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

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
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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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[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
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