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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 694783, 9 pages
http://dx.doi.org/10.1155/2014/694783
Research Article

Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

Graduate School of Education, Mathematics Education, Kyungnam University, Changwon 631-701, Republic of Korea

Received 9 May 2014; Accepted 1 July 2014; Published 20 July 2014

Academic Editor: Jong Kyu Kim

Copyright © 2014 Kyung Soo Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. T. Lau, H. Miyake, and W. Takahashi, “Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 4, pp. 1211–1225, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. S. Reich, “A weak convergence theorem for the alternating method with Bregman distance,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., pp. 313–318, Marcel Dekker, New York, NY, USA, 1996.
  6. D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. D. Butnariu, S. Reich, and A. J. Zaslavski, “Weak convergence of orbits of nonlinear operators in reflexive Banach spaces,” Numerical Functional Analysis and Optimization, vol. 24, no. 5-6, pp. 489–508, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. Y. Censor and S. Reich, “Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization,” Optimization, vol. 37, no. 4, pp. 323–339, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. K. S. Kim, “Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 10, pp. 3413–3419, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. W. Takahashi, Convex Analysis and Approximation Fixed Points, Yokohama Publishers, 2000, (Japanese).
  12. Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A.G. Kartsatos, Ed., vol. 178, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996. View at MathSciNet
  13. Y. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,” Panamerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994. View at MathSciNet
  14. I. Cioranescu, Geometry of Banach Spaces, Duality Mapping and Nonlinear Problems, Kluwer Academic, Amsterdam, The Netherlands, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  15. J. I. Kang, “Fixed points of non-expansive mappings associated with invariant means in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 11, pp. 3316–3324, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. K. S. Kim, “Ergodic theorems for reversible semigroups of nonlinear operators,” Journal of Nonlinear and Convex Analysis, vol. 13, no. 1, pp. 85–95, 2012. View at MathSciNet
  17. A. T. Lau, “Invariant means and fixed point properties of semigroup of nonexpansive mappings,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1525–1542, 2008. View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. A. T. M. Lau, “Semigroup of nonexpansive mappings on a Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 105, no. 2, pp. 514–522, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. A. T. M. Lau and W. Takahashi, “Invariant means and fixed point properties for non-expansive representations of topological semigroups,” Topological Methods in Nonlinear Analysis, vol. 5, no. 1, pp. 39–57, 1995. View at MathSciNet
  20. M. M. Day, “Amenable semigroups,” Illinois Journal of Mathematics, vol. 1, pp. 509–544, 1957. View at Zentralblatt MATH · View at MathSciNet
  21. J. I. Kang, “Fixed point set of semigroups of non-expansive mappings and amenability,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1445–1456, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. A. T. Lau, N. Shioji, and W. Takahashi, “Existence of nonexpansive retractions for amenable semigroups of nonexpansive mappings and nonlinear ergodic theorems in Banach spaces,” Journal of Functional Analysis, vol. 161, no. 1, pp. 62–75, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. R. D. Holmes and A. T. Lau, “Non-expansive actions of topological semigroups and fixed points,” Journal of the London Mathematical Society, vol. 5, no. 2, pp. 330–336, 1972. View at MathSciNet
  24. K. S. Kim, “Nonlinear ergodic theorems of nonexpansive type mappings,” Journal of Mathematical Analysis and Applications, vol. 358, no. 2, pp. 261–272, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. A. T.-M. Lau, “Invariant means on almost periodic functions and fixed point properties,” Rocky Mountain Journal of Mathematics, vol. 3, pp. 69–76, 1973. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. A. T.-M. Lau and P. F. Mah, “Fixed point property for Banach algebras associated to locally compact groups,” Journal of Functional Analysis, vol. 258, no. 2, pp. 357–372, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. A. T. M. Lau and W. Takahashi, “Weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mappings,” Pacific Journal of Mathematics, vol. 126, no. 2, pp. 277–294, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  28. A. T. Lau and Y. Zhang, “Fixed point properties of semigroups of non-expansive mappings,” Journal of Functional Analysis, vol. 254, no. 10, pp. 2534–2554, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. N. Hirano, K. Kido, and W. Takahashi, “Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 12, no. 11, pp. 1269–1281, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. S. Saeidi, “Existence of ergodic retractions for semigroups in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3417–3422, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. W. Takahashi, “A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space,” Proceedings of the American Mathematical Society, vol. 81, no. 2, pp. 253–256, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  32. S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. S. Saeidi, “Strong convergence of Browder's type iterations for left amenable semigroups of Lipschitzian mappings in BANach spaces,” Journal of Fixed Point Theory and Applications, vol. 5, no. 1, pp. 93–103, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  34. F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,” SIAM Journal on Optimization, vol. 19, no. 2, pp. 824–835, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. F. Kohsaka and W. Takahashi, “Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces,” Archiv der Mathematik, vol. 91, no. 2, pp. 166–177, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  36. W. Nilsrakoo and S. Saejung, “Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 14, pp. 6577–6586, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus