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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 [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
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References:

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