Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces (Q879134)

The authors provide a result strictly related to Theorem~1 in [\textit{A.\,Tada} and \textit{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 \textit{R.\,Wittman} [Arch.\ Math.\ 58, No.\,5, 486--491 (1992; Zbl 0797.47036)] and \textit{P.\,L.\thinspace Combettes} and \textit{S.\,A.\thinspace Hirstoaga} [J.~Nonlinear Convex Anal.\ 6, No.\,1, 117--136 (2005; Zbl 1109.90079)].