IMPROVED GRAOIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES

doi:10.1016/S0252-9602(17)30006-1

摘要/Abstract

摘要：

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space C and {A_n}_n∈N be a family of monotone and Lipschitz continuos mappings of C into E^*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming[10] for solving the variational inequality problem for {A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.

关键词: Variational inequality problem, gradient method, monotone operators, 2-uniformly convex Banach space, hybrid method

Abstract:

Key words: Variational inequality problem, gradient method, monotone operators, 2-uniformly convex Banach space, hybrid method

Kazuhide NAKAJO. IMPROVED GRAOIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES[J]. 数学物理学报(英文版), 2017, 37(2): 342-354.

Kazuhide NAKAJO. IMPROVED GRAOIENT METHOD FOR MONOTONE AND LIPSCHITZ CONTINUOUS MAPPINGS IN BANACH SPACES[J]. Acta mathematica scientia,Series B, 2017, 37(2): 342-354.

参考文献

[1] Alber Y I. Metric and generalized projections operators in Banach spaces:properties and applications//Kartasatos A G. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type 15-50; Lecture Notes in Pure and Appl Math, Vol 178. New York:Dekker, 1996
[2] Alber Y I, Reich S. An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamer Math J, 1994, 4:39-54
[3] Baillon J B, Haddad G. Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Israel J Math, 1977, 26:137-150
[4] Bauschke H H, Combettes P L. A weak-to-strong convergence principle for fejér-monotone methods in Hilbert spaces. Math Oper Res, 2001, 26:248-264
[5] Browder F E, Petryshyn W V. Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl, 1967, 20:197-228
[6] Dunn J C. Convexity, mnotonicity, and gradient processes in Hilbert space. J Math Anal Appl, 1976, 53: 145-158
[7] Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Pure and Applied Math, 83. New York:Marcel Dekker, 1984
[8] Goldstein A A. Convex programming in Hilbert space. Bull Amer Math Soc, 1964, 70:709-710
[9] Hartman P, Stampacchia G. On some nonlinear elliptic differential functional equations. Acta Math, 1966, 115:153-188
[10] Haugazeau Y. Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes. Paris, France:Thèse, Université de Paris, 1968
[11] Iiduka H. Takahashi W, Toyoda M. Approximation of solutions of variational inequalities for monotone mappings. Panamer Math J, 2004, 14:49-61
[12] Iiduka H, Takahashi W. Weak converegnce of a projection algorithm for variational inequalities in a Banach space. J Math Anal Appl, 2008, 339:668-679
[13] Iiduka H, Takahashi W. Strong convergence studied by a hybrid type method for monotone operators in a Banach space. Nonlinear Anal, 2008, 68:3679-3688
[14] Kamimura S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim, 2002, 13:938-945
[15] Kimura Y, Nakajo K. Strong convergence to a solution of a variational inequality problem in Banach spaces. J Appl Math, Vol 2014. Article ID 346517, 10 pages. http://dx.doi.org/10.1155/2014/346517
[16] Korpelevich G M. The extragradient method for finding saddle points and other problems. Matecon, 1976, 12:747-756
[17] Levitin E S, Polyak B T. Constrained minimization problems. USSR Comput Math Phys, 1966, 6:1-50
[18] Lions J L, Stampacchia G. Variational inequalities. Comm Pure Appl Math, 1967, 20:493-517
[19] Liu F, Nashed M Z. Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Anal, 1998, 6:313-344
[20] Matsushita S, Takahashi W. A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J Approx Theory, 2005, 134:257-266
[21] Nadezhkina N, Nakajo K, Takahashi W. Applications of extragradient method for solving the combined variational ineqality-fixed point problem in real Hilbert space//Takahashi W, Tanaka T. Nonlinear Analysis and Convex Analysis. Yokohama:Yokohama Publishers, 2007:399-416
[22] Nakajo K, Takahashi W. Strong and weak convergence theorems by an improved splitting method. Commun Appl Nonlinear Anal, 2002, 9:99-107
[23] Nakajo K, Takahashi W. Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J Math Anal Appl, 2003, 279:372-379
[24] Nakajo K, Shimoji K, Takahashi W. Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces. Taiwanese J Math, 2006, 10:339-360
[25] Nakajo K, Shimoji K, Takahashi W. On strong convergence by the hybrid method for families of mappings in Hilbert spaces. Nonlinear Anal, 2009, 71:112-119
[26] Nakajo K, Shimoji K, Takahashi W. Approximations for nonlinear mappings by the hybrid method in Hilbert spaces. Nonlinear Anal, 2011, 74:7025-7032
[27] Popov L D. Introduction to theory, solving methods and economical applications complementarity problems. Ekaterinburg, Russia:Ural State University, 2001(Russian)
[28] Solodov M V, Svaiter B F. Forcing strong convergence of proximal point iterations in a Hilbert space. Math Programming, 2000, 87A:189-202
[29] Takahashi W. Nonlinear Functional Analysis. Yokohama:Yokohama Publishers, 2000
[30] Takahashi W. Convex Analysis and Approximation of Fixed Points. Yokohama:Yokohama Publishers, 2000(Japanese)
[31] Takahashi W, Toyoda M. Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl, 2003, 118:417-428
[32] Xiu N, Zhang J. Some recent advances in projection-type methods for variational inequalities. J Comput Appl Math, 2003, 152:559-585
[33] Xu H K. Inequalities in Banach spaces with applications. Nonlinear Anal, 1991, 16:1127-1138
[34] Z?linescu C. On uniformly convex functions. J Math Anal Appl, 1983, 95344-374
[35] Zhou H. Strong convergence theorems for a family of Lipschitz quasi-pseudo-contractions in Hilbert spaces. Nonlinear Anal, 2009, 71:120-125