Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. (English) Zbl 1122.47056

Takahashi, Satoru; Takahashi, Wataru

J. Math. Anal. Appl. 331, No. 1, 506-515 (2007).

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

Keywords:

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

Citations:

Zbl 1122.47055; Zbl 0797.47036; Zbl 1109.90079

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), Yokohama Publishers Yokohama
[9]	Takahashi, W., Convex analysis and approximation of fixed points, (2000), Yokohama Publishers Yokohama, (in Japanese)
[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