We introduce variational inequality problems on Hilbert $C^*$-modules and we prove several existence results for variational inequalities defined on closed convex sets. Then relation between variational inequalities, $C^*$-valued metric projection and fixed point theory  on  Hilbert $C^*$-modules is studied.

This paper introduces an implicit scheme for a   continuous representation of nonexpansive mappings on a closed convex subset of a Hilbert space with respect to a   sequence of invariant means defined on an appropriate space of bounded, continuous real valued functions of the semigroup.   The main result is to    prove the strong convergence of the proposed implicit scheme to the unique solutio...

The problem ofgeneralized equilibrium problem is very general in the different subjects .Optimization problems, variational inequalities, Nash equilibrium problem and minimax problems are as special cases of generalized equilibrium problem. The purpose of this paper is to investigate the problem of approximating a common element of the set of generalized equilibrium problem, variational inequal...

We consider a multi-step algorithm to approximate a common element of the set of solutions of monotone and Lipschitz-type continuous equilibrium problems, and the set of common fixed points of a finite family of set-valued mappings satisfying condition (E). We prove strong convergence theorems of such an iterative scheme in real Hilbert spaces. This common solution is the unique solution of a v...

this paper introduces an implicit scheme for a   continuous representation of nonexpansive mappings on a closed convex subset of a hilbert space with respect to a   sequence of invariant means defined on an appropriate space of bounded, continuous real valued functions of the semigroup.   the main result is to    prove the strong convergence of the proposed implicit scheme to the unique solutio...

In this paper, we first study a hierarchical problem of Baillon’s type, and we study a strong convergence theorem of this problem. For the special case of this convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon’s type. Our result of the ergodic theorem of Baillon’s type improves and generalizes many existence theorems of this type of problem. Two nume...

In this paper, we introduce a new iterative algorithm for approximating a common solution of certain class of multiple-sets split variational inequality problems. The sequence of the proposed iterative algorithm is proved to converge strongly in Hilbert spaces. As application, we obtain some strong convergence results for some classes of multiple-sets split convex minimization problems.

Based on the very recent work by Censor-Gibali-Reich [7], we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnoselskii-Mann Theorem for averaged operators, we analyze an algorithm for solving a new split monotone inclusions under weaker conditions. Our results improve and develop previously discu...

In this paper, the generalized variational inequality with multi-valued mapping (GVI) is considered. To solve the problem, we first establish a global error bound estimation for GVI with the underlying multi-valued mapping being   strict monotone and Holder continuous. Based on this, we propose a new type of method to solve the GVI, and its global convergence is also established. Keywords-GVI...