Title:
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Strong convergence of an iterative method for variational inequality problems and fixed point problems (English) |
Author:
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Qin, Xiaolong |
Author:
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Kang, Shin Min |
Author:
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Su, Yongfu |
Author:
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Shang, Meijuan |
Language:
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English |
Journal:
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Archivum Mathematicum |
ISSN:
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0044-8753 (print) |
ISSN:
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1212-5059 (online) |
Volume:
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45 |
Issue:
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2 |
Year:
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2009 |
Pages:
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147-158 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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In this paper, we introduce a general iterative scheme to investigate the problem of finding a common element of the fixed point set of a strict pseudocontraction and the solution set of a variational inequality problem for a relaxed cocoercive mapping by viscosity approximate methods. Strong convergence theorems are established in a real Hilbert space. (English) |
Keyword:
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nonexpansive mapping |
Keyword:
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strict pseudocontraction |
Keyword:
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fixed point |
Keyword:
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variational inequality |
Keyword:
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relaxed cocoercive mapping |
MSC:
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47H09 |
MSC:
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47H10 |
MSC:
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47J20 |
MSC:
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47J25 |
idZBL:
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Zbl 1210.47097 |
idMR:
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MR2591671 |
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Date available:
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2009-06-25T18:17:11Z |
Last updated:
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2013-09-19 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/128300 |
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Reference:
|
[1] Browder, F. E.: Fixed point theorems for noncompact mappings in Hilbert spaces.Proc. Natl. Acad. Sci. USA 53 (1965), 1272–1276. MR 0178324, 10.1073/pnas.53.6.1272 |
Reference:
|
[2] Browder, F. E.: Nonlinear operators and nonlinear equations of evolution in Banach spaces.Proc. Sympos. Pure Math. 18 (1976), 78–81. Zbl 0327.47022, MR 0405188 |
Reference:
|
[3] Bruck, R. E.: Properties of fixed point sets of nonexpansive mappings in Banach spaces.Trans. Amer. Math. Soc. 179 (1973), 251–262. Zbl 0265.47043, MR 0324491, 10.1090/S0002-9947-1973-0324491-8 |
Reference:
|
[4] Ceng, L. C., Wang, C. Y., Yao, J. C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities.Math. Methods Oper. Res. 67 (2008), 375–390. Zbl 1147.49007, MR 2403714, 10.1007/s00186-007-0207-4 |
Reference:
|
[5] Ceng, L. C., Yao, J. C.: An extragradient-like approximation method for variational inequality problems and fixed point problems.Appl. Math. Comput. 190 (2007), 205–215. Zbl 1124.65056, MR 2335441, 10.1016/j.amc.2007.01.021 |
Reference:
|
[6] Chen, J. M., Zhang, L. J., Fan, T. G.: Viscosity approximation methods for nonexpansive mappings and monotone mappings.J. Math. Anal. Appl. 334 (2007), 1450–1461. Zbl 1137.47307, MR 2338673, 10.1016/j.jmaa.2006.12.088 |
Reference:
|
[7] Gabay, D.: Applications of the Method of Multipliers to Variational Inequalities, Augmented Lagrangian Methods.North-Holland, Amsterdam. |
Reference:
|
[8] Iiduka, H., Takahashi, W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings.Nonlinear Anal. 61 (2005), 341–350. Zbl 1093.47058, MR 2123081, 10.1016/j.na.2003.07.023 |
Reference:
|
[9] Korpelevich, G. M.: An extragradient method for finding saddle points and for other problems.Ekonomika i matematicheskie metody 12 (1976), 747–756. MR 0451121 |
Reference:
|
[10] Moudafi, A.: Viscosity approximation methods for fixed points problems.Appl. Math. Comput. 241 (2000), 46–55. Zbl 0957.47039, MR 1738332 |
Reference:
|
[11] Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.J. Optim. Theory Appl. 128 (2006), 191–201. Zbl 1130.90055, MR 2201895, 10.1007/s10957-005-7564-z |
Reference:
|
[12] Noor, M. A.: Some developments in general variational inequalities.Appl. Math. Comput. 152 (2004), 199–277. Zbl 1134.49304, MR 2050063 |
Reference:
|
[13] Noor, M. A., Yao, Y.: Three-step iterations for variational inequalities and nonexpansive mappings.Appl. Math. Comput. 190 (2007), 1312–1321. Zbl 1128.65051, MR 2339724, 10.1016/j.amc.2007.02.013 |
Reference:
|
[14] Qin, X., Shang, M., Su, Y.: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems.Math. Comput. Modelling 48 (2008), 1033–1046. Zbl 1187.65058, MR 2458216, 10.1016/j.mcm.2007.12.008 |
Reference:
|
[15] Qin, X., Shang, M., Zhou, H.: Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces.Appl. Math. Comput. 200 (2008), 242–253. Zbl 1147.65048, MR 2421640, 10.1016/j.amc.2007.11.004 |
Reference:
|
[16] Stampacchia, G.: ormes bilineaires coercivites sur les ensembles convexes.C. R. Acad. Sci. Paris Sér. I Math. 258 (1964), 4413–4416. MR 0166591 |
Reference:
|
[17] Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals.J. Math. Anal. Appl. 305 (2005), 227–239. MR 2128124, 10.1016/j.jmaa.2004.11.017 |
Reference:
|
[18] Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings.J. Optim. Theory Appl. 118 (2003), 417–428. Zbl 1055.47052, MR 2006529, 10.1023/A:1025407607560 |
Reference:
|
[19] Verma, R. U.: Generalized system for relaxed cocoercive variational inequalities and its projection methods.J. Optim. Theory Appl. 121 (2004), 203–210. MR 2062977, 10.1023/B:JOTA.0000026271.19947.05 |
Reference:
|
[20] Verma, R. U.: General convergence analysis for two-step projection methods and application to variational problems.Appl. Math. Lett. 18 (2005), 1286–1292. MR 2170885, 10.1016/j.aml.2005.02.026 |
Reference:
|
[21] Xu, H. K.: Iterative algorithms for nonlinear operators.J. London Math. Soc. 66 (2002), 240–256. Zbl 1013.47032, MR 1911872, 10.1112/S0024610702003332 |
Reference:
|
[22] Yao, Y., Yao, J. C.: On modified iterative method for nonexpansive mappings and monotone mappings.Appl. Math. Comput. 186 (2007), 1551–1558. Zbl 1121.65064, MR 2316950, 10.1016/j.amc.2006.08.062 |
Reference:
|
[23] Zhou, H.: Convergence theorems of fixed points for $k$-strict pseudo-contractions in Hilbert spaces.Nonlinear Anal. 69 (2008), 456–462. MR 2426262, 10.1016/j.na.2007.05.032 |
Reference:
|
[24] Zhou, H.: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces.J. Math. Anal. Appl. 343 (2008), 546–556. Zbl 1140.47058, MR 2412149, 10.1016/j.jmaa.2008.01.045 |
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