Composite schemes for variational inequalities over equilibrium problems and variational inclusions
Yonghong Yao, Jung Im Kang, Yeol Je Cho, Yeong‐Cheng Liou · Journal of Inequalities and Applications · 2013
Abstract Let C be a nonempty closed convex subset of a Hilbert space H, and let T : H → H be a nonlinear mapping. It is well known that the following classical variational inequality has been applied in many areas of applied mathematics, modern physical sciences, computerized tomography and many others. Find a point x ∗ ∈ C such that 〈 T x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ C . In this paper, we consider the following variational inequality. Find a point x ∗ ∈ C such that 〈 ( F − γ f ) x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ C , and, for solutions of the variational inequality (B) with the feasibility set C, which is the intersection of the set of solutions of an equilibrium problem and the set of a solutions of a variational inclusion, construct the two composite schemes, that is, the implicit and explicit schemes to converge strongly to the unique solution of the variational inequality (B). Recently, many authors introduced some kinds of algorithms for solving the variational inequality problems, but, in fact, our two schemes are more simple for finding solutions of the variational inequality (B) than others. MSC:49J30, 47H10, 47H17, 49M05.