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

A general iterative method for hierarchical variational inequality problems in Hilbert spaces and... 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 16 (3) – Feb 22, 2012
25 pages

/lp/springer_journal/a-general-iterative-method-for-hierarchical-variational-inequality-sSivXzCJo7
Publisher
SP Birkhäuser Verlag Basel
Subject
Mathematics; Operator Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics; Potential Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0161-0
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Feb 22, 2012

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