On split null point and common fixed point problems for multivalued demicontractive mappings. (English) Zbl 1539.47112
The authors propose a viscosity iterative method for approximating the solution of split null point and common fixed point problems for maximal monotone operators and multivalued demicontractive mappings in Hilbert spaces. Under some standard assumptions, they prove the strong convergence for the sequence generated by their proposed method and, as an application, study the split feasibility and split minimizations problems in real Hilbert spaces. A numerical example is given to illustrate their method.
Reviewer: Mewomo Oluwatosin Temitope (Durban)
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H04 | Set-valued operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
split null point problem; multivalued demicontractive mapping; strong convergence; viscosity methodReferences:
| [1] | Chidume, CE; Bello, AU; Ndambomve, P., Strong and \(####\)-convergence theorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr Appl Anal, 2014 (2014) · Zbl 1469.65099 · doi:10.1155/2014/805168 |
| [2] | Jailoka, P.; Suantai, S., Split null point problems and fixed point problems for demicontractive multivalued mappings, Mediterr J Math, 15, 1-19 (2018) · Zbl 06997035 · doi:10.1007/s00009-018-1251-4 |
| [3] | Censor, Y.; Elfving, T., A multiprojection algorithm using Bregman projections in a product space, Numer Algorithms, 8, 221-239 (1994) · Zbl 0828.65065 · doi:10.1007/BF02142692 |
| [4] | Byrne, C., Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Probl, 18, 441-453 (2002) · Zbl 0996.65048 · doi:10.1088/0266-5611/18/2/310 |
| [5] | Xu, HK., A variable Krasonosel’skii-Mann algorithm and the multiple-set split feasibility problem, Inverse Probl, 22, 2021-2034 (2006) · Zbl 1126.47057 · doi:10.1088/0266-5611/22/6/007 |
| [6] | Xu, HK., Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces, Inverse Probl, 26 (2010) · Zbl 1213.65085 · doi:10.1088/0266-5611/26/10/105018 |
| [7] | Yang, Q., The relaxed CQ algorithm for solving the split feasibility problem, Inverse Probl, 20, 1261-1266 (2004) · Zbl 1066.65047 · doi:10.1088/0266-5611/20/4/014 |
| [8] | Byrne, C.; Censor, Y.; Gibali, A., The split common null point problem, J Nonlinear Convex Anal, 13, 759-775 (2012) · Zbl 1262.47073 |
| [9] | Takahashi, W., The split common null point problem in Banach spaces, Arch Math, 104, 357-365 (2015) · Zbl 1458.47034 · doi:10.1007/s00013-015-0738-5 |
| [10] | 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 · doi:10.1080/02331934.2015.1020943 |
| [11] | Eslamian, M.; Vahidi, J., Split common fixed point problem of nonexpansive semigroup, Mediterr J Math, 13, 1177-1195 (2016) · Zbl 1346.47025 · doi:10.1007/s00009-015-0541-3 |
| [12] | Jailoka, P.; Suantai, S., Split common fixed point and null point problems for demicontractive operators in Hilbert spaces, Optim Methods Softw, 34, 2, 248-263 (2019) · Zbl 07024418 · doi:10.1080/10556788.2017.1359265 |
| [13] | Suantai, S.; Pholasa, N.; Cholamjiak, P., The modified inertial relaxed CQ algorithm for solving the split feasibility problems, J Indust Manage Optim, 14, 4, 1595-1615 (2018) |
| [14] | Moudafi, A., Viscosity approximation methods for fixed-points problems, J Math Anal Appl, 241, 46-55 (2000) · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615 |
| [15] | Song, Y.; Cho, YJ., Some note on Ishikawa iteration for multi-valued mappings, Bull Korean Math Soc, 48, 575-584 (2011) · Zbl 1218.47120 · doi:10.4134/BKMS.2011.48.3.575 |
| [16] | Bauschke, HH; Combettes, PL., Convex analysis and monotone operator theory in Hilbert spaces (2011), New York: Springer, New York · Zbl 1218.47001 |
| [17] | Xu, HK., An iterative approach to quadratic optimization, J Optim Theory Appl, 116, 659-678 (2003) · Zbl 1043.90063 · doi:10.1023/A:1023073621589 |
| [18] | Marino, G.; Xu, HK., A general iterative method for nonexpansive mappings in Hilbert spaces, J Math Anal Appl, 318, 43-52 (2006) · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028 |
| [19] | Tufa, AR; Zegeye, H.; Thuto, M., Convergence theorem for non-self mappings in CAT(0) spaces, Numer Funct Anal Optim, 38, 705-722 (2017) · Zbl 1372.37030 · doi:10.1080/01630563.2016.1261156 |
| [20] | Maingé, PE., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set Valued Anal, 16, 899-912 (2008) · Zbl 1156.90426 · doi:10.1007/s11228-008-0102-z |
| [21] | Chidume, C., Geometric properties of Banach spaces and nonlinear iterations, 1965 (2009), London: Springer Verlag, London · Zbl 1167.47002 |
| [22] | Combettes, PL, Pesquet, J-C.Proximal splitting methods in signal processing. In Bauschke H, Burachik R, Combettes P, Elser V, Luke D, Wolkowicz H, editors. Fixed-point algorithms for inverse problems in science and engineering. New York: Springer; 2011. p. 185-212 (Springer Optim. Appl.; Vol. 49). · Zbl 1217.00018 |
| [23] | Singthong, U.; Suantai, S., Equilibrium problems and fixed point problems for nonspreading-type mappings in Hilbert space, Int J Nonlinear Anal Appl, 2, 51-61 (2011) · Zbl 1281.47060 |
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