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Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces. (English) Zbl 1254.90262
Summary: We introduce an iterative process which converges strongly to a common solution of variational inequality problems for two monotone mappings in Banach spaces. Furthermore, our convergence theorem is applied to the convex minimization problem. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.

90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
90C25Convex programming
Full Text: DOI
[1] Alber Y.: Metric and generalized projection operators in Banach spaces: Properties and Applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. In: Kartsatos, A.G. (ed.) Leture Notes in Pure and Appl Math, vol. 178, pp. 15--50. Dekker, New York (1996) · Zbl 0883.47083
[2] Aoyama, K., Kohsaka, F., Takahash, W.: Strong convergence theorems by shrinking and hybrid projection methods for relatively nonexpansive mappings in Banach spaces. In: Proceedings of the 5th International Conference On Nonlinear and Convex Analysis, YoKohama Publishers, pp. 7--26 (2009) · Zbl 1269.47050
[3] Cioranescu I.: Geometry of Banach spaces, Duality mapping and Nonlinear Problems. Klumer Academic publishers, Amsterdam (1990)
[4] Iiduka H., Takahashi W.: Weak convergence of projection algorithm for variational inequalities in Banach spaces. J. Math. Anal. Appl. 339, 668--679 (2008) · Zbl 1129.49012 · doi:10.1016/j.jmaa.2007.07.019
[5] Iiduka H., Takahashi W.: Strong convergence studied by a hybrid type method for monotone operators in a Banach space. Nonlinear Analysis 68, 3679--3688 (2008) · Zbl 1220.47095 · doi:10.1016/j.na.2007.04.010
[6] Iiduka H., Takahashi W., Toyoda M.: Approximation of solutions of variational inequalities for monotone mappings. Panamer. Math. J. 14, 49--61 (2004) · Zbl 1060.49006
[7] Kacurovskii : On monotone operators and convex functionals. Uspekhi Mat. Nauk 15, 213--215 (1960)
[8] Kamimura S., Takahashi W.: Strong convergence of proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938--945 (2002) · Zbl 1101.90083 · doi:10.1137/S105262340139611X
[9] Kinderlehrer D., Stampaccia G.: An Iteration to Variational Inequalities and Their Applications. Academic Press, New York (1990)
[10] Lions J.L., Stampacchia G.: Variational inequalities. Comm. Pure Appl. Math. 20, 493--517 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302
[11] Matsushita S., Takahashi W.: A strong convergence theorem for relatively nonexpansive mappings in a banach space. J. Approx. Theory 134, 257--266 (2005) · Zbl 1071.47063 · doi:10.1016/j.jat.2005.02.007
[12] Minty G.J.: Monotone operators in Hilbert spaces. Duke Math. J. 29, 341--346 (1962) · Zbl 0111.31202 · doi:10.1215/S0012-7094-62-02933-2
[13] Nakajo K., Takahashi W.: Strong convergence theorems for nonexpansive mappings and nonexpansive semi-groups. J. Math. Anal. Appl. 279, 372--379 (2003) · Zbl 1035.47048 · doi:10.1016/S0022-247X(02)00458-4
[14] Reich S.: A weak convergence theorem for the alternating method with Bergman distance, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. In: Kartsatos, A.G. (ed.) Leture Notes in Pure and Appl Math, vol. 178, pp. 313--318. Dekker, New York (1996) · Zbl 0943.47040
[15] Solodov M.V., Svaiter B.F.: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 87, 189--202 (2000) · Zbl 0971.90062
[16] Takahashi W.: Nonlinear Functional Analysis (Japanese). Kindikagaku, Tokyo (1988) · Zbl 0647.90052
[17] Takahashi W., Zembayashi K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45--57 (2009) · Zbl 1170.47049 · doi:10.1016/j.na.2007.11.031
[18] Vainberg M.M., Kacurovskii R.I.: On the variational theory of nonlinear operators and equations. Dokl. Akad. Nauk 129, 1199--1202 (1959)
[19] Xu H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127--1138 (1991) · Zbl 0757.46033 · doi:10.1016/0362-546X(91)90200-K
[20] Zarantonello E.H.: Solving Functional Equations by Contractive Averaging, Mathematics Research Center, Rep #160. Mathematics Research Centre, Univesity of Wisconsin, Madison (1960)
[21] Zegeye H., Ofoedu E.U., Shahzad N.: Convergence theorems for equilibrium problem, variotional inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl. Math. Comput. 216, 3439--3449 (2010) · Zbl 1198.65100 · doi:10.1016/j.amc.2010.02.054
[22] Zegeye H., Shahzad N.: Strong convergence for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal. 70, 2707--2716 (2009) · Zbl 1223.47108 · doi:10.1016/j.na.2008.03.058
[23] Giannessi, F., Maugeri, A., Pardalos, P.M. (eds): Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, vol. 58. Springer, Berlin (2002) · Zbl 0979.00025
[24] Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems Co-editors. Springer, New York (2010) · Zbl 1178.49001