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Monotone Iterative Technique for Riemann–Liouville Fractional Integro-Differential Equations with Advanced Arguments

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Abstract

In this paper, we consider existence and uniqueness of solutions for nonlinear boundary value problems involving Riemann–Liouville fractional integro-differential equations with advanced arguments. By establishing a new comparison theorem and applying the monotone iterative technique, we show the existence of extremal solutions.

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References

  1. Ahmad B., Nieto J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58(9), 1838–1843 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Belmekki, M., Nieto J.J., Rodriguez-Lopez, R.: Existence of periodic solutions for a nonlinear fractional differential equation. Boundary Value Probl. (2009) (Art. ID. 324561)

  3. Glockle W.G., Nonnenmacher T.F.: A fractional calculus approach of self-similar protein dynamics. Biophys. J. 68, 46–53 (1995)

    Article  Google Scholar 

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

    Book  MATH  Google Scholar 

  5. Jankowski T.: Advanced differential equations with nonlinear boundary conditions. J. Math. Anal. Appl. 304, 490–503 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jankowski T.: First-order impulsive ordinary differential equations with advanced arguments. J. Math. Anal. Appl. 331, 1–12 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kilbsa A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  8. Kosmatov N.: Integral equations and initial value problems for nonlinear differential equations of fractional order. Nonlinear Anal. 70(7), 2521–2529 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lakshmikantham V., Vatsala A.S.: Basic theory of fractional differential equations. Nonlinear Anal. TMA 69(8), 2677–2682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ladde G.S., Lakshmikantham V., Vatsala A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman pub. Co., Boston (1985)

    MATH  Google Scholar 

  11. Lakshmikanthan V., Vatsala A.S.: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21, 828–834 (2008)

    Article  MathSciNet  Google Scholar 

  12. Lakshmikanthan V., Leela S., Vasundhara J.: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge (2009)

    Google Scholar 

  13. Liu, Z.H., Sun, J.H.: Nonlinear boundary value problems of fractional functional integro-differential equations. Comput. Math. Appl. doi:10.1016/j.camwa.2012.02.026

  14. McRae, F.A.: Monotone iterative technique and existence results for fractional differential equations. Nonlinear Anal. (2009). doi:10.1016/j.na.2009.05.074

  15. Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  16. Wang G.T.: Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments. J. Comput. Appl. Math. 236, 2425–2430 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang G.T., Agarwal R.P., Cabada A.: Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 25, 1019–1024 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wei Z.L., Dong W., Che J.L.: Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative. Nonlinear Anal. 73, 3232–3238 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wei Z.L., Li Q.D., Che J.L.: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 367(1), 260–272 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang, S.Q., Su, X.W.: The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order, Comput. Math. Appl. (2011). doi:10.1016/j.camwa.2011.03.008

  21. Zhang S.Q.: Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Anal. 71, 2087–2093 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang S.Q.: Existence of a solution for the fractional differential equation with nonlinear boundary conditions. Comput. Math. Appl. 61, 1202–1208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang S.Q.: Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives. Nonlinear Anal. 71, 2087–2093 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhou Y.: Existence and uniqueness of fractional functional differential equations with unbounded delay. Int. J. Dyn. Syst. Differ. Equ. 1(4), 239–244 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Zhou Y.: Existence and uniqueness of solutions for a system of fractional differential equations. J. Frac. Calc. Appl. Anal. 12, 195–204 (2009)

    MATH  Google Scholar 

  26. Zhou Y., Jiao F.: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal. 11, 4465–4475 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Sciences, Guangxi University for Nationalities, Nanning, 530006, Guangxi, People’s Republic of China

    Zhenhai Liu & Jihua Sun

  2. Departamento de Matemática, Universidad Técnica FedericoSanta María, Casilla 110-V, Valparaíso, Chile

    Iván Szántó

Authors
  1. Zhenhai Liu
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  2. Jihua Sun
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  3. Iván Szántó
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Correspondence to Zhenhai Liu.

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Project supported by NNSF of China Grant No.10971019.

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Liu, Z., Sun, J. & Szántó, I. Monotone Iterative Technique for Riemann–Liouville Fractional Integro-Differential Equations with Advanced Arguments. Results. Math. 63, 1277–1287 (2013). https://doi.org/10.1007/s00025-012-0268-4

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  • DOI: https://doi.org/10.1007/s00025-012-0268-4

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