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Fractional Boundary Value Problems with Integral and Anti-periodic Boundary Conditions

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Abstract

In this paper, we consider a class of boundary value problems of fractional differential equations with integral and anti-periodic boundary conditions, which is a new type of mixed boundary condition. Using the contraction mapping principle, Krasnosel’skii fixed point theorem, and Leray-Schauder degree theory, we obtain some results of existence and uniqueness. Finally, several examples are provided for illustrating the applications of our theoretical analysis.

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References

  1. Bhrawy, A.H., Alghamdi, M.A., Tharwat, M.M.: A new operational matrix of fractional integration for shifted Jacobi polynomials. Bull. Malays. Math. Sci. Soc. 37(4), 983–995 (2014)

  2. Bai, Z., Qiu, T.: Existence of positive solution for singular fractional equations. Appl. Math. Comput. 215, 2761–2767 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, J., Tang, X.H.: Infinitely many solutions for a class of fractional boundary value problem. Bull. Malays. Math. Sci. Soc. 36(4), 1083–1097 (2013)

  4. Hu, Z., Liu, W., Chen, T.: Two-point boundary value problems for fractional differential equations at resonance. Bull. Malays. Math. Sci. Soc. 36(2), 747–755 (2013)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  6. Loghmani, G., Javanmardi, S.: Numerical methods for sequential fractional differential equations for Caputo operator. Bull. Malays. Math. Sci. Soc. 35(2), 315–323 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Liu, Z., Liang, J.: Multiple solutions of nonlinear boundary value problems for fractional differential equations, Bull. Malays. Math. Sci. Soc. 37(1), 239–248 (2014)

  8. Momani, S., Odibat, Z.: Numerical approach to differential equations of fractional order. J. Comput. Appl. Math. 207, 96–110 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Odibat, Z.: A note on phase synchronization in coupled chaotic fractional order systems. Nonlinear Anal. Real World Appl. 13(2), 779–789 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. Song, Y.: Existence of positive solutions for a three-point boundary value problem with fractional q-differences. Bull. Malays. Math. Sci. Soc. 37(4), 955–964 (2014)

  12. Meral, F., Royston, T., Magin, R.: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15(4), 939–945 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Oldham, K.: Ractional differential equations in electrochemistry. Adv. Eng. Softw. 41(1), 9–12 (2010)

    Article  MATH  Google Scholar 

  14. Lee, C., Chang, F.: Fractional-order PID controller optimization via improved electromagnetism-like algorithm. Expert Syst. Appl. 37(12), 8871–8878 (2010)

    Article  Google Scholar 

  15. Ahmed, E., El-Sayed, A., El-Saka, H.: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325(1), 542–553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, F., Burrage, K.: Novel techniques in parameter estimation for fractional dynamical models arising from biological systems. Comput. Math. Appl. 62(3), 822–833 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mophou, G.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61(1), 68–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, J., Zhou, Y., Wei, W.: Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst. Control Lett. 61(4), 472–476 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gorenflo, R., Mainardi, F.: Some recent advances in theory and simulation of fractional diffusion processes. J. Comput. Appl. Math. 229(2), 400–415 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jiang, X., Xu, M., Qi, H.: The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes. Nonlinear Anal. Real World Appl. 11(1), 262–269 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sokolov, I., Chechkin, A., Klafter, J.: Fractional diffusion equation for a power-law-truncated Lévy process. Physica A. 336(3-4), 245–251 (2004)

    Article  Google Scholar 

  22. Nigmatullin, R., Omay, T., Baleanu, D.: On fractional filtering versus conventional filtering in economics. Commun. Nonlinear Sci. Numer. Simul. 15(4), 979–986 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Agrawal, O.P.: Generalized Variational Problems and Euler-Lagrange equations. Comput. Math. Appl. 59(5), 1852–1864 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389(1), 403–411 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lakoud, A., Khaldi, R.: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 75(4), 2692–2700 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Salem, H.: Fractional order boundary value problem with integral boundary conditions involving Pettis integral. Acta Math. Sci. 31(2), 661–672 (2011)

    Article  MathSciNet  Google Scholar 

  27. Xu, Y., He, Z.: Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition. J. Appl. Math. Comput. 44(1–2), 417–435 (2014)

  28. Ahmad, B., Nieto, J.: Anti-periodic fractional boundary value problems. Comput. Math. Appl. 62(3), 1150–1156 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Agrawal, R.P., Ahmad, B.: Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 62(3), 1200–1214 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen, T., Liu, W.: An anti-periodic boundary value problem for the fractional differential equation with a \(p\)-Laplacian operator. Appl. Math. Lett. 25(11), 1671–1675 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, G., Ahmad, B., Zhang, L.: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 74(3), 792–804 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  33. Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980)

    MATH  Google Scholar 

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Acknowledgments

The author is grateful to the Editor and Referees for their valuable suggestions, which significantly improved the quality of the paper. The author also want to thank Prof. Om P. Agrawal for his helpful discussions. This work was partly supported by the ‘2\(+\)6 Program’ of Central South University and finished during the author was a Postdoc research fellow in School of Mathematics and Statistics.

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Correspondence to Yufeng Xu.

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Communicated by Norhashidah M. Ali.

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Xu, Y. Fractional Boundary Value Problems with Integral and Anti-periodic Boundary Conditions. Bull. Malays. Math. Sci. Soc. 39, 571–587 (2016). https://doi.org/10.1007/s40840-015-0126-0

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  • DOI: https://doi.org/10.1007/s40840-015-0126-0

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