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Anti-periodic mild solutions to semilinear fractional differential equations

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Abstract

In this paper, by using the \(\alpha \)-resolvent family theory, Banach contraction mapping principle and Schauder’s fixed point theorem, we investigate the existence of anti-periodic mild solutions to the semilinear fractional differential equations \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t)),\ t\in R,1 \le \alpha \le 2 \) and \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t),u'(t)),\ t\in R,1 < \alpha < 2\), where \(A : D(A)\subset X \rightarrow X\) is the infinitesimal generator of an \(\alpha \)-resolvent family defined on a Banach space \(X\) and \(f\) is a suitable function. Furthermore, an example is given to illustrate our results.

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References

  1. Okochi, H.: On the existence of anti-periodic solutions to a nonlinear evolution equation associated with odd subdifferential operators. J. Funct. Anal. 91, 246–258 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Okochi, H.: On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains. Nonlinear Anal. 14, 771–783 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Haraux, A.: Anti-periodic solutions of some nonlinear evolution equations. Manuscr. Math. 63, 479–505 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aftabizadeh, A.R., Pavel, N.H., Huang, Y.K.: Anti-periodic oscillations of some second-order differential equations and optimal control problems. J. Comp. Appl. Math. 52, 3–21 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aizicovici, S., Pavel, N.H.: Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space. J. Funct. Anal. 99, 387–408 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Aizicovici, S., McKibben, M., Reich, S.: Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities. Nonlinear Anal. 43, 233–251 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Aizicovici, S., Reich, S.: Anti-periodic solutions to a class of non-monotone evolution equations. Discrete Continuous Dyn. Syst. 5, 35–42 (1999)

    MATH  MathSciNet  Google Scholar 

  8. Liu, Z.H.: Anti-periodic solutions to nonlinear evolution equations. J. Funct. Anal. 258, 2026–2033 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, Y.Q., Nieto, J.J., O’Regan, D.: Anti-periodic solutions for full nonlinear first-order differential equations. Math. Comput. Model. 46, 1183–1190 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen, Y.Q., Wang, X.D., Xu, H.X.: Anti-periodic solutions for semilinear evolution equations. J. Math. Anal. Appl. 273, 627–636 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, Q.: Existence of anti-periodic mild solution for semilinear evolution equation. J. Math. Anal. Appl. 377, 110–120 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cao, J., Yang, Q., Huang, Z.: Existence of anti-periodic mild solutions for a class of semilinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 277–283 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wang, Q., Fang, Y.: Existence of anti-periodic mild solutions to fractional differential equations of order \(\alpha \in (0,1)\). Ann. Diff. Equ. 29(3), 346–355 (2013)

    MATH  MathSciNet  Google Scholar 

  14. Wang, R.N., Chen, D.: Anti-periodic problems for semilinear partial neutral evolution equations. Electron. J. Qual. Theory Differ. Equ. 16, 1–16 (2013)

    MATH  Google Scholar 

  15. Chen, Y., O’Regan, D., Agarwal, R.P.: Anti-periodic solutions for semilinear evolution equations in Banach spaces. J. Appl. Math. Comput. 38, 63–70 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. N’Guérékata, G.M., Valmorin, V.: Antiperiodic solutions of semilinear integrodifferential equations in Banach spaces. Appl. Math. Comput. 218, 11118–11124 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. Mophou, G., N’Guérékata, G.M.: Existence of antiperiodic solutions to semilinear evolution equations in intermediate Banach spaces. Advances in Interdisciplinary Mathematical Research, pp. 133–139. Springer, New York (2013)

  18. Al-Islam, N.S., Alsulami, S.M., Diagana, T.: Existence of weighted pseudo anti-periodic solutions to some non-autonomous differential equations. Appl. Math. Comput. 218, 6536–6548 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mophou, G., N’Guérékata, G.M., Valmorin, V.: Asymptotic behavior of mild solutions of some fractional functional integro-differential equations. Afr. Diaspora J. Math. New Ser. 16(1), 70–81 (2013)

    MATH  Google Scholar 

  20. Araya, D., Lizama, C.: Almost automorphic mild solutions to fractional differential equations. Nonlinear Anal. 69(11), 3692–3705 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cuevas, C., Lizam, C.: Almost automorphic solutions to a class of semilinear fractional differential equations. Appl. Math. Lett. 21(12), 1315–1319 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Xia, Z.: Asymptotically periodic solutions of semilinear fractional integro-differential equations. Adv. Differ. Equ. 2014, 1–19 (2014)

    Article  Google Scholar 

  23. Agarwal, R.P., de Andrade, B., C, Cuevas: On type of periodicity and ergodicity to a class of fractional order differential equations. Adv. Differ. Equ. 2010, 1–25 (2010)

    Article  Google Scholar 

  24. Cuevas, C., Henríquez, H.R., Soto, H.: Asymptotically periodic solutions of fractional differential equations. Appl. Math. Comput. 236, 524–545 (2014)

    Article  MathSciNet  Google Scholar 

  25. Wang, J.R., Zhou, Y., Fec̆kan, M.: Alternative results and robustness for fractional evolution equations with periodic boundary conditions. Electron. J. Qual. Theory Differ. Equ. 97, 1–15 (2011)

    Article  MathSciNet  Google Scholar 

  26. Wang, J.R., Fec̆kan, M., Zhou, Y.: Nonexistence of periodic solutions and asymptotically periodic solutions for fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 18(2), 246–256 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  27. Lizama, C.: Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, 278–292 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  28. Henríquez, H., Lizama, C.: Compact almost automorphic solutions to integral equations with infinite delay. Nonlinear Anal. 71(2), 6029–6037 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Precup, R.: Methods in Nonlinear Integral Equations. Kluwer Academic, Dordrecht (2002)

    Book  MATH  Google Scholar 

  30. Martin, R.H.: Nonlinear Operators and Differential Equations in Banach Spaces. Robert E. Krieger Publ. Co., New York (1987)

    Google Scholar 

  31. Li, H., Huang, F., Li, J.: Composition of pseudo almost periodic functions and semilinear differential equations. J. Math. Anal. Appl. 255, 436–446 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank the referee for the valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Physics, Zhengzhou Institute of Aeronautical Industry Management, Zhengzhou , 450015, People’s Republic of China

    Jinghuai Liu, Shaohua Cheng & Litao Zhang

Authors
  1. Jinghuai Liu
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  2. Shaohua Cheng
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  3. Litao Zhang
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Correspondence to Jinghuai Liu.

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Supported by the Tianyuan Special Funds of the National Natural Science Foundation of China (11226337), Science and Technology Project of Henan Province (132300410373) and Scientific Research Fund for Young of Zhengzhou Institute of Aeronautical Industry Management (2013171003).

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Liu, J., Cheng, S. & Zhang, L. Anti-periodic mild solutions to semilinear fractional differential equations. J. Appl. Math. Comput. 48, 381–393 (2015). https://doi.org/10.1007/s12190-014-0808-z

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