COMMON SOLUTION TO GENERALIZED MIXED EQUILIBRIUM PROBLEM AND FIXED POINT PROBLEM FOR A NONEXPANSIVE SEMIGROUP IN HILBERT SPACE

- Journal title : Journal of the Korean Mathematical Society
- Volume 53, Issue 1, 2016, pp.89-114
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2016.53.1.089

Title & Authors

COMMON SOLUTION TO GENERALIZED MIXED EQUILIBRIUM PROBLEM AND FIXED POINT PROBLEM FOR A NONEXPANSIVE SEMIGROUP IN HILBERT SPACE

DJAFARI-ROUHANI, BEHZAD; FARID, MOHAMMAD; KAZMI, KALEEM RAZA;

DJAFARI-ROUHANI, BEHZAD; FARID, MOHAMMAD; KAZMI, KALEEM RAZA;

Abstract

In this paper, we introduce and study an explicit hybrid relaxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.

Keywords

generalized mixed equilibrium problem;fixed-point problem;nonexpansive semigroup;explicit hybrid relaxed extragradient iterative method;

Language

English

Cited by

References

1.

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

2.

F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. (USA) 53 (1965), 1272-1276.

3.

L. C. Ceng, Q. H. Ansari, and J. C. Yao, Some iterative methods for finding fixed points and solving constrained convex minimization problems, Nonlinear Anal. 74 (2011), no. 16, 5286-5802.

4.

L. C. Ceng, T. Tanaka, and J. C. Yao, Iterative construction of fixed points of nonself- mappings in Banach spaces, J. Comput. Appl. Math. 206 (2007), no. 2, 814-825.

5.

F. Cianciaruso, G. Marino, and L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert space, J. Optim. Theory Appl. 146 (2010), no. 2, 491-509.

7.

K. Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990.

8.

K. R. Kazmi and S. H. Rizvi, A hybrid extragradient method for approximating the common solutions of a variational inequality, a system of variational inequalities, a mixed equilibrium problem and a fixed point problem, Appl. Math. Comput. 218 (2012), no. 9, 5439-5452.

9.

K. R. Kazmi, Iterative approximation of a common solution of split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup, Math. Sci. 7 (2013), Article 1.

10.

K. R. Kazmi, Implicit iterative method for approximating a common solution of split equi- librium problem and fixed point problem for a nonexpansive semigroup, Arab J. Math. Sci. 20 (2014), no. 1, 57-75.

11.

G. Marino and H. K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006), no. 1, 43-52.

12.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), no. 4, 595-597.

13.

S. Plubtieng and R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Modelling 48 (2008), no. 1-2, 279-286.

14.

T. Shimizu and W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997), no. 1, 71-83.

15.

X. Xiao, S. Li, L. Li, H. Song, and L. Zhang, Strong convergence of composite general iterative methods for one-parameter nonexpansive semigroup and equilibrium problems, J. Inequal. Appl. 2012 (2012), 131, 19 pp.