American Institute of Mathematical Sciences

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

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

Byung-Soo Lee 1,

1. 

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

Received  April 2011 Revised  June 2011 Published  September 2011

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 and 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), 277-282. doi: 10.1016/j.amc.2010.05.058.

[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), 707-717. doi: 10.1016/S0898-1221(04)90058-2.

[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), 821-829. doi: 10.1016/j.jmaa.2006.05.068.

[4]

N. J. Huang and Y. J. Cho, Fixed point theorems of compatiable mappings in convex metric spaces, Soochow J. Math., 22 (1996), 439-447.

[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), 2990-2995. doi: 10.1016/j.camwa.2010.02.017.

[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), 1-11. doi: 10.1016/j.jmaa.2007.06.051.

[7]

B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metric spaces, Com. Math. Appl., 61 (2011), 3218-3225. doi: 10.1016/j.camwa.2011.04.017.

[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), 883-889. doi: 10.1016/j.amc.2010.01.096.

[9]

K. Nammanee and S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces, Appl. Math. Comput., 187 (2007), 669-679. doi: 10.1016/j.amc.2006.08.081.

[10]

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229. doi: 10.1006/jmaa.2000.7042.

[11]

M. A. Noor and Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput., 187 (2007), 680-685. doi: 10.1016/j.amc.2006.08.088.

[12]

S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005), 506-517. doi: 10.1016/j.jmaa.2005.03.002.

[13]

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.

[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, Article ID 891967, 12pages.

[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), 2067-2071. doi: 10.1016/j.na.2008.02.106.

[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), 1522-1525. doi: 10.1016/j.amc.2009.07.006.

[17]

B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267 (2002), 444-453. doi: 10.1006/jmaa.2001.7649.

[18]

Y. Yao and M. A. Noor, Convergence of three-step iteration for asymptotically nonexpansive mappings, Appl. Math. Comput., 187 (2007), 883-892. doi: 10.1016/j.amc.2006.09.008.

show all references

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), 277-282. doi: 10.1016/j.amc.2010.05.058.

[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), 707-717. doi: 10.1016/S0898-1221(04)90058-2.

[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), 821-829. doi: 10.1016/j.jmaa.2006.05.068.

[4]

N. J. Huang and Y. J. Cho, Fixed point theorems of compatiable mappings in convex metric spaces, Soochow J. Math., 22 (1996), 439-447.

[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), 2990-2995. doi: 10.1016/j.camwa.2010.02.017.

[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), 1-11. doi: 10.1016/j.jmaa.2007.06.051.

[7]

B. S. Lee, Strong convergence theorems with a Noor-type iterative scheme in convex metric spaces, Com. Math. Appl., 61 (2011), 3218-3225. doi: 10.1016/j.camwa.2011.04.017.

[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), 883-889. doi: 10.1016/j.amc.2010.01.096.

[9]

K. Nammanee and S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces, Appl. Math. Comput., 187 (2007), 669-679. doi: 10.1016/j.amc.2006.08.081.

[10]

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217-229. doi: 10.1006/jmaa.2000.7042.

[11]

M. A. Noor and Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Appl. Math. Comput., 187 (2007), 680-685. doi: 10.1016/j.amc.2006.08.088.

[12]

S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005), 506-517. doi: 10.1016/j.jmaa.2005.03.002.

[13]

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.

[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, Article ID 891967, 12pages.

[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), 2067-2071. doi: 10.1016/j.na.2008.02.106.

[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), 1522-1525. doi: 10.1016/j.amc.2009.07.006.

[17]

B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267 (2002), 444-453. doi: 10.1006/jmaa.2001.7649.

[18]

Y. Yao and M. A. Noor, Convergence of three-step iteration for asymptotically nonexpansive mappings, Appl. Math. Comput., 187 (2007), 883-892. doi: 10.1016/j.amc.2006.09.008.

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