## Abstract and Applied Analysis

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

Kyung Soo Kim

#### Abstract

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappings $\mathfrak{I}=\{T(s):s\in S\}$ on a nonempty closed convex subset $C$ of a Banach space with respect to a sequence of asymptotically left invariant means $\{{\mu }_{n}\}$ defined on an appropriate invariant subspace of ${l}^{\mathrm{\infty }}(S)$, where $S$ is a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed points $F(\mathfrak{I})$, where $F(\mathfrak{I})=\bigcap \{F(T(s)):s\in S\}$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 694783, 9 pages.

Dates
First available in Project Euclid: 27 February 2015

https://projecteuclid.org/euclid.aaa/1425048210

Digital Object Identifier
doi:10.1155/2014/694783

Mathematical Reviews number (MathSciNet)
MR3246353

Zentralblatt MATH identifier
07022895

#### Citation

Kim, Kyung Soo. Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 694783, 9 pages. doi:10.1155/2014/694783. https://projecteuclid.org/euclid.aaa/1425048210

#### References

• B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967.
• W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
• S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• W. Takahashi, Convex Analysis and Approximation Fixed Points, Yokohama Publishers, 2000, (Japanese).
• 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.
• 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.
• I. Cioranescu, Geometry of Banach Spaces, Duality Mapping and Nonlinear Problems, Kluwer Academic, Amsterdam, The Netherlands, 1990.
• 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.
• 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.
• 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.
• 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.
• 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.
• M. M. Day, “Amenable semigroups,” Illinois Journal of Mathematics, vol. 1, pp. 509–544, 1957.
• 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.
• 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.
• 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.
• K. S. Kim, “Nonlinear ergodic theorems of nonexpansive type mappings,” Journal of Mathematical Analysis and Applications, vol. 358, no. 2, pp. 261–272, 2009.
• 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.
• 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.
• 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.
• 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.
• 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.
• S. Saeidi, “Existence of ergodic retractions for semigroups in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 10, pp. 3417–3422, 2008.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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. \endinput