Skip to main content
Log in

Concepts of almost periodicity and ergodic theorems in locally convex spaces

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

We wish to investigate mean ergodic theorems for generalizations of almost periodic functions on semigroups, as well as for semigroups of operators in the framework of locally convex spaces. Specially, we present functional characterizations of concepts of almost periodicity for vector-valued functions.

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

Access this article

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

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

Instant access to the full article PDF.

Similar content being viewed by others

Data availability

Not applicable.

References

  1. Amerio, L., Prouse, G.: Almost-Periodic Functions and Functional Equations. Van Nostrand, Reinhold (1971)

    Book  MATH  Google Scholar 

  2. Arendt, W., Batty, C.J.K.: Almost periodic solutions of first- and second-order Cauchy problems. J. Differ. Equ. 137, 363–383 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baillon, J.B.: Un theoreme de type ergodique pour les contractions nonlineaires dans un espace de Hilbert. C. R. Acad. Sci. Paris 280, 1511–1514 (1975)

    MathSciNet  MATH  Google Scholar 

  4. Baillon, J.B.: Quelques proprietes de convergence asymptotique pour les semigroupes de contractions impaires. C. R. Acad. Sci. Paris 283, 75–78 (1976)

    MathSciNet  MATH  Google Scholar 

  5. Baillon, J.B., Brezis, H.: Une remarque sur le comportement asymptotique des semigroupes nonlineaires. Houston J. Math. 2, 5–7 (1976)

    MathSciNet  MATH  Google Scholar 

  6. Batty, C.J.K., Hutter, W., Rabiger, F.: Almost periodicity of mild solutions of inhomogeneous periodic Cauchy problems. J. Differ. Equ. 156, 309–327 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berglund, J.F., Junghenn, H.D., Milnes, P.: Analysis on Semigroups. Wiley, New York (1988)

    MATH  Google Scholar 

  8. Bochner, S.: Beitriige zur theorie der fastperiodischen funktionen. Math. Ann. 96, 119–147 (1927)

    Article  MathSciNet  Google Scholar 

  9. Bochner, S.: Abstrakte fastperiodische funktionen. Acta Math. 61, 149–184 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bohr, H.: Zur Theorie der fastperiodischen Funktionen I. Acta Math. 45, 29–127 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bohr, H.: Zur theorie der fastperiodischen funktionen III. Acta Math. 47, 237–281 (1926)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bruck, R.E.: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Israel J. Math. 32, 107–116 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Day, M.M.: Amenable semigroup. Ill. J. Math. 1, 509–544 (1957)

    MathSciNet  MATH  Google Scholar 

  14. Djafari Rouhani, B., Jamshidnezhad, P., Saeidi, S.: Existence and approximation of zeroes of monotone operators by solutions to nonhomogeneous difference inclusions. J. Math. Anal. Appl. 502, 125268 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eberlein, W.F.: Abstract ergodic theorems and weak almost periodic functions. Trans. Am. Math. Soc. 67, 217–240 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goldberg, S., Irwin, P.: Weakly almost periodic vector-valued functions. Diss. Math. (Rozprawy Mat.) 157, 1–42 (1979)

    MathSciNet  MATH  Google Scholar 

  17. Kada, O.: Existence of ergodic retraction for noncommutative semigroups in Banach spaces. Proc. Am. Math. Soc. 127, 3013–3020 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kada, O.: Strong ergodic theorems for commutative semigroups of operators. Proc. Am. Math. Soc. 127, 3003–3011 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kido, K., Takahashi, W.: Mean ergodic theorems for semigroups of linear operators. J. Math. Anal. Appl. 103, 387–394 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kohlenbach, U.: A uniform quantitative form of sequential weak compactness and Baillon’s nonlinear ergodic theorem. Commun. Contemp. Math. 14, 1250006 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lau, A.T., Shioji, N., Takahashi, W.: Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces. J. Funct. Anal. 161, 62–75 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Leustean, L., Nicolae, A.: Effective results on nonlinear ergodic averages in CAT \((\kappa )\) spaces. Ergod. Theory Dyn. Syst. 36(8), 2580–2601 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Milnes, P.: On vector-valued weakly almost periodic functions. J. Lond. Math. Soc. 22, 467–472 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Miyake, H., Takahashi, W.: Vector-valued weakly almost periodic functions and mean ergodic theorems in Banach spaces. J. Nonlinear Convex Anal. 9, 255–272 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Miyake, H., Takahashi, W.: Mean ergodic theorems for almost periodic semigroups. Taiwanese J. Math. 14, 1079–1091 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Paterson, A.L.T.: Amenability. American Mathematical Society, Providence (1988)

    Book  MATH  Google Scholar 

  27. Pazy, A.: Remarks on nonlinear ergodic theory in Hilbert space. Nonlinear Anal. 3, 863–871 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  28. Reich, S.: Nonlinear evolution equations and nonlinear ergodic theorems. Nonlinear Anal. 1, 319–330 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  29. Reich, S.: Almost convergence and nonlinear ergodic theorems. J. Approx. Theory 24, 269–272 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rode, G.: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space. J. Math. Anal. Appl. 85, 172–178 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ruess, W.M., Phong, V.Q.: Asymptotically almost periodic solutions of evolution equations in Banach spaces. J. Differ. Equ. 122, 282–301 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ruess, W.M., Summers, W.H.: Integration of Asymptotically Almost Periodic Functions and Weak Asymptotic Almost Periodicity. Instytut Matematyczny Polskiej Akademi Nauk, Warszawa (1989)

    MATH  Google Scholar 

  33. Ruess, W.M., Summers, W.H.: Ergodic theorems for semigroups of operators. Proc. Am. Math. Soc. 114, 423–432 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Saeidi, S.: Existence of ergodic retractions for semigroups in Banach spaces. Nonlinear Anal. 69, 3417–3422 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Saeidi, S.: Ergodic retractions for amenable semigroups in Banach spaces with normal structure. Nonlinear Anal. 71, 2558–2563 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Schaefer, H.H.: Topological Vector Spaces. Springer, New York (1971)

    Book  MATH  Google Scholar 

  37. Suzuki, T., Takahashi, W.: Weak and strong convergenc theorems for nonexpansive mappings in Banach spaces. Nonlinear Anal. 47, 2805–2815 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  39. Voigt, J.: A Course on Topological Vector Spaces, Compact Textbooks in Mathematics. Birkhäuser/Springer, New York (2020)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Contributions

All the authors contributed equally and reviewed the manuscript.

Corresponding author

Correspondence to Shahram Saeidi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amini, F., Saeidi, S. Concepts of almost periodicity and ergodic theorems in locally convex spaces. J. Fixed Point Theory Appl. 25, 78 (2023). https://doi.org/10.1007/s11784-023-01081-9

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11784-023-01081-9

Keywords

Mathematics Subject Classification

Navigation