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Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. (English) Zbl 1122.47056
The authors provide a result strictly related to Theorem 1 in [{\it A. Tada} and {\it W. Takahashi}, Proc. NACA (Okinawa, 2005), 609--617 (2007; Zbl 1122.47055), reviewed above]. Indeed, they consider the equilibrium problem: find $x\in C$ such that $$F(x,y)\ge 0\quad \forall y\in C,$$ where $C$ is a nonempty, closed and convex subset of a real Hilbert space $H$, and $F:C\times C\to {\Bbb R}$. The set of solutions is denoted by $EP(F)$. Under the same assumptions of Theorem 1, and given in addition a contraction $f:H\to H$, they find a way to generate two sequences of points, namely $\{x_n\}$ and $\{u_n\}$, approximating in the viscosity sense the equilibria that are also the fixed points $F(S)$ of a nonexpansive map $S$, i.e., both of them converge strongly to a point $z\in EP(F)\cap F(S)$, where $z$ is the projection of $f(z)$ onto $EP(F)\cap F(S)$. As corollaries, they get results previously obtained by {\it R. Wittman} [Arch. Math. 58, No. 5, 486--491 (1992; Zbl 0797.47036)] and {\it P. L.\thinspace Combettes} and {\it S. A.\thinspace Hirstoaga} [J. Nonlinear Convex Anal. 6, No. 1, 117--136 (2005; Zbl 1109.90079)].

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47J20 Inequalities involving nonlinear operators 49J40 Variational methods including variational inequalities 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking” properties 65J15 Equations with nonlinear operators (numerical methods) 90C47 Minimax problems
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##### References:
 [1] Blum, E.; Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. student 63, 123-145 (1994) · Zbl 0888.49007 [2] Combettes, P. L.; Hirstoaga, S. A.: Equilibrium programming in Hilbert spaces. J. nonlinear convex anal. 6, 117-136 (2005) · Zbl 1109.90079 [3] Flam, S. D.; Antipin, A. S.: Equilibrium programming using proximal-like algorithms. Math. program. 78, 29-41 (1997) · Zbl 0890.90150 [4] Moudafi, A.: Viscosity approximation methods for fixed-point problems. J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039 [5] Opial, Z.: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. amer. Math. soc. 73, 561-597 (1967) · Zbl 0179.19902 [6] Shioji, N.; Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. amer. Math. soc. 125, 3641-3645 (1997) · Zbl 0888.47034 [7] A. Tada, W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in: W. Takahashi, T. Tanaka (Eds.), Nonlinear Analysis and Convex Analysis, Yokohama Publishers, Yokohama, 2006, in press · Zbl 1122.47055 [8] Takahashi, W.: Nonlinear functional analysis. (2000) · Zbl 0997.47002 [9] Takahashi, W.: Convex analysis and approximation of fixed points. (2000) · Zbl 1089.49500 [10] Xu, H. K.: An iterative approach to quadratic optimization. J. optim. Theory appl. 116, 659-678 (2003) · Zbl 1043.90063 [11] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. math. 58, 486-491 (1992) · Zbl 0797.47036