An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces
详细信息 查看全文
- 作者：Abebe R. Tufa ; H. Zegeye
- 关键词：47H09 ; 47H10 ; 65J15
- 刊名：Arabian Journal of Mathematics
- 出版年：2015
- 出版时间：September 2015
- 年：2015
- 卷：4
- 期：3
- 页码：199-213
- 全文大小：823 KB
- 参考文献：1.Alber, Ya.: Metric and generalized projection operators in Banach spaces: properties and applications, In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp. 15-0. Dekker, New York (1996)
2.Alber Ya., Iusem A.N.: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set-Valued Anal. 9, 315-35 (2001)MATH MathSciNet CrossRef
3.Aoyama K., Kohsaka F.: Strongly relatively nonexpansive sequences generated by firmly nonexpansive-like mappings. Fixed Point Theory Appl. 2014, 95 (2014)CrossRef
4.Bello Cruz J.Y., Iusem A.N.: A strongly convergent direct method for monotone variational inequalities in Hilbert space. Numer. Funct. Anal. Optim. 30, 23-6 (2009)MATH MathSciNet CrossRef
5.Cai G., Bu S.: An iterative algorithm for a general system of variational inequalities and fixed point problems in q-uniformly smooth Banach spaces. Optim. Lett. 7, 267-87 (2013)MATH MathSciNet CrossRef
6.Censor, Y.; Gibali, A.; Reich S.: Extensions of Korpelevichs extragradient method for the variational inequality problem in Euclidean space. Optimization (2010). doi:10.-080/-2331934.-010.-39689
7.Censor Y., Gibali A., Reich S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318-35 (2011)MATH MathSciNet CrossRef
8.Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers, Dordrecht (1990). MR 91m:46021
9.He B.: A new method for a class of linear variational inequalities. Math. Progr. 66, 137-44 (1994)MATH CrossRef
10.Iiduka H., Takahashi W.: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 61, 341-50 (2005)MATH MathSciNet CrossRef
11.Iiduka H., Takahashi W., Toyoda M.: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14, 49-1 (2004)MATH MathSciNet
12.Kamimura S., Takahashi W.: Strong convergence of proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938-45 (2002)MathSciNet CrossRef
13.Kassay, G.; Reich, S.; Sabach, S.: Iterative methods for solving systems of variational inequalities in refelxive Banach spaces. SIAM J. Optim. 21, 1319-344 (2011)
14.Korpelevich G.M.: The extragradient method for finding saddle points and other problems. Matecon. 12, 747-56 (1976)MATH
15.Kumam, P.: A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive. Nonlinear Anal. Hybrid syst. (2008). doi:10.-016/?j.?nahs.-008.-9.-17
16.Mainge, P.E.: Strong convergence of projected subgradient methods for nonsmooth and non-strictly convex minimization. Set-Valued Anal. 16, 899-12 (2008)
17.Nadezhkina, N.; Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191-01 (2006)
18.Noor, M.A.: A class of new iterative methods for solving mixed variational inequalities. Math. Comput. Modell. 31, 11-9 (2000)
19.Reich, S.: A weak convergence theorem for the alternating method with Bregman distances. In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, pp. 313-18. Marcel Dekker, New York (1996)
20.Reich S., Sabach S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225-40 (2012)MathSciNet CrossRef
21.Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75-8 (1970)
22.Shahzad, N.; Zegeye, H.: Approximation methods for a common minimum-norm point of a solution of variational inequality and fixed point problems in Banach spaces. Bull. Korean Math. Soc. 51, 1 (2014)
23.Tada, A.; Takahashi, W.: Weak and strong convergence theorems for nonexpansive mappings and equilbrium problems. J. Optim. Theory Appl. 133, 359-70 (2007)
24.Takahashi, W.: Nonlinear Functional Analysis (Japanese). Kindikagaku, Tokyo (1988)
25.Takahashi, W.; Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118, 417-28 (2003)
26.Takahashi W., Zembayashi K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 70, 45-7 (2009)MATH MathSciNet CrossRef
27.Xu H.K.: Inequalities in Banach spaces with applictions. Nonlinear Anal. 16, 1127-138 (1991)MATH MathSciNet CrossRef
28.Xu H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109-13 (2002)MATH CrossRef
29.Yamada, I.: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In: But - 作者单位：Abebe R. Tufa (1)
H. Zegeye (1)
1. Department of Mathematics, University of Botswana, Pvt. Bag 00704, Gaborone, Botswana - 刊物主题：Mathematics, general;
- 出版者：Springer Berlin Heidelberg
- ISSN：2193-5351
文摘
Let C be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth real Banach space E. Let T: C?C be relatively nonexpansive mapping and let A i : C?E* be L i -Lipschitz monotone mappings, for i = 1,2. In this paper, we introduce and study an iterative process for finding a common point of the fixed point set of a relatively nonexpansive mapping and the solution set of variational inequality problems for A 1 and A 2. Under some mild assumptions, we show that the proposed algorithm converges strongly to a point in \({F(T)\cap VI(C, A_1)\cap VI(C, A_2)}\). Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators. Mathematics Subject Classification 47H09 47H10 65J15