Advertisement

Fréchet phase space for nonlinear infinite delay equations in Banach spaces

  • G DivyabharathiEmail author
  • T Sengadir
Article
  • 57 Downloads

Abstract

In this paper, we prove the existence of solutions to nonlinear differential equations with infinite delay in Banach spaces. We construct a Fréchet space and the unique solutions are given by semigroups in this Fréchet space. Applications to partial differential equations with infinite delay are given.

Keyword

Infinite delay Fréchet space semigroup 

Mathematics Subject Classification

35R10 47H20 

References

  1. 1.
    Adimy M, Bollazahirb H and Ezzinbi K, Existence for a class of partial functional differential equations with infinite delay, Nonlinear Anal.  46 (2001) 91–112MathSciNetCrossRefGoogle Scholar
  2. 2.
    Arino O A, Burton T A and Haddock J R, Periodic solution to functional differental equations, Proc. R. Soc. Edinburgh  101A (1985) 253–271CrossRefGoogle Scholar
  3. 3.
    Atkinson F V and Haddock J R, On determining phase spaces for functional differential equations, Funkcialaj Ekvacioj  31 (1988) 331–347MathSciNetzbMATHGoogle Scholar
  4. 4.
    Burton T A and Zhang B, Periodic solutions of abstract differential equations with infinite delay, J. Diff. Equ.  90 (1991) 357–396MathSciNetCrossRefGoogle Scholar
  5. 5.
    Burton T A, Dwiggins D P and Feng Y, Periodic solutions of functional differential equations with infinite delay, J. London Math. Soc.  40 (1989) 81–88MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burton T A, Feng Y, Continuity in functional differential equation with infinite delay, Acta. Math. Appl. Sinica (English Ser.)  7 (1991) 229–244MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burton T A and Heirng R H, Boundedness in infinite delay systems, J. Math. Anal. Appl.  144 (1989) 486–502MathSciNetCrossRefGoogle Scholar
  8. 8.
    Burton T A, Zhang Bo, Uniform ultimate boundedness and periodicity in functioal differential equations, Tohoku. Math. J.  42 (1990) 93–100MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fitzgibbon W E, Delay equations of parabolic type in Banach space, Nonlinear Equations in Abstract Spaces, Proc. Internat. Sympos. (1978) pp. 81–93Google Scholar
  10. 10.
    Haddock J R, Nkashama M N and Jianhong Wu, Asymptotic constancy for linear neutral Volterra integro differential equations, Tohoku Math. J.  41 (1989) 689–710MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hassane B, Semigroup approach to semilinear partial functional differential equations with infinite delay, J. Inequalities Appl. 2007 (2007) 1–13MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hassane B, Honglian Y and Rong Y, Global attractor for some partial functional differential equations with infinite delay, Funkcialaj Ekvacioj  54 (2011) 139–156MathSciNetCrossRefGoogle Scholar
  13. 13.
    Jack K and Kato J, Phase space for retarded equations with infinite delay, Funkcial. Ekvac  21 (1978) 11–41MathSciNetzbMATHGoogle Scholar
  14. 14.
    Jan Pruss, On linear Volterra equations of parabolic type in Banach spaces, Trans. Amer. Math. Soc. 301 (1987) 691–721MathSciNetCrossRefGoogle Scholar
  15. 15.
    Janhong Wu, Theory and applications of partial functional differential equations, Appl. Math. Sci. (1996) (New York: Springer-Verlag) vol. 119Google Scholar
  16. 16.
    Kunisch K and Schappacher W, Necessary conditions for partial differential equation with delay to generate \(C_{0}\)-semigroups, J. Diff. Equ.  50 (1983) 49–79CrossRefGoogle Scholar
  17. 17.
    Mouataz B M, Abdelouaheb A and Ahcene D, Periodic solutions and stability in a nonlinear neutral system of differential equations with infinite delay, Bol. Soc. Math. Mex (2016)zbMATHGoogle Scholar
  18. 18.
    Piriyadarshini D and Sengadir T, Existence of solutions and semi-discretization for partial differential equation with infinite delay, Diff. Equ. Appl. 7(3) (2015) 313–331Google Scholar
  19. 19.
    Royden H L, Real Analysis (1988) (New Delhi: Pearson Education)zbMATHGoogle Scholar
  20. 20.
    Sengadir T, Semigroups on Fréchet spaces and equations with infinite delays, Proc. Indian Acad. Sci. (Math. Sci.) 117(1) (2007) 71–84MathSciNetCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of MathematicsCentral University of Tamil NaduThiruvarurIndia

Personalised recommendations