Skip to main content
SpringerLink
Log in

On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study the solvability and optimal controls of a class of fractional integrodifferential evolution systems with infinite delay in Banach spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and continuous dependence of mild solutions are investigated by utilizing the techniques of a priori estimation and extension of step by steps. Finally, existence of optimal controls for system governed by fractional integrodifferential evolution systems with infinite delay is proved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: North-Holland Mathematics Studies, vol. 204. Elsevier Science, Amsterdam (2006)

    Google Scholar 

  2. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Lakshmikantham, V., Leela, S., Devi, J.V.: Theory of Fractional Dynamic Systems. Scientific Publishers Cambridge, Cambridge (2009)

    MATH  Google Scholar 

  5. Agarwal, R.P., Benchohra, M., Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Appl. Math. 109, 973–1033 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Agarwal, R.P., Belmekki, M., Benchohra, M.: A survey on semilinear differential equations and inclusions involving Riemann–Liouville fractional derivative. Adv. Differ. Equ. 2009, e1–e47 (2009)

    MathSciNet  Google Scholar 

  7. Agarwal, R.P., Zhou, Y., He, Y.: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 59, 1095–1100 (2010)

    MATH  MathSciNet  Google Scholar 

  8. Belmekki, M., Benchohra, M.: Existence results for fractional order semilinear functional differential with nondense domain. Nonlinear Anal. TMA 72, 925–932 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional functional differential inclusions with infinite delay and application to control theory. Fract. Calc. Appl. Anal. 11, 35–56 (2008)

    MATH  MathSciNet  Google Scholar 

  11. Henderson, J., Ouahab, A.: Fractional functional differential inclusions with finite delay. Nonlinear Anal. TMA 70, 2091–2105 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hu, L., Ren, Y., Sakthivel, R.: Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays. Semigroup Forum 79, 507–514 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mophou, G.M., N’Guérékata, G.M.: Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay. Appl. Math. Comput. 216, 61–69 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ren, Y., Qin, Y., Sakthivel, R.: Existence results for fractional order semilinear integro-differential evolution equations with infinite delay. Integral Equ. Oper. Theory 67, 33–49 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chang, Y.K., Kavitha, V., Arjunan, M.M.: Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. Nonlinear Anal. TMA 71, 5551–5559 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jaradat, O.K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlinear Anal. TMA 69, 3153–3159 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. El-Borai, M.M.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. El-Borai, M.M.: The fundamental solutions for fractional evolution equations of parabolic type. J. Appl. Math. Stoch. Anal. 3, 197–211 (2004)

    Article  MathSciNet  Google Scholar 

  19. Hernández, E., O’Regan, D., Balachandran, K.: On recent developments in the theory of abstract differential equations with fractional derivatives. Nonlinear Anal. TMA 73, 3462–3471 (2010)

    Article  MATH  Google Scholar 

  20. Wang, J., Zhou, Y.: Study of an approximation process of time optimal control for fractional evolution systems in Banach spaces. Adv. Differ. Equ. 2011, e1–e16 (2011)

    Article  Google Scholar 

  21. Wang, J., Yang, Y., Wei, W.: Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces. Opusc. Math. 30, 361–381 (2010)

    MATH  MathSciNet  Google Scholar 

  22. Wang, J., Wei, W., Yang, Y.: On some impulsive fractional differential equations in Banach spaces. Opusc. Math. 30, 507–525 (2010)

    MATH  MathSciNet  Google Scholar 

  23. Wang, J., Wei, W., Yang, Y.: Fractional nonlocal integrodifferential equations and its optimal control in Banach spaces. J. SIAM 14, 79–91 (2010)

    MathSciNet  Google Scholar 

  24. Wang, J., Zhou, Y., Wei, W.: A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces. Commun. Nonlinear Sci. Numer. Simul. 16, 4049–4059 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  25. Wang, J., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl. 12, 262–272 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  27. Zhou, Y., Jiao, F., Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. TMA 71, 3249–3256 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  28. Zhou, Y., Jiao, F.: Existence of extremal solutions for discontinuous fractional functional differential equations. Int. J. Dyn. Differ. Equ. 2, 237–252 (2009)

    MATH  MathSciNet  Google Scholar 

  29. Zhou, Y., Jiao, F.: Existence of mild solutions for fractional neutral evolution equations. Comput. Math. Appl. 59, 1063–1077 (2010)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Hale, J., Kato, J.: Phase spaces for retarded equations with infinite delay. Funkc. Ekvacioj 21, 11–41 (1978)

    MATH  MathSciNet  Google Scholar 

  32. Henry, D.: Geometric Theory of Semilinear Parabolic Equations. LNM, vol. 840. Springer, Berlin (1981)

    MATH  Google Scholar 

  33. Ye, H., Gao, J., Ding, Y.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328, 1075–1081 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Zeidler, E.: Nonlinear Functional Analysis and Its Application II/A. Springer, New York (1990)

    Book  Google Scholar 

  35. Balder, E.: Necessary and sufficient conditions for L 1-strong-weak lower semicontinuity of integral functional. Nonlinear Anal., Real World Appl. 11, 1399–1404 (1987)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Guizhou University, Guiyang, Guizhou, 550025, China

    JinRong Wang

  2. Department of Mathematics, Xiangtan University, Xiangtan, Hunan, 411105, China

    Yong Zhou

  3. Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia

    Milan Medveď

Authors
  1. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milan Medveď
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Zhou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J., Zhou, Y. & Medveď, M. On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay. J Optim Theory Appl 152, 31–50 (2012). https://doi.org/10.1007/s10957-011-9892-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9892-5

Keywords

Search

Navigation