A PARALLEL HYBRID METHOD FOR EQUILIBRIUM PROBLEMS, VARIATIONAL INEQUALITIES AND NONEXPANSIVE MAPPINGS IN HILBERT SPACE

- Journal title : Journal of the Korean Mathematical Society
- Volume 52, Issue 2, 2015, pp.373-388
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2015.52.2.373

Title & Authors

A PARALLEL HYBRID METHOD FOR EQUILIBRIUM PROBLEMS, VARIATIONAL INEQUALITIES AND NONEXPANSIVE MAPPINGS IN HILBERT SPACE

Hieu, Dang Van;

Hieu, Dang Van;

Abstract

In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. Strong convergence theorem is proved for the sequence generated by the scheme. Finally, a parallel iterative algorithm for two finite families of variational inequalities and nonexpansive mappings is established.

Keywords

hybrid method;equilibrium problem;variational inequality;parallel computation;

Language

English

Cited by

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References

1.

P. K. Anh, Ng. Buong, and D. V. Hieu, Parallel methods for regularizing systems of equations involving accretive operators, Appl. Anal. 93 (2014), no. 10, 2136-2157.

2.

P. K. Anh and C. V. Chung, Parallel hybrid methods for a finite family of relatively nonexpansive mappings, Numer. Funct. Anal. Optim. 35 (2014), no. 6, 649-664.

3.

P. K. Anh and D. V. Hieu, Parallel and sequential hybrid methods for a finite fam- ily of asymptotically quasi $\phi$ -nonexpansive mappings, J. Appl. Math. Comput. (2014), DOI:10.1007/s12190-014-0801-6.

4.

H. H. Bauschke, J. M. Borwein, and A. S. Lewis, The method of cyclic projections for closed convex sets in Hilbert space, Recent developments in optimization theory and nonlinear analysis (Jerusalem, 1995), 1-38, Contemp. Math., 204, Amer. Math. Soc., Providence, RI, 1997.

5.

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Program. 63 (1994), no. 1-4, 123-145.

6.

M. Burger and B. Kaltenbacher, Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Anal. 44 (2006), no. 1, 153-182.

7.

P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), no. 1, 117-136.

8.

A. De Cezaro, M. Haltmeier, A. Leitao, and O. Scherzer, On steepest-descent-Kaczmarz method for regularizing systems of nonlinear ill-posed equations, Appl. Math. Comput. 202 (2008), no. 2, 596-607.

9.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Math., vol. 28, Cambridge University Press, Cambridge, 1990.

10.

M. Haltmeier, R. Kowar, A. Leitao, and O. Scherzer, Kaczmarz methods for regularizing nonlinear ill-posed equations, Inverse Probl. Imaging 1 (2007), no. 2, 289-298.

11.

H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive nonself- mappings and inverse-strongly-monotone mappings, J. Convex Anal. 11 (2004), no. 1, 69-79.

12.

S. Saeidi, Iterative methods for equilibrium problems, variational inequalities and fixed points, Bull. Iranian Math. Soc. 36 (2010), no. 1, 117-135.

13.

M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. 87 (2000), no. 1, 189-202.

14.

W. Takahashi, Weak and strong convergence theorems for families of nonexpansive map- pings and their applications, Ann. Univ. Mariae Curie-Sklodowska Sect. A 51 (1997), no. 2, 277-292.

15.

W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.

16.

S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium prob- lems and fixed point in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), no. 1, 506-515.

17.

W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417-428.

18.

X. Yu, Y. Yao, and Y. C. Liou, Strong convergence of a hybrid method for pseudomono- tone variational inequalities and fixed point problem, An. St. Univ. "Ovidius" Constanta Ser. Mat. 20 (2012), no. 1, 489-504.