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Viscosity iteration methods for a split feasibility problem and a mixed equilibrium problem in a Hilbert space. (English) Zbl 1292.47044

Summary: In this paper, we consider and analyze two viscosity iteration algorithms (one implicit and one explicit) for finding a common element of the solution set \(\mathrm{MEP}(F_1, F_2)\) of a mixed equilibrium problem and the set \(\Gamma\) of a split feasibility problem in a real Hilbert space. Furthermore, we derive the strong convergence of a viscosity iteration algorithm to an element of \(\mathrm{MEP}(F_1, F_2) \cap \Gamma\) under mild assumptions.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
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[1] Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space.Numer. Algorithms 1994, 8:221-239. · Zbl 0828.65065 · doi:10.1007/BF02142692
[2] Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem.Inverse Probl. 2002, 18:441-453. · Zbl 0996.65048 · doi:10.1088/0266-5611/18/2/310
[3] Byrne C: A unified treatment of some iterative algorithms in signal processing and image reconstruction.Inverse Probl. 2004, 18:103-120. · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[4] Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications for inverse problems.Inverse Probl. 2005, 21:2071-2084. · Zbl 1089.65046 · doi:10.1088/0266-5611/21/6/017
[5] Lopez, G.; Martin, V.; Xu, HK; Censor, Y. (ed.); Jiang, M. (ed.); Wang, G. (ed.), Iterative algorithms for the multiple-sets split feasibility problem, 243-279 (2009), Madison
[6] Qu B, Xiu N: A note on the CQ algorithm for the split feasibility problem.Inverse Probl. 2005, 21:1655-1665. · Zbl 1080.65033 · doi:10.1088/0266-5611/21/5/009
[7] Wang, F.; Xu, HK, Cyclic algorithms for split feasibility problems in Hilbert spaces (2011) · Zbl 1308.47079
[8] Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem.Inverse Probl. 2006, 22:2021-2034. · Zbl 1126.47057 · doi:10.1088/0266-5611/22/6/007
[9] Jaiboona C, Kumam P: A general iterative method for addressing mixed equilibrium problems and optimization problems.Nonlinear Anal. 2010, 73:1180-1202. · Zbl 1205.49011 · doi:10.1016/j.na.2010.04.041
[10] Kumam, P.; Jaiboon, C., Approximation of common solutions to system of mixed equilibrium problems, variational inequality problem, and strict pseudo-contractive mappings, No. 2011 (2011) · Zbl 1215.47075
[11] Saewan S, Kumam P: A modified hybrid projection method for solving generalized mixed equilibrium problems and fixed point problems in Banach spaces.Comput. Math. Appl. 2011, 62:1723-1735. · Zbl 1232.47049 · doi:10.1016/j.camwa.2011.06.014
[12] Saewan S, Kumam P: Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-asymptotically nonexpansive mappings.Appl. Math. Comput. 2011, 218:3522-3538. · Zbl 1246.65085 · doi:10.1016/j.amc.2011.08.099
[13] Xie DP: Auxiliary principle and iterative algorithm for a new system of generalized mixed equilibrium problems in Banach spaces.Appl. Math. Comput. 2011, 218:3507-3514. · Zbl 1259.65082 · doi:10.1016/j.amc.2011.08.097
[14] Yao YH, Noor MA, Lioud YC, Kang SM: Some new algorithms for solving mixed equilibrium problems.Comput. Math. Appl. 2010, 60:1351-1359. · Zbl 1201.65116 · doi:10.1016/j.camwa.2010.06.016
[15] Geobel K, Kirk WA: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.[Cambridge Studies in Advanced Mathematics 28] · doi:10.1017/CBO9780511526152
[16] Cianciaruso, F.; Marino, G.; Muglia, L.; Hong, Y., A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, No. 2010 (2010) · Zbl 1203.47043
[17] Xu, HK, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, No. 26 (2010)
[18] Nadezhkina N, Takahashi W: Weak convergence theorem by an extra gradient method for nonexpansive mappings and monotone mappings.J. Optim. Theory Appl. 2006, 128:191-201. · Zbl 1130.90055 · doi:10.1007/s10957-005-7564-z
[19] Browder FE: Fixed point theorems for noncompact mappings in Hilbert spaces.Proc. Natl. Acad. Sci. USA 1965, 43:1272-1276. · Zbl 0125.35801 · doi:10.1073/pnas.53.6.1272
[20] Xu HK: Iterative algorithms for nonlinear operators.J. Lond. Math. Soc. 2002, 2:1-17.
[21] Xu HK: Viscosity approximation methods for nonexpansive mappings.J. Math. Anal. Appl. 2004, 298:279-291. · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[22] Yao, YH; Chen, RD; Marino, G.; Liou, YC, Applications of fixed-point and optimization methods to the multiple-set split feasibility problem, No. 2012 (2012) · Zbl 1245.49051
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