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Positivity

, Volume 16, Issue 3, pp 429–453 | Cite as

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

  • Lai-Jiu Lin
  • Wataru Takahashi
Article

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.

Keywords

Equilibrium problem Fixed point Inverse-strongly monotone mapping Hierarchical variational inequality problems Iteration procedure Maximal monotone operator Resolvent Strict pseudo-contraction 

Mathematics Subject Classification (2000)

47H05 47H10 58E35 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsNational Changhua University of EducationChanghuaTaiwan
  2. 2.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyTokyoJapan

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