class of Nonlinear Singular Riemann–Liouville Fractional Differential Equations

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Abstract

By using the Guo–Krasnosel’skii fixed point theorem and some height functions defined on special bounded sets, we investigate the existence and multiplicity of positive solutions for a class of nonlinear singular Riemann–Liouville fractional differential equations with sign-changing nonlinearities, subject to Riemann–Stieltjes boundary conditions which contain fractional derivatives.

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References

  1. Ahmad, B., Luca, R.: Existence of solutions for a sequential fractional integro-differential system with coupled integral boundary conditions. Chaos Solitons Fractals 104, 378–388 (2017)

    Article  MathSciNet  Google Scholar 

  2. Ahmad, B., Ntouyas, S.K.: Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Appl. Math. Comput. 266, 615–622 (2015)

    MathSciNet  Google Scholar 

  3. Aljoudi, S., Ahmad, B., Nieto, J.J., Alsaedi, A.: A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions. Chaos Solitons Fractals 91, 39–46 (2016)

    Article  MathSciNet  Google Scholar 

  4. Alsaedi, A., Ntouyas, S.K., Agarwal, R.P., Ahmad, B.: On Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Adv. Differ. Equ. 2015(33), 1–12 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Arafa, A.A.M., Rida, S.Z., Khalil, M.: Fractional modeling dynamics of HIV and CD4\(^+\) T-cells during primary infection. Nonlinear Biomed. Phys. 6(1), 1–7 (2012)

    Article  Google Scholar 

  6. Caballero, J., Cabrera, I., Sadarangani, K.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. Abstr. Appl. Anal. 2012, 1–11 (2012). Article ID 303545

    Article  MathSciNet  Google Scholar 

  7. Cole, K.: Electric conductance of biological systems. In: Proceedings of Cold Spring Harbor Symposia on Quantitative Biology, pp. 107–116. Col Springer Harbor Laboratory Press, New York (1993)

    Article  Google Scholar 

  8. Das, S.: Functional Fractional Calculus for System Identification and Controls. Springer, New York (2008)

    MATH  Google Scholar 

  9. Ding, Y., Ye, H.: A fractional-order differential equation model of HIV infection of CD4\(^+\) T-cells. Math. Comput. Model. 50, 386–392 (2009)

    Article  MathSciNet  Google Scholar 

  10. Djordjevic, V., Jaric, J., Fabry, B., Fredberg, J., Stamenovic, D.: Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31, 692–699 (2003)

    Article  Google Scholar 

  11. Ge, Z.M., Ou, C.Y.: Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals 35, 705–717 (2008)

    Article  Google Scholar 

  12. Graef, J.R., Kong, L., Kong, Q., Wang, M.: Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions. Fract. Calc. Appl. Anal. 15(3), 509–528 (2012)

    Article  MathSciNet  Google Scholar 

  13. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)

    MATH  Google Scholar 

  14. Guo, L., Liu, L., Wu, Y.: Iterative unique positive solutions for singular p-Laplacian fractional differential equation system with several parameters. Nonlinear Anal. Model. Control 23(2), 182–203 (2018)

    Article  MathSciNet  Google Scholar 

  15. Henderson, J., Luca, R.: Boundary Value Problems for Systems of Differential, Difference and Fractional Equations. Positive solutions. Elsevier, Amsterdam (2016)

    MATH  Google Scholar 

  16. Henderson, J., Luca, R.: Existence of positive solutions for a singular fractional boundary value problem. Nonlinear Anal. Model. Control 22(1), 99–114 (2017)

    Article  MathSciNet  Google Scholar 

  17. Henderson, J., Luca, R.: Systems of Riemann–Liouville fractional equations with multi-point boundary conditions. Appl. Math. Comput. 309, 303–323 (2017)

    MathSciNet  Google Scholar 

  18. Henderson, J., Luca, R., Tudorache, A.: On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 18(2), 361–386 (2015)

    Article  MathSciNet  Google Scholar 

  19. Henderson, J., Luca, R., Tudorache, A.: Existence and nonexistence of positive solutions for coupled Riemann–Liouville fractional boundary value problems. Discrete Dyn. Nat. Soc. 2016, 1–12 (2016). Article ID 2823971

    Article  Google Scholar 

  20. 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 

  21. Klafter, J., Lim, S.C., Metzler, R. (eds.): Fractional Dynamics in Physics. World Scientific, Singapore (2011)

    Google Scholar 

  22. Liu, L., Li, H., Liu, C., Wu, Y.: Existence and uniqueness of positive solutions for singular fractional differential systems with coupled integral boundary value problems. J. Nonlinear Sci. Appl. 10, 243–262 (2017)

    Article  MathSciNet  Google Scholar 

  23. Liu, S., Liu, J., Dai, Q., Li, H.: Uniqueness results for nonlinear fractional differential equations with infinite-point integral boundary conditions. J. Nonlinear Sci. Appl. 10, 1281–1288 (2017)

    Article  MathSciNet  Google Scholar 

  24. Luca, R.: Positive solutions for a system of Riemann–Liouville fractional differential equations with multi-point fractional boundary conditions. Bound. Value Prob. 2017(102), 1–35 (2017)

    MathSciNet  Google Scholar 

  25. Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  Google Scholar 

  26. Povstenko, Y.Z.: Fractional Thermoelasticity. Springer, New York (2015)

    Book  Google Scholar 

  27. Pu, R., Zhang, X., Cui, Y., Li, P., Wang, W.: Positive solutions for singular semipositone fractional differential equation subject to multipoint boundary conditions. J. Funct. Spaces 2017, 1–7 (2017). Article ID 5892616

    Article  MathSciNet  Google Scholar 

  28. Sabatier, J., Agrawal, O.P., Machado, J.A.T. (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  30. Shen, C., Zhou, H., Yang, L.: Positive solution of a system of integral equations with applications to boundary value problems of differential equations. Adv. Differ. Equ. 2016(260), 1–26 (2016)

    MathSciNet  Google Scholar 

  31. Xu, J., Wei, Z.: Positive solutions for a class of fractional boundary value problems. Nonlinear Anal. Model. Control 21, 1–17 (2016)

    MathSciNet  Google Scholar 

  32. Zhang, X.: Positive solutions for a class of singular fractional differential equation with infinite-point boundary conditions. Appl. Math. Lett. 39, 22–27 (2015)

    Article  MathSciNet  Google Scholar 

  33. Zhang, X., Zhong, Q.: Triple positive solutions for nonlocal fractional differential equations with singularities both on time and space variables. Appl. Math. Lett. 80, 12–19 (2018)

    Article  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Gh. Asachi Technical University, 11 Blvd. Carol I, 700506, Iasi, Romania

    Rodica Luca

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  1. Rodica Luca
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Luca, R. On a Class of Nonlinear Singular Riemann–Liouville Fractional Differential Equations. Results Math 73, 125 (2018). https://doi.org/10.1007/s00025-018-0887-5

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  • DOI: https://doi.org/10.1007/s00025-018-0887-5

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