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:
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
Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space which are useful in nonlinear analysis and optimization.
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Lin, LJ., Takahashi, W. A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications. Positivity 16, 429–453 (2012). https://doi.org/10.1007/s11117-012-0161-0
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DOI: https://doi.org/10.1007/s11117-012-0161-0
Keywords
- Equilibrium problem
- Fixed point
- Inverse-strongly monotone mapping
- Hierarchical variational inequality problems
- Iteration procedure
- Maximal monotone operator
- Resolvent
- Strict pseudo-contraction