## 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 [A. Tada and 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 $F(x,y)\geq 0\quad \forall y\in C,$ where $$C$$ is a nonempty, closed and convex subset of a real Hilbert space $$H$$, and $$F:C\times C\to {\mathbb R}$$. 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 R. Wittman [Arch. Math. 58, No. 5, 486–491 (1992; Zbl 0797.47036)] and P. L. Combettes and S. A. Hirstoaga [J. Nonlinear Convex Anal. 6, No. 1, 117–136 (2005; Zbl 1109.90079)].
Reviewer: Rita Pini (Milano)

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47J20 Variational and other types of inequalities involving nonlinear operators (general) 49J40 Variational inequalities 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 65J15 Numerical solutions to equations with nonlinear operators 90C47 Minimax problems in mathematical programming

### Citations:

Zbl 1122.47055; Zbl 0797.47036; Zbl 1109.90079
Full Text:

### References:

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