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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 & 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.  doi: 10.1016/j.na.2006.08.032.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.05.058.  Google Scholar

[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.  doi: 10.1090/S0002-9904-1967-11725-7.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.05.068.  Google Scholar

[5]

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings,, Proc. Amer. Math. Soc., 35 (1972), 171.  doi: 10.1090/S0002-9939-1972-0298500-3.  Google Scholar

[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.  doi: 10.1016/j.camwa.2010.02.017.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.06.051.  Google Scholar

[8]

B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metrix spaces,, Com. Math. Appl., 61 (2011), 3218.  doi: 10.1016/j.camwa.2011.04.017.  Google Scholar

[9]

Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings,, J. Math. Anal. Appl., 259 (2001), 1.  doi: 10.1006/jmaa.2000.6980.  Google Scholar

[10]

W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mapping and its applications,, Nonlinear Analysis, 69 (2008), 2695.  doi: 10.1016/j.na.2007.08.044.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2009.03.002.  Google Scholar

[12]

Y. Song and Y. Zheng, Strong convergence of iteration algorithms for a countable family of nonexpansive mappings,, Nonlinear Analysis, 71 (2009), 3072.  doi: 10.1016/j.na.2009.01.219.  Google Scholar

[13]

W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces,, Nonlinear Analysis, 70 (2009), 719.  doi: 10.1016/j.na.2008.01.005.  Google Scholar

[14]

S. Temir and O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space,, J. Math. Anal. Appl., 329 (2007), 759.  doi: 10.1016/j.jmaa.2006.07.004.  Google Scholar

[15]

Y. X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings,, Comput. Math. Appl., 49 (2005), 1905.  doi: 10.1016/j.camwa.2004.05.017.  Google Scholar

show all references

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.  doi: 10.1016/j.na.2006.08.032.  Google Scholar

[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.  doi: 10.1016/j.amc.2010.05.058.  Google Scholar

[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.  doi: 10.1090/S0002-9904-1967-11725-7.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2006.05.068.  Google Scholar

[5]

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings,, Proc. Amer. Math. Soc., 35 (1972), 171.  doi: 10.1090/S0002-9939-1972-0298500-3.  Google Scholar

[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.  doi: 10.1016/j.camwa.2010.02.017.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.06.051.  Google Scholar

[8]

B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metrix spaces,, Com. Math. Appl., 61 (2011), 3218.  doi: 10.1016/j.camwa.2011.04.017.  Google Scholar

[9]

Q. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings,, J. Math. Anal. Appl., 259 (2001), 1.  doi: 10.1006/jmaa.2000.6980.  Google Scholar

[10]

W. Nilsrakoo and S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mapping and its applications,, Nonlinear Analysis, 69 (2008), 2695.  doi: 10.1016/j.na.2007.08.044.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2009.03.002.  Google Scholar

[12]

Y. Song and Y. Zheng, Strong convergence of iteration algorithms for a countable family of nonexpansive mappings,, Nonlinear Analysis, 71 (2009), 3072.  doi: 10.1016/j.na.2009.01.219.  Google Scholar

[13]

W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces,, Nonlinear Analysis, 70 (2009), 719.  doi: 10.1016/j.na.2008.01.005.  Google Scholar

[14]

S. Temir and O. Gul, Convergence theorem for I-asymptotically quasi-nonexpansive mapping in Hilbert space,, J. Math. Anal. Appl., 329 (2007), 759.  doi: 10.1016/j.jmaa.2006.07.004.  Google Scholar

[15]

Y. X. Tian, Convergence of an Ishikawa type iterative scheme for asymptotically quasi-nonexpansive mappings,, Comput. Math. Appl., 49 (2005), 1905.  doi: 10.1016/j.camwa.2004.05.017.  Google Scholar

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