Skip to main content
SpringerLink
Log in

Almost Automorphic and Almost Periodic Dynamics for Quasimonotone Non-Autonomous Functional Differential Equations

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

Abstract

The occurrence of almost automorphic dynamics for monotone non-autonomous recurrent finite-delay functional differential equations is analyzed. Topological methods are used to ensure its presence in the case of existence of semicontinuous semi-equilibria. When these semi-equilibria are continuous and strong, the presence of almost automorphic extensions is persistent under small perturbations. The above method provides a minimal set isomorphic to the base in the case of a convex semiflow. Some examples show the applicability of these results.

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. A.I. Alonso R. Obaya A.M. Sanz (2005) ArticleTitleA note on non-autonomous scalar functional differential equations with small delay C. R. Acad. Sci. Paris, Sér. I Math., 340 155–160

    Google Scholar 

  2. H. Amann (1976) ArticleTitleFixed point equations and nonlinear eigenvalue problems in ordered Banach spaces SIAM Rev. 18 620–709 Occurrence Handle10.1137/1018114

    Article  Google Scholar 

  3. L. Arnold I.D. Chueshov (1998) ArticleTitleOrder-preserving random dynamical systems: equilibria, attractors, applications Dynam. Stability Syst. 13 265–280

    Google Scholar 

  4. L. Arnold I.D. Chueshov (2001) ArticleTitleCooperative random and stochastic differential equations Discrete Contin. Dyn. Syst. 7 1–33

    Google Scholar 

  5. J.-P. Aubin H. Frankowska (1990) Set-Valued Analysis Birkhäuser Boston, Basel, Berlin

    Google Scholar 

  6. G. Choquet (1969) Lectures on Analysis. Integration and Topological Vector Spaces, Vol. I, Math Lecture Notes Benjamin Reading, MA.

    Google Scholar 

  7. I.D. Chueshov (2002) Monotone Random Systems Theory and Applications, Lecture Notes in Math. 1779 Springer-Verlag Berlin, Heidelberg

    Google Scholar 

  8. R. Ellis (1969) Lectures on Topological Dynamics Benjamin New York

    Google Scholar 

  9. A.M. Fink P.O. Frederickson (1971) ArticleTitleUntimately boundedness does not imply almost periodicity J. Differential Equations 9 280–284 Occurrence Handle10.1016/0022-0396(71)90081-7

    Article  Google Scholar 

  10. J.K. Hale S.M. Verduyn Lunel (1993) Introduction to Functional Differential Equations, Applied Mathematical Sciences 99 Springer-Verlag Berlin, Heidelberg, New York

    Google Scholar 

  11. Y. Hino S. Murakami T. Naiko (1991) Functional Differential Equations with Infinite Delay, Lecture Notes in Math 1473 Springer-Verlag Berlin, Heidelberg

    Google Scholar 

  12. R. Johnson (1980) ArticleTitleOn a Floquet theory for almost-periodic, two-dimensional linear systems J. Differential Equations 37 184–205 Occurrence Handle10.1016/0022-0396(80)90094-7

    Article  Google Scholar 

  13. R. Johnson (1981) ArticleTitleA linear, almost periodic equation with an almost automorphic solution Proc. Amer. Math. Soc. 82 IssueID2 199–205

    Google Scholar 

  14. R. Johnson (1982) ArticleTitleOn almost periodic linear differential systems of Millionscikov and Vinograd J. Math. Anal. Appl. 85 452–460 Occurrence Handle10.1016/0022-247X(82)90011-7

    Article  Google Scholar 

  15. M.A. Krasnoselskii J.A. Lisfshits A.V. Sobolev (1989) Positive Linear Systems: The Method of Positive Operators, Sigma Series in Appl. Math. 5 Heldermann Verlag Berlin

    Google Scholar 

  16. U. Krause P. Ranft (1992) ArticleTitleA limit set trichotomy for monotone nonlinear dynamical systems Nonlinear Anal. 19 375–392 Occurrence Handle10.1016/0362-546X(92)90182-E

