Advertisement

Journal of Applied Mathematics and Computing

, Volume 34, Issue 1–2, pp 101–112 | Cite as

Approximation of solutions to impulsive functional differential equations

  • M. MuslimEmail author
  • Ravi P. Agarwal
Article

Abstract

In this paper we shall study a semilinear impulsive functional differential equation in a separable Hilbert space. We shall use the analytic semigroups theory of linear operators and fixed point technique to establish the existence, uniqueness, and the convergence of approximate solutions to the given problem. We will also prove the existence and convergence of finite-dimensional approximate solutions to the given problem. An example is also illustrated.

Keywords

Impulsive functional differential equations Banach fixed point theorem Analytic semigroup 

Mathematics Subject Classification (2000)

34A60 34K05 34K45 35K70 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bazley, N.: Approximation of wave equations with reproducing nonlinearities. Nonlinear Anal. 3, 539–546 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bazley, N.: Global convergence of Faedo–Galerkin approximations to nonlinear wave equations. Nonlinear Anal. 4, 503–507 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benchohra, M., Ouahabi, A.: Some uniqueness results for impulsive semilinear neutral functional differential equations. Georgian Math. J. 9(3), 423–430 (2002) zbMATHMathSciNetGoogle Scholar
  4. 4.
    Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Impulsive functional differential equations with variable times and infinite delay. Int. J. Appl. Math. Sci. 2(1), 130–148 (2005) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Dhage, B.C., Ntouyas, S.K.: Existence results for impulsive neutral functional differential inclusions. Topol. Methods Nonlinear Anal. 25(2), 349–361 (2005) zbMATHMathSciNetGoogle Scholar
  6. 6.
    Goethel, R.: Faedo–Galerkin approximation in equations of evolution. Math. Methods Appl. Sci. 6, 41–54 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Heinz, E., von Wahl, W.: Zu einem Satz von F.W. Browder über nichtlineare Wellengleichungen. Math. Z. 141, 33–45 (1974) CrossRefGoogle Scholar
  8. 8.
    Laxshmikantham, V., Bainov, D.D., Simenov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) Google Scholar
  9. 9.
    Miletta, P.D.: Approximation of solutions to evolution equations. Math. Methods Appl. Sci. 17, 753–763 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Murakami, H.: On linear ordinary and evolution equations. Funkcial. Ekvac. 9, 151–162 (1966) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Ntouyas, S.K.: Existence results for impulsive partial neutral functional differential inclusions. Electron. Differ. Equ. 30, 1–11 (2005) Google Scholar
  12. 12.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) zbMATHGoogle Scholar
  13. 13.
    Segal, I.: Nonlinear semigroups. Ann. Math. 78, 339–364 (1963) CrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2009

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA

Personalised recommendations