A note on the split common fixed-point problem for quasi-nonexpansive operators. (English) Zbl 1232.49017

Moudafi, A.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 12, 4083-4087 (2011).

Summary: Based on the very recent work by Y. Censor and A. Segal [“The split common fixed point problem for directed operators”, J. Convex Anal. 16, No. 2, 587-600 (2009; Zbl 1189.65111)] and inspired by H.-K. Xu [“A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem”, Inverse Probl. 22, No. 6, 2021-2034 (2006; Zbl 1126.47057)], J. Zhao and Q. Yang [“Several solution methods for the split feasibility problem”, Inverse Probl. 21, No. 5, 1791-1799 (2005; Zbl 1080.65035)], and H.H. Bauschke and P. L. Combettes [“A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces”, Math. Oper. Res. 26, No. 2, 248-264 (2001; Zbl 1082.65058)], we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms.

MSC:

49J53	Set-valued and variational analysis
65K10	Numerical optimization and variational techniques
49M37	Numerical methods based on nonlinear programming
90C25	Convex programming

Keywords:

feasibility problem; quasi-nonexpansive operator; Fejér monotonicity

Citations:

Zbl 1189.65111; Zbl 1126.47057; Zbl 1080.65035; Zbl 1082.65058

References:

[1]	Censor, Y.; Segal, A., The split common fixed point problem for directed operators, J. Convex Anal., 16, 587-600 (2009) · Zbl 1189.65111
[2]	Bauschke, H. H.; Combettes, P. L., A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. Oper. Res., 26, 2, 248-264 (2001) · Zbl 1082.65058
[3]	Yamada, I., Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optim., 25, 619-655 (2004) · Zbl 1095.47049
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[7]	Bauschke, H. H.; Borwein, J. M., On projection algorithms for solving convex feasibility problems, SIAM Rev., 38, 3, 367-426 (1996) · Zbl 0865.47039
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[9]	Xu, H. K., A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22, 2021-2034 (2006) · Zbl 1126.47057
[10]	Zhao, J.; Yang, Q., Several solution methods for the split feasibility problem, Inverse Problems, 21, 5, 1791-1799 (2005) · Zbl 1080.65035