Convergence of almost orbits of semigroups

Abstract

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of \({\ell }^\infty (S)\) which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let \((E, \Vert \cdot \Vert _E)\) be a uniformly convex Banach space and let C be a nonempty, bounded, closed and convex subset of E. Assume that C has nonempty interior, is locally uniformly rotund and \(\mathcal {T}=\left\{ T\left( s\right) \right\} _{s \in S}\) is an asymptotically nonexpansive semigroup which acts on C. Assume also that \(\mathcal {T}\) has a unique fixed point \(\tilde{x}\) and that, in addition, this point \(\tilde{x}\) lies on the boundary \(\partial C\) of C. Then \(\{u(s) \}_{s \in S}\) converges strongly to \(\tilde{x}\) for each almost orbit \(u \in AO_Y (\mathcal {T})\).

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References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

    Clarkson, J.A.: Unifomly convex spaces. Trans. Am. Math. Soc. 78, 396–414 (1936)

    Article  Google Scholar 

  3. 3.

    Day, M.M.: Ergodic theorems for Abelian semigroups. Trans. Am. Math. Soc. 51, 399–412 (1942)

    MathSciNet  MATH  Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

    Edelstein, M.: The construction of an asymptotic center with a fixed point property. Bull. Am. Math. Soc. 78, 206–208 (1972)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  8. 8.

    Goebel, K., Kirk, W.A., Thele, R.L.: Uniformly Lipschitzian families of transformations in Banach spaces. Can. J. Math. 26, 1245–1256 (1974)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    Google Scholar 

  10. 10.

    Grzesik, A., Kaczor, W., Kuczumow, T., Reich, S.: Convergence of iterates of nonexpansive mappings and orbits of nonexpansive semigroups. J. Math. Anal. Appl. 475, 519–531 (2019)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Grzesik, A., Kaczor, W., Kuczumow, T., Reich, S.: Means and convergence of semigroup orbits. Fixed Point Theory 21, 495–506 (2020)

    Article  Google Scholar 

  12. 12.

    Kim, K.J.: Asymptotic behavior of an asymptotically nonexpansive semigroup in Banach spaces with Opial’s condition. Commun. Korean Math. Soc. 12, 237–250 (1997)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Kirk, W.A.: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 17, 339–346 (1974)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kobayashi, K., Miyadera, I.: On the asymptotic behavior of almost-orbits of nonlinear contraction semigroups in Banach spaces. Nonlinear Anal. 6, 349–365 (1982)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lovaglia, A.R.: Locally uniformly convex Banach spaces. Trans. Am. Math. Soc. 78, 225–238 (1955)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Park, J.Y., Takahashi, W.: On the asymptotic behavior of almost-orbits of commutative semigroups in Banach spaces. In: Lin, B.-L., Simons, S. (eds.) Nonlinear and Convex Analysis, pp. 271–293. Marcel Dekker, New York (1987)

    Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

    Takahashi, W.: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama (2000)

    Google Scholar 

  19. 19.

    Zanco, C., Zucchi, A.: Moduli of rotundity and smoothness for convex bodies. Boll. Un. Mat. Ital 7, 833–855 (1993)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Zhu, L., Huang, Q., Li, G.: Nonlinear ergodic theorems and weak convergence theorems for reversible semigroup of asymptotically nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2013, 231 (2013)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The third author was partially supported by the Israel Science Foundation (Grant Nos. 389/12 and 820/17), the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. All the authors are grateful to an anonymous referee for his/her useful comments and helpful suggestions.

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Correspondence to Simeon Reich.

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Kaczor, W., Kuczumow, T. & Reich, S. Convergence of almost orbits of semigroups. Anal.Math.Phys. 11, 51 (2021). https://doi.org/10.1007/s13324-021-00495-3

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Keywords

  • Almost orbit
  • Asymptotically nonexpansive mapping
  • Fixed point
  • Locally uniformly rotund set
  • Semigroup of mappings
  • Uniform convexity

Mathematics Subject Classification

  • 41A65
  • 47H10
  • 47H20