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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 {\it Y. Censor} and {\it 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 {\it 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)], {\it J. Zhao} and {\it Q. Yang} [“Several solution methods for the split feasibility problem”, Inverse Probl. 21, No. 5, 1791-1799 (2005; Zbl 1080.65035)], and {\it H.H. Bauschke} and {\it 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 Optimization techniques (numerical methods) 49M37 Methods of nonlinear programming type in calculus of variations 90C25 Convex programming
Full Text:
##### 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 · http://www.heldermann.de/JCA/JCA16/JCA162/jca16031.htm [2] Bauschke, H. H.; Combettes, P. L.: 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 · doi:10.1287/moor.26.2.248.10558 [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 · doi:10.1081/NFA-200045815 [4] I. Yamada, Signal processing applications of a pair of simple fixed point algorithms, Approximation and optimization in image restoration and reconstruction, Poquerolles, France, 2009. [5] Maingé, P. -E.: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. math. Appl. 59, No. 1, 74-79 (2010) · Zbl 1189.49011 · doi:10.1016/j.camwa.2009.09.003 [6] Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem, Inverse problems 18, 441-453 (2002) · Zbl 0996.65048 · doi:10.1088/0266-5611/18/2/310 [7] Bauschke, H. H.; Borwein, J. M.: On projection algorithms for solving convex feasibility problems, SIAM rev. 38, No. 3, 367-426 (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710 [8] Maruster, S.; Popirlan, C.: On the Mann-type iteration and convex feasibility problem, J. comput. Appl. math. 212, 390-396 (2008) · Zbl 1135.65027 · doi:10.1016/j.cam.2006.12.012 [9] Xu, H. K.: A variable Krasnoselskii--Mann algorithm and the multiple-set split feasibility problem, Inverse problems 22, 2021-2034 (2006) · Zbl 1126.47057 · doi:10.1088/0266-5611/22/6/007 [10] Zhao, J.; Yang, Q.: Several solution methods for the split feasibility problem, Inverse problems 21, No. 5, 1791-1799 (2005) · Zbl 1080.65035 · doi:10.1088/0266-5611/21/5/017