Cyclic hybrid methods for finding common fixed points of a finite family of nonexpansive mappings

Volume 9, Issue 5, pp 2000--2005 http://dx.doi.org/10.22436/jnsa.009.05.06

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Authors

Qiao-Li Dong - College of Science, Civil Aviation University of China, Tianjin 300300, China. - Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China. Yan-Yan Lu - College of Science, Civil Aviation University of China, Tianjin 300300, China. Jinfeng Yang - Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China.

Abstract

In this paper, we propose a cyclic hybrid method for computing a common fixed point of a finite family of nonexpansive mappings. The strong convergence of the method is established. Numerical examples illustrate that the proposed method has an advantage in computing.

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Keywords

• Common fixed point
• hybrid method
• cyclic computation
• nonexpansive mapping.

MSC

•  47H05
•  47H07
•  47H10

References

• [1] P. K. Anh, C. V. Chung , Parallel hybrid methods for a finite family of relatively nonexpansive mappings, Numer. Func. Anal. Optim., 35 (2014), 649-664.


• [2] H. H. Bauschke, P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, Berlin (2011)


• [3] L. C. Ceng, N. C. Wong, J. C. Yao , Strong and weak convergence theorems for an infinite family of nonexpansive mappings and applications, Fixed Point Theory Appl., 2012 (2012), 21 pages.


• [4] Y. Censor, T. Elfving , A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.


• [5] Y. Censor, T. Elfving, N. Kopf, T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems , Inverse Problems, 21 (2005), 2071-2084.


• [6] J. P. Chancelier , Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces , J. Math. Anal. Appl., 353 (2009), 141-153.


• [7] Q. L. Dong, Y. Y. Lu , A new hybrid algorithm for a nonexpansive mapping, Fixed Point Theory Appl., 2015 (2015), 7 pages.


• [8] Q. L. Dong, H. B. Yuan, Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping , Fixed Point Theory Appl., 2015 (2015), 12 pages.


• [9] S. He, C. Yang, P. Duan , Realization of the hybrid method for Mann iterations , Appl. Math. Comput., 217 (2010), 4239-4247.


• [10] K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)


• [11] C. Martinez-Yanes, H. K. Xu , Strong convergence of the CQ method for fixed point iteration processes , Nonlinear Anal., 64 (2006), 2400-2411.


• [12] K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.


• [13] K. Nammanee, R. Wangkeeree, New iterative approximation methods for a countable family of nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2011 (2011), 24 pages.


• [14] W. Nilsrakoo, S. Saejung, Weak and strong convergence theorems for countable Lipschitzian mappings and its applications, Nonlinear Anal., 69 (2008), 2695-2708.


• [15] M. V. Solodov, B. F. Svaiter, Forcing strong convergence of proximal point iterations in Hilbert space , Math. Program, 87 (2000), 189-202.


• [16] W. Takahashi , Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces , Nonlinear Anal., 70 (2009), 719-734.


• [17] W. Takahashi, Y. Takeuchi, R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341 (2008), 276-286.


• [18] L. Wei, Y. J. Cho, H. Y. Zhou, A strong convergence theorem for common fixed points of two relatively nonexpansive mappings and its applications, J. Appl. Math. Comput., 29 (2009), 95-103.


• [19] Y. Yao, Y. C. Liou, N. C. Wong, Iterative algorithms based on the implicit midpoint rule for nonexpansive mappings, J. Nonlinear Convex A., (preprint),


• [20] Y. Yao, M. Postolache, Y. C. Liou, Z. Yaz , Construction algorithms for a class of monotone variational inequalities, Optim. Lett., (in press),


• [21] H. Zhou, Y. Su , Strong convergence theorems for a family of quasi-asymptotic pseudo-contractions in Hilbert spaces, Nonlinear Anal., 70 (2009), 4047-4052.