On the strong convergence of an algorithm about firmly pseudo-demicontractive mappings for the split common fixed-point problem. (English) Zbl 1318.47101

Yu, Yanrong; Sheng, Delei

J. Appl. Math. 2012, Article ID 256930, 9 p. (2012).

Summary: Based on the recent work by Y. Censor and A. Segal [J. Convex Anal. 16, No. 2, 587–600 (2009; Zbl 1189.65111)] and inspired by A. Moudafi [Inverse Probl. 26, No. 5, Article ID 055007, 6 p. (2010; Zbl 1219.90185)], we modify the algorithm of demicontractive operators proposed by Moudafi and study the modified algorithm for the class of firmly pseudodemicontractive operators to solve the split common fixed-point problem in a Hilbert space. We also give a strong convergence theorem under some appropriate conditions. Our work improves and/or develops the work of Moudafi, Censor and Segal, and other results.

MSC:

47J25	Iterative procedures involving nonlinear operators
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

demicontractive operators; pseudodemicontractive operators; split common fixed-point problem; strong convergence

Citations:

Zbl 1189.65111; Zbl 1219.90185

References:

[1]	Y. Censor and T. Elfving, “A multiprojection algorithm using Bregman projections in a product space,” Numerical Algorithms, vol. 8, no. 2-4, pp. 221-239, 1994. · Zbl 0828.65065
[2]	C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Problems, vol. 18, no. 2, pp. 441-453, 2002. · Zbl 0996.65048
[3]	Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, “The multiple-sets split feasibility problem and its applications for inverse problems,” Inverse Problems, vol. 21, no. 6, pp. 2071-2084, 2005. · Zbl 1089.65046
[4]	Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, “A unified approach for inversion problems in intensity-modulated radiation therapy,” Physics in Medicine and Biology, vol. 51, no. 10, pp. 2353-2365, 2006.
[5]	Y. Censor and A. Segal, “The split common fixed point problem for directed operators,” Journal of Convex Analysis, vol. 16, no. 2, pp. 587-600, 2009. · Zbl 1189.65111
[6]	A. Moudafi, “The split common fixed-point problem for demicontractive mappings,” Inverse Problems, vol. 26, no. 5, Article ID 055007, p. 6, 2010. · Zbl 1219.90185
[7]	A. Moudafi, “A note on the split common fixed-point problem for quasi-nonexpansive operators,” Nonlinear Analysis, vol. 74, no. 12, pp. 4083-4087, 2011. · Zbl 1232.49017
[8]	F. Wang and H. K. Xu, “Cyclic algorithms for split feasibility problems in Hilbert spaces,” Nonlinear Analysis, vol. 74, no. 12, pp. 4105-4111, 2011. · Zbl 1308.47079
[9]	D. l. Sheng and R. D. Chen, “On the strong convergence of an algorithm about pseudo-demicontractive mappings for the split common fixed-point problem,” 2011, http://www.paper.edu.cn.
[10]	I. Yamada, “Signal processing applications of a pair of simple fixed-point algorithms,” in Approximation and Optimization in Image Restoration and Reconstruction, Poquerolles, France, 2009.
[11]	P. L. Combettes, “The convex feasibility problem in image recovery,” Advances in Imaging and Electron Physics C, vol. 95, pp. 155-270, 1996.
[12]	K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301-308, 1993. · Zbl 0895.47048