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

作者：Lin Lai Jiu^*; Takahashi Wataru

来源：Positivity, 2012, 16(3): 429-453.

DOI：10.1007/s11117-012-0161-0

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

Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. Let alpha %26gt; 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 %26lt; k %26lt; 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 %26gt; 0. Take mu, gamma is an element of Ras follows %26lt;br%26gt;0 %26lt; mu %26lt; 2 (gamma) over bar /L-2, 0 %26lt; gamma %26lt; (gamma) over bar - L-2 mu/2/k. %26lt;br%26gt;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 %26lt;br%26gt;%26lt;(V - gamma g)z(0), q - z(0)%26gt; %26gt;= 0, for all(q) is an element of (A + B)(-1)0 boolean AND F(-1)0. %26lt;br%26gt;Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.

• 出版日期2012-9