×

Iterative methods for pseudomonotone variational inequalities and fixed-point problems. (English) Zbl 1267.90158

This paper presents an iterative scheme that combines two well-known schemes: extragradient and approximate proximal methods. The authors suggest an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. Some necessary and sufficient conditions are derived for strong convergence of the sequences generated by the proposed scheme. The results presented in this paper extend and improve some corresponding results in the literature.

MSC:

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bnouhachem, A.: An additional projection step to He and Liao’s method for solving variational inequalities. J. Comput. Appl. Math. 206, 238–250 (2007) · Zbl 1136.49024 · doi:10.1016/j.cam.2006.07.001
[2] Bnouhachem, A., Noor, M.A., Hao, Z.: Some new extragradient iterative methods for variational inequalities. Nonlinear Anal. 70, 1321–1329 (2009) · Zbl 1242.49076 · doi:10.1016/j.na.2008.02.014
[3] Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) · Zbl 0536.65054
[4] He, B.S., Yang, Z.H., Yuan, X.M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004) · Zbl 1068.65087 · doi:10.1016/j.jmaa.2004.04.068
[5] Iusem, A.N.: An iterative algorithm for the variational inequality problem. Comput. Appl. Math. 13, 103–114 (1994) · Zbl 0811.65049
[6] Korpelevich, G.M.: An extragradient method for finding saddle points and for other problems. Ekonom. Mat. Metody 12, 747–756 (1976) · Zbl 0342.90044
[7] Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–517 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302
[8] Aslam Noor, M.: Some development in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004) · Zbl 1134.49304 · doi:10.1016/S0096-3003(03)00558-7
[9] Aslam Noor, M.: Projection-proximal methods for general variational inequalities. J. Math. Anal. Appl. 318, 53–62 (2006) · Zbl 1086.49005 · doi:10.1016/j.jmaa.2005.05.024
[10] Aslam Noor, M., Inayat Noor, K.: Self-adaptive projection algorithms for general variational inequalities. Appl. Math. Comput. 151, 659–670 (2004) · Zbl 1053.65048 · doi:10.1016/S0096-3003(03)00368-0
[11] Aslam Noor, M., Inayat Noor, K., Rassias, T.M.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993) · Zbl 0788.65074 · doi:10.1016/0377-0427(93)90058-J
[12] Aslam Noor, M., Huang, Z.: Wiener–Hopf equations technique for variational inequalities and nonexpansive mappings. Appl. Math. Comput. 1, 504–510 (2007) · Zbl 1193.49009 · doi:10.1016/j.amc.2007.02.117
[13] Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Math. 258, 4413–4416 (1964) · Zbl 0124.06401
[14] Yao, J.C.: Variational inequalities with generalized monotone operators. Math. Oper. Res. 19, 691–705 (1994) · Zbl 0813.49010 · doi:10.1287/moor.19.3.691
[15] Yao, Y., Noor, M.A.: On viscosity iterative methods for variational inequalities. J. Math. Anal. Appl. 325, 776–787 (2007) · Zbl 1115.49024 · doi:10.1016/j.jmaa.2006.01.091
[16] Yao, Y., Noor, M.A.: On modified hybrid steepest-descent methods for general variational inequalities. J. Math. Anal. Appl. 334, 1276–1289 (2007) · Zbl 1123.49009 · doi:10.1016/j.jmaa.2007.01.036
[17] Ceng, L.C., Yao, J.C.: An extragradient-like approximation method for variational inequality problems and fixed point problems. Appl. Math. Comput. 1906, 206–215 (2007) · Zbl 1124.65056
[18] Ceng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 10, 1293–1303 (2006) · Zbl 1110.49013
[19] Ceng, L.C., Al-Homidan, S., Ansari, Q.H., Yao, J.-C.: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 223, 967–974 (2009) · Zbl 1167.47307 · doi:10.1016/j.cam.2008.03.032
[20] Yao, Y., Yao, J.C.: On modified iterative method for nonexpansive mappings and monotone mappings. Appl. Math. Comput. 186, 1551–1558 (2007) · Zbl 1121.65064 · doi:10.1016/j.amc.2006.08.062
[21] Takahashi, W., Zembayashi, K.: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory Appl. 2008, 528476 (2008), 11 pp. · Zbl 1187.47054 · doi:10.1155/2008/528476
[22] Cholamjiak, P.: A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces. Fixed Point Theory Appl. 2009, 719360 (2009), 18 pp. doi: 10.1155/2009/719360 · Zbl 1167.65379
[23] Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006) · Zbl 1143.47047 · doi:10.1137/050624315
[24] Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[25] Ceng, L.C., Teboulle, M., Yao, J.C.: Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed point problems. J. Optim. Theory Appl. 146, 19–31 (2010) · Zbl 1222.47091 · doi:10.1007/s10957-010-9650-0
