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

Abstract

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let α > 0 and let A be an α-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 ${\overline{\gamma}}$-strongly monotone and L-Lipschitzian continuous operator with ${\overline{\gamma} >0 }$ and L > 0. Take ${\mu, \gamma \in \mathbb R}$ as follows:

$${0 < \mu < \frac{2\overline{\gamma}}{L^2}, \quad 0 < \gamma < \frac{\overline{\gamma}-\frac{L^2 \mu}{2}}{k}.}$$

In this paper, under the assumption ${(A+B)^{-1}0 \cap F^{-1}0 \neq \emptyset}$, we prove a strong convergence theorem for finding a point ${z_0\in (A+B)^{-1}0\cap F^{-1}0}$ which is a unique solution of the hierarchical variational inequality

$${\langle (V-\gamma g)z_0, q-z_0 \rangle \geq 0, \quad \forall q\in (A+B)^{-1}0 \cap F^{-1}0.}$$

Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.