A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems
The author investigates the equilibrium problems given as: (1) Find \(x^* \subset C\) such that \(f(x^*,y)0\geq0\) \(\forall y \in C\) where \(C\) is a nonempty closed convex subset of a real Hilbert space \(H\) and \(f\) is a bifunction from \(\langle C \times C \rangle\) to \(\mathbb R\). If \(f(x,y) = \langle F(x), y-x \rangle\) for every \([x,y] \in C\), where \(F\) is a mapping from \(C\) to \(H\), then problem (1) becomes the following variational inequalities: (2) Find \(x^* \subset C\) such that \(\langle F(x^*) ,y-x\rangle \geq 0\) \(\forall y \in C\). The author introduces a new iterative algorithm for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a pseudomonotone, Lipschitz-type continuous bifunction. This method can be considered as an improvement of the Peng iterative method (iterative algorithm for mixed equilibrium problems). Main results: For \(x^* \in (3)\) (the set of solutions of (2)), if \(f(x, \cdot)\) is convex and subdifferentiable on \(C\) for all \(x \in C\) and if \(f\) is pseudomonotone on \(C\), one then has \(\|t^n - x^*\|^2 \leq \|x^n - x^*\|^2 - (1-2 \lambda_nC_1) \|x^n - y^n\|^2 - (1-2 \lambda_nC_2) \|t^n - y^n\|^2\) \(\forall n \geq 0\). (A strong convergence theorem of sequences \(\{x^n\}\), \(\{y^n\}\), \(\{z^n\} \) and \(\{t^n\}\) defined by the algorithm, based on the extragradient method which solves the problem of finding a common element of two sets (3) and Fix(T) for a monotone, Lipschitz-type continuous bifunction \(f\) in a real Hilbert space \(H\)). Further, if the assumptions: \(f\) is pseudomonotone on \(C\), \(f\) is Lipschitz-type continuous on \(C\), for each \(x \in C\), \(y \rightarrow f(x,y)\) is convex and subdifferentiable on \(C\), \(Sol(f,C) \cap Fix(T) \neq 0\), holds and \(T \) is nonexpansive on \(C\), then the sequences \(\{x^n\}\), \(\{y^n\}\), \(\{z^n\} \) and \(\{t^n\}\) generated by the presented algorithm converge strongly to the same point \(x^* = Pr_{Sol(f,C) \cap Fix(T)}(x^o)\).
scientific article; zbMATH DE number 6143137