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A new approach to solving split equality problems in Hilbert spaces. (English) Zbl 07632504

Summary: Using a product space approach, we propose and study several iterative methods for solving certain types of split equality problems in Hilbert spaces.

MSC:

47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
49J53 Set-valued and variational analysis
90C25 Convex programming
Full Text: DOI

References:

[1] Censor, Y.; Elfving, T., A multi projection algorithm using Bregman projections in a product space, Numer Algor, 8, 221-239 (1994) · Zbl 0828.65065
[2] Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl, 18, 441-453 (2002) · Zbl 0996.65048
[3] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl, 18, 103-120 (2004) · Zbl 1051.65067
[4] Butnariu, D.; Resmerita, E., Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr Appl Anal, 2006, 1-39 (2006) · Zbl 1130.47046
[5] Byrne, C.; Censor, Y.; Gibali, A., The split common null point problem, J Nonlinear Convex Anal, 13, 759-775 (2012) · Zbl 1262.47073
[6] Ceng, LC; Ansari, QH; Yao, JC., An extragradient method for solving split feasibility and fixed point problems, Comput Math Appl, 64, 633-642 (2012) · Zbl 1252.65102
[7] Ceng, LC; Wong, NC; Yao, JC., Hybrid extragradient methods for finding minimum-norm solutions of split feasibility problems, J Nonlinear Convex Anal, 16, 1965-1983 (2015) · Zbl 1338.47082
[8] Ceng, LC; Wong, MM; Yao, JC., A hybrid extragradient-like approximation method with regularization for solving split feasibility and fixed point problems, J Nonlinear Convex Anal, 14, 163-182 (2013) · Zbl 1301.49021
[9] Ceng, LC; Ansari, QH; Yao, JC., Relaxed extragradient methods for finding minimum-norm solutions ofthe split feasibility problem, Nonlinear Anal, 75, 2116-2125 (2012) · Zbl 1236.47066
[10] Ceng, LC; Yao, JC., Relaxed and hybrid viscosity methods for general system of variational inequalities with split feasibility problem constraint, Fixed Point Theory Appl, 2013 (2013) · Zbl 1355.49006
[11] Censor, Y.; Elfving, T.; Kopf, N., The multiple-sets split feasibility problem and its application, Inverse Probl, 21, 2071-2084 (2005) · Zbl 1089.65046
[12] Censor, Y.; Gibali, A.; Reich, S., Algorithms for the split variational inequality problems, Numer Algor, 59, 301-323 (2012) · Zbl 1239.65041
[13] Chen, JZ; Ceng, LC; Qiu, YQ, Extra-gradient methods for solving split feasibility and fixed point problems, Fixed Point Theory Appl, 2015 (2015) · Zbl 1346.47035
[14] Cui, HH; Ceng, LC., Iterative solutions of the split common fixed point problem for strictly pseudocontractive mappings, J Fixed Point Theory Appl, 20, 2 (2018) · Zbl 1490.65101
[15] Eslamian, M.; Eslamian, P., Strong convergence of a split common fixed point problem, Numer Funct Anal Optim, 37, 10, 1248-1266 (2016) · Zbl 1356.49026
[16] Guan, JL; Ceng, LC; Hu, B., Strong convergence theorem for split monotone variational inclusion with constraints of variational inequalities and fixed point problems, J Inequal Appl, 2018 (2018) · Zbl 1498.47122
[17] Masad, E.; Reich, S., A note on the multiple-set split convex feasibility problem in Hilbert space, J Nonlinear Convex Anal, 8, 367-371 (2007) · Zbl 1171.90009
[18] Reich, S.; Truong, MT; Mai, TNH., The split feasibility problem with multiple output sets in Hilbert spaces, Optim Lett, 14, 2335-2353 (2020) · Zbl 1460.90201
[19] Reich. T. M. Tuyen, S., Iterative methods for solving the generalized split common null point problem in Hilbert spaces, Optimization, 69, 1013-1038 (2020) · Zbl 1445.47043
[20] Reich. T. M. Tuyen, S.; Trang, NM., Parallel iterative methods for solving the split common fixed point problem in Hilbert spaces, Numer Funct Anal Optim, 41, 778-805 (2020) · Zbl 1442.47064
[21] Reich. T. M. Tuyen, S., Two projection methods for solving the multiple-set split common null point problem in Hilbert spaces, Optimization, 69, 1913-1934 (2020) · Zbl 07249879
[22] Reich, S.; Tuyen, TM., A new algorithm for solving the split common null point problem in Hilbert spaces, Numer Algor, 83, 789-805 (2020) · Zbl 1513.47128
[23] Reich, S.; Tuyen, TM., Two new self-adaptive algorithms for solving the split common null point problem with multiple output sets in Hilbert spaces, J Fixed Point Theory Appl, 23 (2021) · Zbl 1521.47109
[24] Takahashi, S.; Takahashi, W., The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65, 281-287 (2016) · Zbl 1338.47110
[25] Takahashi, W., The split feasibility problem and the shrinking projection method in Banach spaces, J Nonlinear Convex Anal, 16, 1449-1459 (2015) · Zbl 1343.47074
[26] Takahashi, W., The split common null point problem in Banach spaces, Arch Math, 104, 357-365 (2015) · Zbl 1458.47034
[27] Wang, F.; Xu, H-K., Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal, 74, 4105-4111 (2011) · Zbl 1308.47079
