SpringerLink
Log in

Retarded equations with infinite delays

  • Conference paper
  • First Online:
Functional Differential Equations and Approximation of Fixed Points

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 730))

Abstract

It is the purpose of these notes to describe the theory of Hale and Kato for functional differential equations based on a space of initial data which satisfy some very reasonable axioms. We also indicate some recent results of Naito showing how extensive the theory of linear systems can be developed in an abstract setting in particular, the characterization of the spectrum of the infinitesimal generator together with the decomposition theory and exponential estimates of solutions.

This research was supported in part by the Air Force Office of Scientific Research under AF-AFOSR 76-3092A, in part by the National Science Foundation under NSF-MCS 78-18858, and in part by the United States Army under ARO-D-31-124-73-G130.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free ship** worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Banks, H.T. and J. Burns, An abstract framework for approximate solutions to optimal control problems governed by hereditary systems. Proc. Int. Conf. Diff. Eqns. Univ. Southern California, 1974, Academic Press, 1975.

    Google Scholar 

  2. Banks, H.T. and M. Jacobs, An attainable sets approach to optimal control of functional differential equations with function space side conditions. J. Differential Equations, 13(1973), 127–149.

    Article  MathSciNet  MATH  Google Scholar 

  3. Barbu, V. and S. Grossman, Asymptotic behavior of linear integral differential systems. Trans. Am. Math. Soc., 173(1972), 277–289.

    Article  MathSciNet  MATH  Google Scholar 

  4. Coleman, B.D. and V.J. Mizel, Norms and semigroups in the theory of fading memory. Arch. Rat. Mech. Ana., 23 (1966), 87–123.

    MathSciNet  MATH  Google Scholar 

  5. Coleman, B.D. and V.J. Mizel, On the general theory of fading memory. Arch. Rat. Mech. Ana., 29(1968), 18–31.

    MathSciNet  MATH  Google Scholar 

  6. Coleman, B.D. and V.J. Mizel, On the stability of solutions of functional differential equations. Arch. Rat. Mech. Ana., 30(1968), 173–196.

    MathSciNet  MATH  Google Scholar 

  7. Coleman, B.D. and D. Owen, On the initial value problem for a class of functional differential equations. Arch. Rat. Mech. Ana., 55(1974), 275–299.

    MathSciNet  MATH  Google Scholar 

  8. Cooperman, G., α-condensing maps and dissipative systems. Ph.D. Thesis, Brown University, Providence, Rhode Island, 1978.

    Google Scholar 

  9. Delfour, M.C. and S.K. Mitter, Hereditary differential systems with constant delays, I. General case. J. Differential Equations, 12(1972), 213–255.

    Article  MathSciNet  MATH  Google Scholar 

  10. Hale, J.K., Dynamical systems and stability. J. Math. Ana. Appl., 26(1969), 39–59.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hale, J.K., Functional differential equations with infinite delay. J. Math. Ana. Appl., 48(1974), 276–283.

    Article  MathSciNet  MATH  Google Scholar 

  12. Hale, J.K., Theory of Functional Differential Equations. Appl. Math. Sci., Vol. 3, Springer-Verlag, 1977.

    Google Scholar 

  13. Hale, J.K. and J. Kato, Phase space for retarded equations with infinite delay. Funk. Ekv., 21(1978), 11–41.

    MathSciNet  MATH  Google Scholar 

  14. Hale, J.K. and O. Lopes, Fixed point theorems and dissipative processes. J. Differential Equations, 13(1973), 391–402.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hino, Y., Asymptotic behavior of solutions of some functional differential equations. Tôhoku Math. J., 22(1970), 98–108.

