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

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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 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.