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2013, 3(3): 557-565. doi: 10.3934/naco.2013.3.557

## A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces

 1 Department of Mathematics, Kyungsung University, Busan 608-736

Received  July 2012 Revised  April 2013 Published  July 2013

In this paper, we introduce a countably infinite iterative scheme and consider a sufficient and necessary condition for the existence of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces.
Citation: Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557
##### References:
 [1] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Analysis, 67 (2007), 2350-2360. doi: 10.1016/j.na.2006.08.032. [2] S. S. Chang, L. Yang and X. R. Wang, Stronger convergence theorem for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comput., 217 (2010), 277-282. doi: 10.1016/j.amc.2010.05.058. [3] S. B. Diaz and F. B. Metcalf, On the structure of the set of sequential limit points of successive approximations, Bull. Amer. Math. Soc., 73 (1967), 516-519. doi: 10.1090/S0002-9904-1967-11725-7. [4] H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829. doi: 10.1016/j.jmaa.2006.05.068. [5] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174. doi: 10.1090/S0002-9939-1972-0298500-3. [6] A. R. Khan and M. A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Com. Math. Appl., 59 (2010), 2990-2995. doi: 10.1016/j.camwa.2010.02.017. [7] A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1-11. doi: 10.1016/j.jmaa.2007.06.051. [8] B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metrix spaces, Com. Math. Appl., 61 (2011), 3218-3225. doi: 10.1016/j.camwa.2011.04.017. [9] Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 259 (2001), 1-7. doi: 10.1006/jmaa.2000.6980. [10] W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mapping and its applications, Nonlinear Analysis, 69 (2008), 2695-2708. doi: 10.1016/j.na.2007.08.044. [11] W. Nilsrakoo and S. Saejung, Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications, J. Math. Anal. Appl., 356 (2009), 154-167. doi: 10.1016/j.jmaa.2009.03.002. [12] Y. Song and Y. Zheng, Strong convergence of iteration algorithms for a countable family of nonexpansive mappings, Nonlinear Analysis, 71 (2009), 3072-3082. doi: 10.1016/j.na.2009.01.219. [13] W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 719-734. doi: 10.1016/j.na.2008.01.005. [14] S. Temir and O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space, J. Math. Anal. Appl., 329 (2007), 759-765. doi: 10.1016/j.jmaa.2006.07.004. [15] Y. X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 49 (2005), 1905-1912. doi: 10.1016/j.camwa.2004.05.017.

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##### References:
 [1] K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Analysis, 67 (2007), 2350-2360. doi: 10.1016/j.na.2006.08.032. [2] S. S. Chang, L. Yang and X. R. Wang, Stronger convergence theorem for an infinite family of uniformly quasi-Lipschitzian mappings in convex metric spaces, Appl. Math. Comput., 217 (2010), 277-282. doi: 10.1016/j.amc.2010.05.058. [3] S. B. Diaz and F. B. Metcalf, On the structure of the set of sequential limit points of successive approximations, Bull. Amer. Math. Soc., 73 (1967), 516-519. doi: 10.1090/S0002-9904-1967-11725-7. [4] H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829. doi: 10.1016/j.jmaa.2006.05.068. [5] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174. doi: 10.1090/S0002-9939-1972-0298500-3. [6] A. R. Khan and M. A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, Com. Math. Appl., 59 (2010), 2990-2995. doi: 10.1016/j.camwa.2010.02.017. [7] A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1-11. doi: 10.1016/j.jmaa.2007.06.051. [8] B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metrix spaces, Com. Math. Appl., 61 (2011), 3218-3225. doi: 10.1016/j.camwa.2011.04.017. [9] Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 259 (2001), 1-7. doi: 10.1006/jmaa.2000.6980. [10] W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mapping and its applications, Nonlinear Analysis, 69 (2008), 2695-2708. doi: 10.1016/j.na.2007.08.044. [11] W. Nilsrakoo and S. Saejung, Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications, J. Math. Anal. Appl., 356 (2009), 154-167. doi: 10.1016/j.jmaa.2009.03.002. [12] Y. Song and Y. Zheng, Strong convergence of iteration algorithms for a countable family of nonexpansive mappings, Nonlinear Analysis, 71 (2009), 3072-3082. doi: 10.1016/j.na.2009.01.219. [13] W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Analysis, 70 (2009), 719-734. doi: 10.1016/j.na.2008.01.005. [14] S. Temir and O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space, J. Math. Anal. Appl., 329 (2007), 759-765. doi: 10.1016/j.jmaa.2006.07.004. [15] Y. X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 49 (2005), 1905-1912. doi: 10.1016/j.camwa.2004.05.017.
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