A viscosity extragradient method for an equilibrium problem and fixed point problem in Hilbert space
Let \(H\) be a real Hilbert space and \(C\) be its nonempty, closed and convex subset. Let \(T:C\to C\) be a nonexpansive mapping and \(f:C\times C\to \mathbb{R}\) be a function satisfying the following conditions: (1)~\(f\) is pseudomonotone and \(f(x,x)=0\) for each \(x\in C\); (2)~\(f\) is Lipschitz-type continuous on~\(C\); (3)~\(f(x,\cdot)\) is convex and subdifferentiable, for each \(x\in C\); (4)~\(f(x,y)\) is jointly weakly continuous on \(C\times C\). Assume that \(A=\operatorname{Fix}(T)\cap EP(f)\neq\emptyset\), where \(\operatorname{Fix}(T)\) is the set of fixed points of \(T\) and \(EP(f)=\{z\in C:f(z,y)\geq0\text{ for all }y\in C\}\) is the set of solutions of the equilibrium problem of~\(f\). The authors present an algorithm to find an element in \(A\). A numerical example illustrates the algorithm.
scientific article; zbMATH DE number 6858711