    Article  MathSciNet  MATH  Google Scholar 

  16. Hino, Y., Continuous dependence for functional differential equations. Tôhoku Math. J., 23(1971), 565–571.

    Article  MathSciNet  MATH  Google Scholar 

  17. Hino, Y., On the stability of the solution of some functional differential equations. Funk. Ekv., 14(1971), 47–60.

    MathSciNet  MATH  Google Scholar 

  18. Hino, Y., Stability and existence of almost periodic solutions of some functional differential equations. Tôhoku Math. J. 28(1976), 389–409.

    Article  MathSciNet  MATH  Google Scholar 

  19. Hino, Y., Favard's separation theorem in functional differential equations with infinite retardations. Tôhoku Math. J., 30(1978), 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  20. Hino, Y., Almost periodic solutions of functional differential equations with infinite retardation. Preprint.

    Google Scholar 

  21. Ize, A. and J.G. dos Reis, Contributions to stability of neutral functional differential equations. J. Differential Equations, 29(1978), 58–65.

    Article  MathSciNet  MATH  Google Scholar 

  22. Kappel, F., The invariance of limit sets for autonomous functional differential equations. SIAM J. Appl. Math., 19(1970), 408–419.

    Article  MathSciNet  MATH  Google Scholar 

  23. Kappel, F. and N. Schappacher, Nonlinear functional differential equations and abstract integral equations. Proc. Royal Soc. Edinburgh, Ser. A. To appear.

    Google Scholar 

  24. Leitman, M.J. and V.J. Mizel, On fading memory spaces and hereditary integral equations. Arch. Rat. Mech. Ana., 55(1974), 18–51.

    MathSciNet  MATH  Google Scholar 

  25. Lima, P., Hopf bifurcation in equations with infinite delays. Ph.D. Thesis, Brown University, Providence, Rhode Island, 1977.

    Google Scholar 

  26. Naito, T., Integral manifolds for linear functional differential equations on some Banach space. Funk. Ekv., 13(1970), 199–213.

    MathSciNet  MATH  Google Scholar 

  27. Naito, T., On autonomous linear functional differential equations with infinite retardations. J. Differential Eqns., 2(1976), 297–315.

    Article  MathSciNet  MATH  Google Scholar 

  28. Naito, T., On linear autonomous retarded equations with an abstract space for infinite delay. J. Differential Equations. To appear.

    Google Scholar 

  29. Nussbaum, R., The radius of the essential spectrum. Duke Math. J., 37(1970), 473–488.

    Article  MathSciNet  MATH  Google Scholar 

  30. Palmer, J.W., Liapunov stability theory for nonautonomous functional differential equations. Ph.D. Thesis, Brown University, Providence, Rhode Island, 1978.

    Google Scholar 

  31. Reber, D., Approximation and optimal control of linear hereditary systems. Ph.D. Thesis, Brown University, Providence, Rhode Island, 1978.

    Google Scholar 

  32. Schumacher, K., Existence and continuous dependence for functional differential equations with unbounded delay. Arch. Rat. Mech. Ana., 67(1978), 315–335.

    MathSciNet  MATH  Google Scholar 

  33. Schumacher, K., Dynamical systems with memory on history spaces with monotonic seminorms. Preprints.

    Google Scholar 

  34. Stech, H., Contribution to the theory of functional differential equations with infinite delay. J. Differential Equations, 27(1978), 421–443.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lefschetz Center for Dynamical Systems Division of Applied Mathematics, Brown University, 02912, Providence, Rhode Island

    Jack K. Hale

Authors
  1. Jack K. Hale
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Heinz-Otto Peitgen Hans-Otto Walther

Rights and permissions

Reprints and permissions

Copyright information

© 1979 Springer-Verlag

About this paper

Cite this paper

Hale, J.K. (1979). Retarded equations with infinite delays. In: Peitgen, HO., Walther, HO. (eds) Functional Differential Equations and Approximation of Fixed Points. Lecture Notes in Mathematics, vol 730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0064317

Download citation

  • DOI: https://doi.org/10.1007/BFb0064317

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09518-7

  • Online ISBN: 978-3-540-35129-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation