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http://dx.doi.org/10.7858/eamj.2011.27.1.035

CONVERGENCE THEOREMS OF A FINITE FAMILY OF ASYMPTOTICALLY QUASI-NONEXPANSIVE TYPE MAPPINGS IN BANACH SPACES  

Saluja, Gurucharan Singh (Department of Mathematics & Information Technology Govt. Nagarjuna P. G. College of Science)
Publication Information
East Asian mathematical journal / v.27, no.1, 2011 , pp. 35-49 More about this Journal
Abstract
In this paper, we study multi-step iterative algorithm with errors and give the necessary and sufficient condition to converge to com mon fixed points for a finite family of asymptotically quasi-nonexpansive type mappings in Banach spaces. Also we have proved a strong convergence theorem to converge to common fixed points for a finite family said mappings on a nonempty compact convex subset of a uniformly convex Banach spaces. Our results extend and improve the corresponding results of [2, 4, 7, 8, 9, 10, 12, 15, 20].
Keywords
Asymptotically quasi-nonexpansive type mapping; common fixed point; multi-step iterative algorithm with errors; uniformly convex Banach space; uniformly (L, ${\alpha}$)-Lipschitz mapping; strong convergence;
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1 D. R. Sahu, S. C. Shrivastava and B. L. Malager, Approximation of common fixed points of a family of asymptotically quasi-nonexpansive mappings, Demonstratio Math. 41 (2008), no. 3, 625-632.
2 N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, Vol.2006, Article ID 18909, pages 1-10.
3 J. Schu, Weak and strong convergence theorems to fixed points of asymptotically non-expansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153-159.   DOI
4 K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301-308.   DOI   ScienceOn
5 K. K. Tan and H. K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994), 733-739.   DOI   ScienceOn
6 Y. C. Tang and J. G. Peng, Approximation of common fixed points for a finite family of uniformly quasi-Lipschitzian mappings in Banach spaces, Thai. J. Maths. 8 (2010), no. 1, 63-70.
7 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   ScienceOn
8 S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.   DOI   ScienceOn
9 A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points of Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 341 (2008), 1-11.   DOI   ScienceOn
10 Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 259 (2001), 1-7.   DOI   ScienceOn
11 Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl. 259 (2001), 18-24.   DOI   ScienceOn
12 Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member of uniformly convex Banach spaces, J. Math. Anal. Appl. 266 (2002), 468-471.   DOI   ScienceOn
13 W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.   DOI   ScienceOn
14 W. V. Petryshyn and T. E. Williamson, Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43 (1973), 459-497.   DOI
15 J. Quan, S. S. Chang and X. J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive-type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications Volume 2006, Article ID 70830, pages 1-8.
16 M. K. Ghosh and L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl. 207 (1997), 96-103.   DOI   ScienceOn
17 R. E. Bruck, T. Kuczumow and S. Reich, Convergence of iterates of asymptotically non-expansive mappings in Banach spaces with the uniform Opial property, Colloq. Math. 65 (1993), 169-179.   DOI
18 C. E. Chidume and Bashir Ali, Convergence theorems for finite families of asymptotically quasi-nonexpansive mappings, J. Inequalities and Applications, Vol.2007, Article ID 68616, 10 pages.
19 C. E. Chidume and E. U. Ofoedu, Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings, J. Math. Anal. Appl. 333 (2007), 128-141.   DOI   ScienceOn
20 K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.   DOI   ScienceOn