Weak convergence theorem for a class of split variational inequality problems and applications in a Hilbert space

Abstract

In this paper, we consider the algorithm proposed in recent years by Censor, Gibali and Reich, which solves split variational inequality problem, and Korpelevich’s extragradient method, which solves variational inequality problems. As our main result, we propose an iterative method for finding an element to solve a class of split variational inequality problems under weaker conditions and get a weak convergence theorem. As applications, we obtain some new weak convergence theorems by using our weak convergence result to solve related problems in nonlinear analysis and optimization.

Iterative algorithms for split variational Inequalities and generalized split feasibility problems with applications

“…In this paper, motivated by the results announced in Censor et al [3], we propose an extragradient and a modified extragradient iterative algorithms, and prove strong convergence theorems in a real Hilbert space. Our theorems improve and complement the related results in Censor et al [3], Tian and Jiang [19]. We present numerical experiments to illustrate the convergence of our algorithms.…”

“…Assume that D ⊂ C i n for some n ≥ 1. Let u ∈ D. By a similar argument as in Claim 1 of Theorem 3.1, we obtain that 19) which implies that D ⊂ C i n+1 . Hence,…”

“…Set N = 1. Even for a single operator, the algorithms in Theorems 3.1 and 3.2 are slightly different from the algorithms studied by Tian and Jiang [19].…”

“…Recently, Tian and Jiang [19] studied the following algorithm for approximating a solution of the SFP in a real Hilbert space:…”

“…Lemma 2.6. (Tiang and Jiang[19]) Let M be a nonempty closed convex subset of a real Hilbert space H. Let Θ be a convex function of H into R. If Θ is differentiable, then z is a solutions of constrained convex minimization problem(2.3) if and only if z ∈ V I(M, Θ ).3. MAIN RESULTS3.1.…”