[26] Bauschke, H.H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202, 150–159 (1996) · Zbl 0956.47024 · doi:10.1006/jmaa.1996.0308
[27] Browder, F.E.: Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces. Arch. Ration. Mech. Anal. 24, 82–90 (1967) · Zbl 0148.13601 · doi:10.1007/BF00251595
[28] Chancelier, J.P.: Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 353, 141–153 (2009) · Zbl 1166.65026 · doi:10.1016/j.jmaa.2008.11.041
[29] Chang, S.S.: Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 323, 1402–1416 (2006) · Zbl 1111.47057 · doi:10.1016/j.jmaa.2005.11.057
[30] Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 44, 506–510 (1953) · Zbl 0050.11603 · doi:10.1090/S0002-9939-1953-0054846-3
[31] Cianciaruso, F., Marino, G., Muglia, L., Yao, Y.: On a two-step algorithm for hierarchical fixed point problems and variational inequalities. J. Inequal. Appl. 2009, 208692 (2009), 13 pp. doi: 10.1155/2009/208692 · Zbl 1180.47040 · doi:10.1155/2009/208692
[32] Cianciaruso, F., Colao, V., Muglia, L., Xu, H.K.: On an implicit hierarchical fixed point approach to variational inequalities. Bull. Aust. Math. Soc. 80, 117–124 (2009) · Zbl 1168.49005 · doi:10.1017/S0004972709000082
[33] Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990) · Zbl 0708.47031
[34] Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318(1), 43–52 (2006) · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028
[35] Martinez-Yanes, C., Xu, H.K.: Strong convergence of the CQ method for fixed point processes. Nonlinear Anal. 64, 2400–2411 (2006) · Zbl 1105.47060 · doi:10.1016/j.na.2005.08.018
[36] Opial, Z.: Weak convergence of the sequence of successive approximations of nonexpansive mappings. Bull. Am. Math. Soc. 73, 595–597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[37] Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979) · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6
[38] Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwan. J. Math. 5, 387–404 (2001) · Zbl 0993.47037
[39] Suzuki, T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103–123 (2005) · Zbl 1123.47308 · doi:10.1155/FPTA.2005.103
[40] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992) · Zbl 0797.47036 · doi:10.1007/BF01190119
[41] Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[42] Xu, H.K.: A variable Krasnoselski–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
[43] Yao, Y., Liou, Y.C., Chen, R.: Convergence theorems for fixed point problems and variational inequality problems in Hilbert spaces. Math. Nachr. 282(12), 1827–1835 (2009) · Zbl 1225.47119 · doi:10.1002/mana.200610817
[44] Yao, Y., Liou, Y.-C., Yao, J.-C.: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Appl. 2007, 64363 (2007), 12 pp. · Zbl 1156.47056 · doi:10.1155/2007/32870
[45] Zegeye, H., Shahzad, N.: Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings. Appl. Math. Comput. 191, 155–163 (2007) · Zbl 1194.47089 · doi:10.1016/j.amc.2007.02.072
[46] Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[47] Takahashi, S., Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008) · Zbl 1142.47350 · doi:10.1016/j.na.2008.02.042
[48] Qin, X., Cho, Y.J., Kang, S.M.: An iterative method for an infinite family of nonexpansive mappings in Hilbert spaces. Bull. Malays. Math. Soc. 32(2), 161–171 (2009) · Zbl 1223.47091
[49] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) · Zbl 0888.49007
[50] Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52, 301–316 (2003) · Zbl 1176.90640 · doi:10.1080/0233193031000120039
[51] Iusem, A.N., Sosa, W.: New existence results for equilibrium problems. Nonlinear Anal. 52, 621–635 (2003) · Zbl 1017.49008 · doi:10.1016/S0362-546X(02)00154-2
[52] Iusem, A.N., Nasri, M.: Inexact proximal point methods for equilibrium problems in Banach spaces. Numer. Funct. Anal. Optim. 28(11–12), 1279–1308 (2007) · Zbl 1144.65041 · doi:10.1080/01630560701766668
[53] Bianchi, M., Pini, R.: A note on equilibrium problems with properly quasimonotone bifunctions. J. Glob. Optim. 20, 67–76 (2001) · Zbl 0985.90090 · doi:10.1023/A:1011234525151
[54] Bianchi, M., Pini, R.: Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124, 79–92 (2005) · Zbl 1064.49004 · doi:10.1007/s10957-004-6466-9
[55] Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996) · Zbl 0903.49006 · doi:10.1007/BF02192244
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.