Common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space

다국어 초록 (Multilingual 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.

참고문헌 (Reference)

1 Z. Opial, "Weak convergence of the sequence of successive approximations for nonexpansive mappings" 73 (73): 595-597, 1967

2 K. Goebel, "Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics" Cambridge University Press 28-, 1990

3 T. Shimizu, "Strong convergence to common fixed points of families of nonexpansive mappings" 211 (211): 71-83, 1997

4 X. Xiao, "Strong convergence of composite general iterative methods for one-parameter nonexpansive semigroup and equilibrium problems" 2012 (2012): 19-, 2012

5 L. C. Ceng, "Some iterative methods for finding fixed points and solving constrained convex minimization problems" 74 (74): 5286-5802, 2011

6 F. Cianciaruso, "Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert space" 146 (146): 491-509, 2010

7 L. C. Ceng, "Iterative construction of fixed points of nonselfmappings in Banach spaces" 206 (206): 814-825, 2007

8 K. R. Kazmi, "Iterative approximation of a common solution of split generalized equilibrium problem and a fixed point problem for a nonexpansive semigroup" 7 : 2013

9 H. K. Xu, "Iterative algorithms for nonlinear operators" 66 (66): 240-256, 2002

10 K. R. Kazmi, "Implicit iterative method for approximating a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup" 20 (20): 57-75, 2014

11 E. Blum, "From optimization and variational inequalities to equilibrium problems" 63 (63): 123-145, 1994

12 F. E. Browder, "Fixed point theorems for noncompact mappings in Hilbert space" 53 : 1272-1276, 1965

13 S. Plubtieng, "Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces" 48 (48): 279-286, 2008

14 K. R. Kazmi, "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" 218 (218): 5439-5452, 2012

15 K. Fan, "A generalization of Tychonoff ’s fixed-point theorem" 142 : 305-310, 1961

16 G. Marino, "A general iterative method for nonexpansive mappings in Hilbert spaces" 318 (318): 43-52, 2006