[28] Xu, H-K., A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl, 22, 2021-2034 (2006) · Zbl 1126.47057
[29] Xu, H-K., Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces, Inverse Probl, 26 (2010) · Zbl 1213.65085
[30] Yang, Q., The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl, 20, 1261-1266 (2004) · Zbl 1066.65047
[31] Moudafi, A., Alternating CQ-algorithms for convex feasibility and split fixed-point problems, J Nonlinear Convex Anal, 15, 809-818 (2014) · Zbl 1393.47034
[32] Attouch, H.; Redont, P.; Soubeyran, A., A new class of alternating proximal minimization algorithms with costs-to-move, SIAM J Optim, 18, 1061-1081 (2007) · Zbl 1149.65039
[33] Attouch, H.; Bolte, J.; Redont, P., Alternating proximal algorithms for weakly coupled convex minimization problems, applications to dynamical games and PDE’s, J Convex Anal, 15, 485-506 (2008) · Zbl 1154.65044
[34] Attouch, H.; Cabot, A.; Frankel, F., Alternating proximal algorithms for linearly constrained variational inequalities: application to domain decomposition for PDE’s, Nonlinear Anal, 74, 7455-7473 (2011) · Zbl 1228.65100
[35] Censor, Y.; Bortfeld, T.; Martin, B., A unified approach for inversion problems in intensity modulated radiation therapy, Phys Med Biol, 51, 2353-2365 (2006)
[36] Byrne, C, Moudafi, A. Extensions of the CQ algorithm for the split feasibility and split equality problems. hal-00776640-version 1. 2013.
[37] Chang, S-S; Yang, L.; Qin, L., Strongly convergent iterative methods for split equality variational inclusion problems in banach spaces, Acta Math Sci, 36, 1641-1650 (2016) · Zbl 1374.47077
[38] Chidume, CE; Romanus, OM; Nnyaba, UV., An iterative algorithm for solving split equality fixed point problems for a class of nonexpansive-type mappings in Banach spaces, Numer Algor, 82, 987-1007 (2019) · Zbl 07128074
[39] Dong, Q-L; He, S.; Zhao, J., Solving the split equality problem without prior knowledge of operator norms, Optimization, 64, 9, 1887-1906 (2015) · Zbl 1337.47086
[40] Eslamian, M.; Shehu, Y.; Iyiola, OS., A strong convergence theorem for a general split equality problem with applications to optimization and equilibrium problem, Calcolo, 55 (2018) · Zbl 1482.65078
[41] Kazmi, KR; Ali, R.; Furkan, M., Common solution to a split equality monotone variational inclusion problem, a split equality generalized general variational-like inequality problem and a split equality fixed point problem, Fixed Point Theory, 20, 1, 211-232 (2019) · Zbl 1491.47066
[42] Moudafi, A.; Al-Shemas, E., Simultaneous iterative methods for split equality problems and application, Tran Math Prog Appl, 1, 1-11 (2013)
[43] Nnakwe, MO., Solving split generalized mixed equality equilibrium problems and split equality fixed point problems for nonexpansive-type maps, Carpathian J Math, 36, 1, 119-126 (2020) · Zbl 1484.47164
[44] Shehu, Y.; Ogbuisi, FU; Iyiola, OS., Strong convergence theorem for solving split equality fixed point problem which does not involve the prior knowledge of operator norms, Bull Iranian Math Soc, 43, 2, 349-371 (2017) · Zbl 07006700
[45] Ugwunnadi, GC., Iterative algorithm for the split equality problem in Hilbert spaces, J Appl Anal, 22, 1, 81-89 (2016) · Zbl 1345.90109
[46] Vuong, PT; Strodiot, JJ; Nguyen, VH., A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces, Optimization, 64, 11, 2321-2341 (2015) · Zbl 1329.65129
[47] Zegeye, H., The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaces, J Fixed Point Theory Appl, 20 (2018) · Zbl 1491.47074
[48] Zhao, J., Solving split equality fixed point problem of quasi-nonexpansive mappings without prior knowledge of operator norms, Optimizations, 64, 2619-2630 (2015) · Zbl 1326.47104
[49] Zhao, J.; Wang, S., Viscosity approximation methods for the split equality common fixed point problem of quasi-nonexpansive operators, Acta Math Sci, 36B, 5, 1474-1486 (2016) · Zbl 1374.47083
[50] Goebel, K.; Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings (1984), New York: Marcel Dekker, New York · Zbl 0537.46001
[51] Rockafellar, RT., On the maximal monotonicity of subdifferential mappings, Pacific J Math, 33, 209-216 (1970) · Zbl 0199.47101
[52] López, G.; Márquez, MM; Wang, F., Solving the split feasibility problem without prior knowledge of matrix norms, Inverse Probl, 28 (2012) · Zbl 1262.90193
[53] Wang, F., A new iterative method for the split common fixed point problem in Hilbert spaces, Optimization, 66, 3, 407-415 (2017) · Zbl 1365.90250
[54] Moudafi, A., The split common fixed-point problem for demicontractive mappings, Inverse Probl, 26 (2010) · Zbl 1219.90185
[55] Yao, Y.; Liou, Y-C; Postolache, M., Self-adaptive algorithms for the split problem of the demicontractive operators, Optimization, 67, 9, 1309-1319 (2018) · Zbl 1433.90188
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