Strong and weak convergence theorems for general mixed equilibrium problems and variational inequality problems and fixed point problems in Hilbert spaces

Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems

“…Then they obtained some weak and strong convergence theorems for the proposed iterative algorithms. Very recently, motivated by Yao et al [26], Cai and Bu [3] introduced two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically -strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. Then they proved some strong and weak convergence theorems for the proposed iterative algorithms under appropriate conditions.…”

“…The GMEP (1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games. The GMEP is further considered and studied in [2][3][4][5].…”

On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions and systems of generalized equilibria

“…The literature on the VIP is vast and Korpelevich's extragradient method has received great attention given by many authors, who improved it in various ways; see e.g., [8,9,10,11,12,14,18,20,23,24,25,28,29,30,34,35,36] and references therein, to name but a few.…”

“…The GMEP (1.2) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problems in noncooperative games and others. The GMEP is further considered and studied; see e.g., [20,25,26,29,36].…”

A new convergence theorem in a reflexive Banach space

“…In such problems, strong convergence or norm convergence is often much more desirable than the weak convergence. To obtain the strong convergence of the Krasnoselskii-Mann iteration, regularization techniques recently have been extensively investigated; see [7]- [11], [18], [21]- [23], [27]- [32] and the references therein.…”