APPROXIMATION METHODS FOR A COMMON MINIMUM-NORM POINT OF A SOLUTION OF VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS IN BANACH SPACES

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 51, Issue 3, 2014, pp.773-788
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2014.51.3.773

Title & Authors

APPROXIMATION METHODS FOR A COMMON MINIMUM-NORM POINT OF A SOLUTION OF VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS IN BANACH SPACES

Shahzad, N.; Zegeye, H.;

Shahzad, N.; Zegeye, H.;

Abstract

We introduce an iterative process which converges strongly to a common minimum-norm point of solutions of variational inequality problem for a monotone mapping and fixed points of a finite family of relatively nonexpansive mappings in Banach spaces. Our theorems improve most of the results that have been proved for this important class of nonlinear operators.

Keywords

monotone mappings;relatively nonexpansive mappings;strong convergence;variational inequality problems;

Language

English

Cited by

References

1.

Y. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and applications of nonlinear operators of accretive and monotone type, 15-50, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996.

2.

K. Aoyama, F. Kohsaka, and W. Takahashi, Proximal point method for monotone operators in Banach spaces, Taiwanese J. Math. 15 (2011), no. 1, 259-281.

3.

H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive and ${\alpha}$ -inverse strongly monotone mappings, Nonlinear Anal. 61 (2005), no. 3, 341-350.

4.

H. Iiduka and W. Takahashi, Strong convergence studied by a hybrid type method for monotone operators in a Banach space, Nonlinear Anal. 68 (2008), no. 12, 3679-3688.

5.

H. Iiduka and W. Takahashi, Weak convergence of projection algorithm for variational inequalities in Banach spaces, J. Math. Anal. Appl.; DOI:10.1016/j.jmaa.2007.07.019.

6.

H. Iiduka, W. Takahashi, and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, Panamer. Math. J. 14 (2004), no. 2, 49-61.

7.

S. Kamimura, F. Kohsaka, and W. Takahashi, Weak and strong convergence theorems for maximal monotone operators in a Banach space, Set-Valued Anal. 12 (2004), no. 4, 417-429.

8.

S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. Optim. 13 (2002), no. 3, 938-945.

9.

D. Kinderlehrer and G. Stampaccia, An Iteration to Variational Inequalities and Their Applications, Academic Press, New York, 1990.

10.

P. Kumam, A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear Anal. Hybrid Syst. 2 (2008), no. 4, 1245-1255.

11.

J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-517.

12.

P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), no. 7-8, 899-912.

13.

S. Y. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005), no. 2, 257-266.

14.

S. Reich, A weak convergence theorem for the alternating method with Bregman distances, Theory and applications of nonlinear operators of accretive and monotone type, 313-318, Lecture Notes in Pure and Appl. Math., 178, Dekker, New York, 1996.

15.

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optimization 14 (1976), no. 5, 877-898.

16.

A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl. 133 (2007), no. 3, 359-370.

17.

W. Takahashi, Nonlinear Functional Analysis, Kindikagaku, Tokyo, 1988.

18.

W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009), no. 1, 45-57.

19.

H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002), no. 1, 109-113.

20.

H. Zegeye and E. U. Ofoedu, and N. Shahzad, Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasinonexpansive mappings, Appl. Math. Comput. 216 (2010), no. 12, 3439-3449.

21.

H. Zegeye and N. Shahzad, Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal. 70 (2009), no. 7, 2707-2716.

22.

H. Zegeye and N. Shahzad, A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems, Nonlinear Anal. 74 (2011), no. 1, 263-272.

23.

H. Zegeye and N. Shahzad, A hybrid approximation method for equilibrium, variational inequality and fixed point problems, Nonlinear Anal. Hybrid Syst. 4 (2010), no. 4, 619-630.