Iiduka, Hideaki; Takahashi, Wataru Strong convergence studied by a hybrid type method for monotone operators in a Banach space. (English) Zbl 1220.47095 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 12, 3679-3688 (2008). Summary: We study strong convergence for monotone operators. We first introduce a hybrid type algorithm for monotone operators. Next, we obtain a strong convergence theorem (Theorem 3.3) for finding a zero point of an inverse-strongly monotone operator in a Banach space. Finally, we apply our convergence theorem to the problem of finding a minimizer of a convex function. Cited in 26 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H05 Monotone operators and generalizations 47J05 Equations involving nonlinear operators (general) Keywords:generalized projection; inverse-strongly monotone operator; zero point; variational inequality problem; \(p\)-uniformly convex; strong convergence PDFBibTeX XMLCite \textit{H. Iiduka} and \textit{W. Takahashi}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 12, 3679--3688 (2008; Zbl 1220.47095) Full Text: DOI References: [1] Alber, Y. I., Metric and generalized projection operators in Banach spaces: Properties and applications, (Kartsatos, A. G., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math., vol. 178 (1996), Dekker: Dekker New York), 15-50 · Zbl 0883.47083 [2] Baillon, J. B.; Haddad, G., Quelques propriétés des opérateurs angle-bornés et \(n\)-cycliquement monotones, Israel J. Math., 26, 137-150 (1977) · Zbl 0352.47023 [3] Ball, K.; Carlen, E. A.; Lieb, E. H., Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., 115, 463-482 (1994) · Zbl 0803.47037 [4] Beauzamy, B., Introduction to Banach Spaces and Their Geometry (1985), North Holland · Zbl 0585.46009 [5] Browder, F. E.; Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20, 197-228 (1967) · Zbl 0153.45701 [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] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim., 13, 938-945 (2002) · Zbl 1101.90083 [8] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001 [9] Lions, J. L.; Stampacchia, G., Variational inequalities, Comm. Pure Appl. Math., 20, 493-517 (1967) · Zbl 0152.34601 [10] Liu, F.; Nashed, M. Z., Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal., 6, 313-344 (1998) · Zbl 0924.49009 [11] Martinet, B., Réglarisation d’inéuations variationnelles par approximations successives, Rev. Francaise Informat. Recherche Opétationnelle, 4, 154-158 (1970) · Zbl 0215.21103 [12] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279, 372-379 (2003) · Zbl 1035.47048 [13] Reich, S., A weak convergence theorem for the alternating method with Bregman distances, (Kartsatos, A. G., Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math., vol. 178 (1996), Dekker: Dekker New York), 313-318 · Zbl 0943.47040 [14] Rockafellar, R. T., On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc., 149, 75-88 (1970) · Zbl 0222.47017 [15] Rockafellar, R. T., Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14, 877-898 (1976) · Zbl 0358.90053 [16] 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 [17] Takahashi, Y.; Hashimoto, K.; Kato, M., On sharp uniform convexity, smoothness, and strong type, cotype inequalities, J. Nonlinear Convex Anal., 3, 267-281 (2002) · Zbl 1030.46012 [18] Takahashi, W., Nonlinear Functional Analysis (2000), Yokohama Publishers: Yokohama Publishers Yokohama [19] Xu, H. K., Inequalities in Banach spaces with applications, Nonlinear Anal., 16, 1127-1138 (1991) · Zbl 0757.46033 [20] Zeidler, E., Nonlinear Functional Analysis and its Applications II/B (1990), Springer: Springer New York, NY [21] Zălinescu, C., On uniformly convex functions, J. Math. Anal. Appl., 95, 344-374 (1983) · Zbl 0519.49010 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.