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Zbl 1122.47056
Takahashi, Satoru; Takahashi, Wataru
Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.
(English)
[J] J. Math. Anal. Appl. 331, No. 1, 506-515 (2007). ISSN 0022-247X

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)].
[Rita Pini (Milano)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47J20 Inequalities involving nonlinear operators
49J40 Variational methods including variational inequalities
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties
65J15 Equations with nonlinear operators (numerical methods)
90C47 Minimax problems

Keywords: viscosity approximation method; equilibrium problem; fixed point; nonexpansive mapping; strong convergence

Citations: Zbl 1122.47055; Zbl 0797.47036; Zbl 1109.90079

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