Skip to main content
Log in

The Favard Separation Condition for Almost Periodic Linear Systems

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

For linear systems which depend almost periodically on time, the Favard separation condition is shown to be equivalent to the following dimensional fact: all the systems in the hull have the same number of independent bounded solutions.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Cheban, D.N.: Bounded solutions of linear almost periodic systems of differential equations. Izv. Math. 62, 581–600 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bochner, S.: A new approach to almost periodicity. Proc. Natl. Acad. Sci. USA. 48, 2039–2043 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  3. Favard, J.: Sur les equations différentielles linéaires à coefficients presque-périodiques. Acta Math. 51, 31–81 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  4. Favard, J.: Sur certains systmes différentiels scalaires linéaires et homogénes coefficients presque-périodiques. Ann. Mat. Pura Appl. 61, 297–316 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fink, A.M.: Almost periodic differential equations, Lecture Notes in Mathematics, 377th edn. Springer, Berlin (1974)

  6. Johnson, R.A.: On a Floquet theory for almost-periodic, two-dimensional linear systems. J. Diff. Equ. 37, 184–205 (1980)

    Article  MATH  Google Scholar 

  7. Johnson, R.A.: A linear almost periodic equation with an almost automorphic solution. Proc. Am. Math. Soc. 82, 199–205 (1981)

    Article  MATH  Google Scholar 

  8. Ortega, R., Tarallo, M.: Almost periodic linear differential equations with non-separated solutions. J. Funct. Anal. 237, 402–426 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Palmer, K.J.: On bounded solutions of almost periodic linear differential systems. J. Math. Anal. Appl. 103, 16–25 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Rockafellar, R.T., Wets, R.J.: Variational analysis, Grundlehren der Mathematischen Wissenschaften, 317th edn. Springer, Berlin (1998)

  11. Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems I. J. Diff. Equ. 15, 429–458 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sacker, R.J., Sell, G.R.: Existence of dichotomies and invariant splittings for linear differential systems III. J. Diff. Equ. 22, 497–522 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sacker, R.J., Sell, G.: Lifting properties in skew-product flows with applications to differential equations. Mem. Am. Math. Soc. 11(190), (1977)

  14. Sell, G.R.: Topological dynamics and ordinary differential equations. Lecture Notes. Van Nostrand-Reinhold, New York (1971)

  15. Sell, G.R.: Linear differential systems. Lecture Notes, Univ. of Minnesota (1975)

  16. Vinograd, R.E.: A projected trace and partwise isometric transformations fo almost periodic systems. Proc. Am. Math. Soc. 86, 299–304 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhikov, V.V., Levitan, B.M.: Favard theory. Russ. Math. Surv. 32, 129–180 (1977)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Massimo Tarallo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tarallo, M. The Favard Separation Condition for Almost Periodic Linear Systems. J Dyn Diff Equat 25, 291–304 (2013). https://doi.org/10.1007/s10884-013-9309-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-013-9309-2

Keywords

Mathematics Subject Classification (2000)

Navigation