A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let alpha > 0 and let A be an alpha-inverse-strongly monotone mapping of C into H and let B be a maximal monotone operator on H. Let F be a maximal monotone operator on H such that the domain of F is included in C. Let 0 < k < 1 and let g be a k-contraction of H into itself. Let V be a -strongly monotone and L-Lipschitzian continuous operator with and L > 0. Take mu, gamma is an element of Ras follows 0 < mu < 2 (gamma) over bar /L-2, 0 < gamma < (gamma) over bar - L-2 mu/2/k. In this paper, under the assumption (A + B)(-1)0 boolean AND F(-1)0 not equal empty set, we prove a strong convergence theorem for finding a point z(0) is an element of (A + B)(-1)0 boolean AND F(-1)0 which is a unique solution of the hierarchical variational inequality <(V - gamma g)z(0), q - z(0)> >= 0, for all(q) is an element of (A + B)(-1)0 boolean AND F(-1)0. Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.