Strong convergence theorems with three-step iteration in star-shaped metric spaces

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2011, 1(3): 371-379. doi: 10.3934/naco.2011.1.371

Strong convergence theorems with three-step iteration in star-shaped metric spaces

1.	Department of Mathematics, Kyungsung University, Busan 608-736, South Korea

Received April 2011 Revised June 2011 Published September 2011

Abstract
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The author considers a Noor-type three-step iterative scheme including Ishikawa-type scheme as a special case to approximate common fixed points of an infinite family of uniformly quasi-Lipschitzian mappings and an infinite family of nonexpansive mappings in star-shaped metric spaces. His results are cases of star-shaped metric space of results shown in [7].

Keywords: Noor-type iteration, Star-shaped metric space, uniformly quasi-$\sup(f)$-Lipschitzian mapping., $f$-expansive mapping.

Mathematics Subject Classification: 47H10, 47H17, 49J4.

Citation: Byung-Soo Lee. Strong convergence theorems with three-step iteration in star-shaped metric spaces. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 371-379. doi: 10.3934/naco.2011.1.371

References:

[1]	S. S. Chang, L. Yang and X. R. Wang, Stronger convergence theroem 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
[2]	Y. J. Cho, H. Y. Zhou and G. Guo, Weak and strong convengence theorems for three-step iteration with errors for asymptotically nonexpansive mappings,, Comput. Math. Appl., 47 (2004), 707. doi: 10.1016/S0898-1221(04)90058-2. Google Scholar
[3]	Hafiz Fukhar-ud-din and Safeer Hussin 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
[4]	N. J. Huang and Y. J. Cho, Fixed point theorems of compatiable mappings in convex metric spaces,, Soochow J. Math., 22 (1996), 439. Google Scholar
[5]	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
[6]	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
[7]	B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metric spaces,, Com. Math. Appl., 61 (2011), 3218. doi: 10.1016/j.camwa.2011.04.017. Google Scholar
[8]	Q. Y. Liu, Z. B. Liu and N. J. Huang, Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces,, Appl. Math. Comput., 216 (2010), 883. doi: 10.1016/j.amc.2010.01.096. Google Scholar
[9]	K. Nammanee and S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces,, Appl. Math. Comput., 187 (2007), 669. doi: 10.1016/j.amc.2006.08.081. Google Scholar
[10]	M. A. Noor, New approximation schemes for general variational inequalities,, J. Math. Anal. Appl., 251 (2000), 217. doi: 10.1006/jmaa.2000.7042. Google Scholar
[11]	M. A. Noor and Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities,, Appl. Math. Comput., 187 (2007), 680. doi: 10.1016/j.amc.2006.08.088. Google Scholar
[12]	S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings,, J. Math. Anal. Appl., 311 (2005), 506. doi: 10.1016/j.jmaa.2005.03.002. Google Scholar
[13]	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
[14]	Y. X. Tian and C. D. Yang, Convergence theorems of three-step iterative scheme for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces,, Fixed Point Theory and Applications vol. 2009, (2009). Google Scholar
[15]	C. Wang and L. W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschizian mappings in convex metric spaces,, Nonlinear Anal. TMA, 70 (2009), 2067. doi: 10.1016/j.na.2008.02.106. Google Scholar
[16]	C. Wang, J. H. Zhu, B. Damjanovic and L. G. Hu, Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces,, Appl. Math. Comput., 215 (2009), 1522. doi: 10.1016/j.amc.2009.07.006. Google Scholar
[17]	B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces,, J. Math. Anal. Appl., 267 (2002), 444. doi: 10.1006/jmaa.2001.7649. Google Scholar
[18]	Y. Yao and M. A. Noor, Convergence of three-step iteration for asymptotically nonexpansive mappings,, Appl. Math. Comput., 187 (2007), 883. doi: 10.1016/j.amc.2006.09.008. Google Scholar

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References:

[1]	S. S. Chang, L. Yang and X. R. Wang, Stronger convergence theroem 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
[2]	Y. J. Cho, H. Y. Zhou and G. Guo, Weak and strong convengence theorems for three-step iteration with errors for asymptotically nonexpansive mappings,, Comput. Math. Appl., 47 (2004), 707. doi: 10.1016/S0898-1221(04)90058-2. Google Scholar
[3]	Hafiz Fukhar-ud-din and Safeer Hussin 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
[4]	N. J. Huang and Y. J. Cho, Fixed point theorems of compatiable mappings in convex metric spaces,, Soochow J. Math., 22 (1996), 439. Google Scholar
[5]	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
[6]	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
[7]	B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metric spaces,, Com. Math. Appl., 61 (2011), 3218. doi: 10.1016/j.camwa.2011.04.017. Google Scholar
[8]	Q. Y. Liu, Z. B. Liu and N. J. Huang, Approximating the common fixed points of two sequences of uniformly quasi-Lipschitzian mappings in convex metric spaces,, Appl. Math. Comput., 216 (2010), 883. doi: 10.1016/j.amc.2010.01.096. Google Scholar
[9]	K. Nammanee and S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces,, Appl. Math. Comput., 187 (2007), 669. doi: 10.1016/j.amc.2006.08.081. Google Scholar
[10]	M. A. Noor, New approximation schemes for general variational inequalities,, J. Math. Anal. Appl., 251 (2000), 217. doi: 10.1006/jmaa.2000.7042. Google Scholar
[11]	M. A. Noor and Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities,, Appl. Math. Comput., 187 (2007), 680. doi: 10.1016/j.amc.2006.08.088. Google Scholar
[12]	S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings,, J. Math. Anal. Appl., 311 (2005), 506. doi: 10.1016/j.jmaa.2005.03.002. Google Scholar
[13]	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
[14]	Y. X. Tian and C. D. Yang, Convergence theorems of three-step iterative scheme for a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces,, Fixed Point Theory and Applications vol. 2009, (2009). Google Scholar
[15]	C. Wang and L. W. Liu, Convergence theorems for fixed points of uniformly quasi-Lipschizian mappings in convex metric spaces,, Nonlinear Anal. TMA, 70 (2009), 2067. doi: 10.1016/j.na.2008.02.106. Google Scholar
[16]	C. Wang, J. H. Zhu, B. Damjanovic and L. G. Hu, Approximating fixed points of a pair of contractive type mappings in generalized convex metric spaces,, Appl. Math. Comput., 215 (2009), 1522. doi: 10.1016/j.amc.2009.07.006. Google Scholar
[17]	B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces,, J. Math. Anal. Appl., 267 (2002), 444. doi: 10.1006/jmaa.2001.7649. Google Scholar
[18]	Y. Yao and M. A. Noor, Convergence of three-step iteration for asymptotically nonexpansive mappings,, Appl. Math. Comput., 187 (2007), 883. doi: 10.1016/j.amc.2006.09.008. Google Scholar

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