Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces
The authors study two iterative algorithms for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of finite variational inequalities for inverse strongly monotone mappings and the set of common fixed points of an asymptotically \(K\)-strictly pseudocontractive mapping in the intermediate sense in a real Hilbert space. Here, the general mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases. Numerous problems in physics, optimization and economics reduce to find a solution of problem: \(F(x,y) + \varphi (y) - \varphi(x) + \langle Ax,y-x\rangle \geq 0\) \(\forall y \in \mathbb C\), where a mapping \(A:\mathbb C \rightarrow H\) is \(\alpha\)-inverse strongly monotone (a nonlinear mapping), \(\varphi :\mathbb C \rightarrow R\) is a real-valued function. If \(\varphi =0\), then the proposed problem reduces to the generalized equilibrium problem which is to study \(x \in \mathbb C\) such that: \(F(x,y) + \langle Ax,y-x\rangle \geq 0\) \(\forall y \in \mathbb C\). Furthermore, as \(A=0\), then the mixed equilibrium problem (find \(x \in \mathbb C\) such that \(F(x,y) + \varphi (y) - \varphi(x) \geq 0\) \(\forall y \in \mathbb C\)), is considered. The authors prove some strong and weak convergence theorems by using the proposed schemes under some appropriate conditions. The results obtained in this paper improve and extend the corresponding results announced by many others.
scientific article; zbMATH DE number 6155440