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

Dong, Qiao-Li¹ ; Lu, Yan-Yan² ; Yang, Jinfeng³

¹ 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
² College of Science, Civil Aviation University of China, Tianjin 300300, China
³ Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Journal of nonlinear sciences and its applications, Tome 9 (2016) no. 5, p. 2000-2005.

Résumé

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.

Détail

DOI : 10.22436/jnsa.009.05.06

Classification : 47H05, 47H07, 47H10
Keywords: Common fixed point, hybrid method, cyclic computation, nonexpansive mapping.

Bibliographie

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

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

[3] Ceng, L. C.; Wong, N. C.; J. C. Yao Strong and weak convergence theorems for an infinite family of nonexpansive mappings and applications, Fixed Point Theory Appl., Volume 2012 (2012), pp. 1-21

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

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

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

[7] Dong, Q. L.; Y. Y. Lu A new hybrid algorithm for a nonexpansive mapping, Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-7

[8] Dong, Q. L.; Yuan, H. B. Accelerated Mann and CQ algorithms for finding a fixed point of a nonexpansive mapping , Fixed Point Theory Appl., Volume 2015 (2015), pp. 1-12

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

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

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

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

[13] Nammanee, K.; Wangkeeree, R. New iterative approximation methods for a countable family of nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., Volume 2011 (2011), pp. 1-24

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

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

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

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

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

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

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

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