Common Solutions of a System of Variational Inequality Problems

The theory of variational inequalities has played an important role in the study of a wide class of problems arising in pure and applied sciences including mechanics, optimization and optimal control, partial differential equation, operations research and engineering sciences. During the last decades this problem has been studied by many authors, (see [1-15]). Recently, Censor, Gibali and Reich [16, 17] (see also [18]) introduced the Common Solutions to Variational Inequality Problem (CSVIP) which consists of finding common solutions to unrelated variational inequality. The general form of the CSVIP is the following: Let H be a Hilbert space. Let there be given, for each i = 1, 2, ..., N , an operator hi : H −→ H and a nonempty, closed and convex subset Ci ⊂ H, with ∩N i=1 Ci ̸= ∅. The CSVIP (for single-valued operators) is to find a point z ∈ ∩N i=1 Ci such that, for each i=1,2,...,N, ⟨hiz, x− z⟩ ≥ 0, ∀x ∈ Ci, 1 ≤ i ≤ N. (1.1) For 1 ≤ i ≤ N , we denote by SOL(Ci, hi) the solution set of (1). We note that in CSVIP, if we choose all hi = 0, then the problem reduces to that of finding a point z ∈ ∩N i=1 Ci in the nonempty intersection of a finite family of closed and convex set, which 1, Corresponding author,Department of Mathematics, Mazandaran University of Science and Technology, Behshahr, Iran, Email: [email protected] 2Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand, Email: [email protected] 3 Department of Mathematics, Iran University of Science and Technology, Noor, Iran, Email: [email protected]