    Article  Google Scholar 

  17. B. Levitan V. Zhikov (1982) Almost Periodic Functions and Differential Equations Cambridge University Press Cambridge

    Google Scholar 

  18. V.M. Millionscikov (1968) ArticleTitleProof of the existence of irregular systems of linear differential equations with almost periodic coefficients Differ. Uravn. 4 IssueID3 391–396

    Google Scholar 

  19. S. Novo R. Obaya (2004) ArticleTitleStrictly ordered minimal subsets of a class of convex monotone skewproduc semiflows J. Differential Equations 196 249–288 Occurrence Handle10.1016/S0022-0396(03)00152-9

    Article  Google Scholar 

  20. S. Novo R. Obaya A.M. Sanz (2005) ArticleTitleAttractor minimal sets for cooperative and strongly convex delay differential equations J. Differential Equations 208 IssueID1 86–123 Occurrence Handle10.1016/j.jde.2004.01.002

    Article  Google Scholar 

  21. S. Novo R. Obaya A.M. Sanz (2004) ArticleTitleAlmost periodic and almost automorphic dynamics for scalar convex differential equations Israel J. Math. 144 157–189 Occurrence HandleMR2121539

    MathSciNet  Google Scholar 

  22. Novo S., Obaya R., and Sanz, A.M. Attractor minimal sets for non-autonomous delay functional differential equations with applications to neural networks, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., to appear.

  23. Z. Opial (1961) ArticleTitleSur une équation différentielle presque-périodique sans solution presquepériodic Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 9 673–676

    Google Scholar 

  24. R. Ortega M. Tarallo (2003) ArticleTitleAlmost periodic upper and lower solutions J. Differential Equations 193 343–358 Occurrence Handle10.1016/S0022-0396(03)00130-X

    Article  Google Scholar 

  25. Sacker R.J., and Sell G.R., (1977). Lifting properties in skew-products .ows with applications to differential equations, Mem. Amer. Math. Soc. 190, Amer. Math. Soc., Providence. .

  26. J.F. Selgrade (1980) ArticleTitleAsymptotic behavior of solutions to single loop positive feedback systems J. Differential Equations 38 80–103 Occurrence Handle10.1016/0022-0396(80)90026-1

    Article  Google Scholar 

  27. W. Shen Y. Yi (1995) ArticleTitleDynamics of almost periodic scalar parabolic equations J. Differential Equations 122 114–136 Occurrence Handle10.1006/jdeq.1995.1141

    Article  Google Scholar 

  28. Shen W., and Yi Y., (1998). Almost automorphic and almost periodic dynamics in Skewproducts semiflows. Mem. Amer. Math. Soc. 647, Amer. Math. Soc., Providence.

  29. Smith H.L., (1995). Monotone Dynamical Systems. An introduction to the Theory of Competitive and Cooperative Systems. Amer. Math. Soc., Providence.

  30. H.L. Smith H.R. Thieme (1991) ArticleTitleStrongly order preserving semi.ows generated by functional differential equations J. Differential Equations 93 332–363 Occurrence Handle10.1016/0022-0396(91)90016-3

    Article  Google Scholar 

  31. P. Takáç (1992) ArticleTitleLinearly stable subharmonic orbits in strongly monotone time-periodic dynamical systems Proc. Amer. Math. Soc. 115 IssueID3 691–698

    Google Scholar 

  32. R.E. Vinograd (1975) ArticleTitleA problem suggested by N.P. Erugin Differ. Uravn. 11 IssueID4 632–638

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, E.T.S. Ingenieros Industriales, Paseo del Cauce s/n, 47011, Valladolid, Spain

    Sylvia Novo, Carmen Núñez & Rafael Obaya

Authors
  1. Sylvia Novo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carmen Núñez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rafael Obaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sylvia Novo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Novo, S., Núñez, C. & Obaya, R. Almost Automorphic and Almost Periodic Dynamics for Quasimonotone Non-Autonomous Functional Differential Equations. J Dyn Diff Equat 17, 589–619 (2005). https://doi.org/10.1007/s10884-005-5814-2

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-005-5814-2

Keywords

Search

Navigation