Skip to main content
SpringerLink
Log in

A Class of Nonlocal Impulsive Problems for Integrodifferential Equations in Banach Spaces

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we study the existence and uniqueness of the PC-mild solution for a class of nonlinear integrodifferential impulsive differential equations with nonlocal conditions

$$\left\{\begin{array}{l} x'(t)=Ax(t)+f\left(t,x(t), \int_{0}^{t}k(t,s,x(s))ds\right), \quad t\in J=[0,b], \,\, t\neq t_{i},\\ x(0)=g(x)+x_{0},\\ \Delta x(t_{i})=I_{i}(x(t_{i})), \quad i=1,2,\ldots,p, \,\, 0=t_{0} < t_{1} < \cdots < t_{p} < t_{p+1}=b.\end{array} \right.$$

Using the generalized Ascoli-Arzela theorem given by us, some fixed point technique including Schaefer fixed point theorem and Krasnoselskii fixed point theorem, and theory of operators semigroup, some new results are obtained. At last, some examples are given to illustrate the theory.

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. Ahmed N.U.: Optimal impulsive control for impulsive systems in Banach space. Int. J. Differ. Equ. Appl. 1, 37–52 (2000)

    MATH  MathSciNet  Google Scholar 

  2. Ahmed N.U.: Some remarks on the dynamics of impulsive systems in Banach space. Math. Anal. 8, 261–274 (2001)

    MATH  Google Scholar 

  3. Ahmed N.U., Teo K.L., Hou S.H.: Nonlinear impulsive systems on infinite dimensional spaces. Nonlinear Anal. 54, 907–925 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ahmed N.U.: Existence of optimal controls for a general class of impulsive systems on Banach space. SIAM J. Control Optimal 42, 669–685 (2003)

    Article  MATH  Google Scholar 

  5. Ahmed N.U.: Optimal feedback control for impulsive systems on the space of finitely additive measures. Publ. Math. Debrecen 70, 371–393 (2007)

    MATH  MathSciNet  Google Scholar 

  6. Benchohra M., Henderson J., Ntouyas S.K., Ouahab A.: Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space. J. Inequal. Appl. 2005, 189–205 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Byszewski L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Byszewski L.: Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems. Dyn. Syst. Appl. 5, 595–605 (1996)

    MATH  MathSciNet  Google Scholar 

  9. Byszewski L., Akca H.: Existence of solutions of a semilinear functional differential evolution nonlocal problem. Nonlinear Anal. 34, 65–72 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Byszewski L., Akca H.: On a mild solution of a semilinear functional-differential evolution nonlocal problem. J. Appl. Math. Stoch. Anal. 10, 265–271 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Balachandran K., Park D.G., Kwun Y.C.: Nonlinear integrodifferential equations of Sobolev type with nonlocal conditions in Banach spaces. Commun. Korean Math. Soc. 14, 223–231 (1999)

    MATH  MathSciNet  Google Scholar 

  12. Balachandran K., Uchiyama K.: Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces. Proc. Indian Acad. Sci. (Math. Sci.) 110, 225–232 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Balachandran K., Chandrasekaran M.: The nonlocal Cauchy problem for semilinear integrodifferential equation with devating argument. Proc. Edinburgh Math. Soc. 44, 63–70 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Balachandran K., Kumar R.R.: Existence of solutions of integrodifferential evolution equations with time varying delays. Appl. Math. E Notes 7, 1–8 (2007)

    MATH  MathSciNet  Google Scholar 

  15. Bainov D.D., Simeonov P.S.: Impulsive Differential Equations: Periodic Solutions and Applications. Longman Scientific and Technical Group. Limited, New York (1993)

    Google Scholar 

  16. Lakshmikantham V., Bainov D.D., Simeonov P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore, London (1989)

    MATH  Google Scholar 

  17. Lin Y., Liu J.H.: Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear Anal. 26, 1023–1033 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Liu J.H., Ezzinbi K.: Non-autonomous integrodifferential equations with nonlocal conditions. J. Integr. Equ. Appl. 15, 79–93 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liang J., Liu J.H., Xiao T.-J.: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal. 57, 183–189 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Liu J.: Nonlinear impulsive evolution equations. Dyn. Contin. Discrete Impuls. Syst. 6, 77–85 (1999)

    MATH  Google Scholar 

  21. Liang J., Liu J.H., Xiao T.-J.: Nonlocal problems for integrodifferential equations. Dyn. Contin. Discrete Impuls. Syst. 15, 815–824 (2008)

    MATH  MathSciNet  Google Scholar 

  22. Liang J., Liu J.H., Xiao T.-J.: Nonlocal impulsive problems for nonlinear differential equations in Banach spaces. Math. Comput. Model. 49, 798–804 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mophou G.M.: Existence and uniqueness of mild solution to impulsive fractional differential equations. Nonlinear Anal. 72, 1604–1615 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Krasnoselskii M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, New York (1964)

    Google Scholar 

  25. Schaefer H.: Über die methode der priori Schranhen. Math. Ann. 129, 415–416 (1955)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang, J., Xiang, X., Wei, W., Chen, Q.: The generalized Gronwall inequality and its application to periodic solutions of integrodifferential impulsive periodic system on Banach space. J. Inequal. Appl. 2008, Article ID 430521, 22 p. (2008)

  27. Wang, J., Xiang, X., Wei, W.: Existence of periodic solutions for integrodifferential impulsive periodic system on Banach space. Abstract Appl. Anal. 2008, Article ID 939062, 19 p. (2008)

  28. Wang J., Xiang X., Wei W.: Periodic solutions of a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space. Electr. J. Qual. Theory Differ. Equ. 4, 1–17 (2009)

    Google Scholar 

  29. Wang, J., Xiang, X., Wei, W.: A class of nonlinear integrodifferential impulsive periodic systems of mixed type and optimal controls on Banach space. J. Appl. Math. Comput (in press). doi:10.1007/s12190-009-0332-8

  30. Wei W., Xiang X., Peng Y.: Nonlinear impulsive integro-differential equation of mixed type and optimal controls. Optimization 55, 141–156 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Xiang X., Wei W.: Mild solution for a class of nonlinear impulsive evolution inclusion on Banach space. Southeast Asian Bull. Math. 30, 367–376 (2006)

    MATH  MathSciNet  Google Scholar 

  32. Xiang X., Wei W., Jiang Y.: Strongly nonlinear impulsive system and necessary conditions of optimality. Dyn. Cont. Discrete Impuls. Syst. A Math. Anal. 12, 811–824 (2005)

    MATH  MathSciNet  Google Scholar 

  33. Yu X.L., Xiang X., Wei W.: Solution bundle for class of impulsive differential inclusions on Banach spaces. J. Math. Anal. Appl. 327, 220–232 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Yang T.: Impulsive control theory. Springer, Berlin (2001)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science of Guizhou University, Guizhou University, Guiyang, 550025, People’s Republic of China

    JinRong Wang & W. Wei

Authors
  1. JinRong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to JinRong Wang.

Additional information

JinRong Wang acknowledge support from the National Natural Science Foundation of Guizhou Province (2010, No.2142) and Introducing Talents Foundation for the Doctor of Guizhou University (2009, No.031). W. Wei acknowledge support from the National Natural Science Foundation of China (No. 10961009).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, J., Wei, W. A Class of Nonlocal Impulsive Problems for Integrodifferential Equations in Banach Spaces. Results. Math. 58, 379–397 (2010). https://doi.org/10.1007/s00025-010-0057-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-010-0057-x

Mathematics Subject Classification (2000)

Keywords

Search

Navigation