Abstract
Let C be a nonempty closed convex subset of a real Hilbert space H. Let Q:C→C be a fixed contraction and S,T:C→C be two nonexpansive mappings such that Fix(T)≠∅. Consider the following two-step iterative algorithm:
It is proven that under appropriate conditions, the above iterative sequence {x n } converges strongly to \(\tilde{x}\in \mathrm{Fix}(T)\) which solves some variational inequality depending on a given criterion S, namely: find \(\tilde{x}\in H\) ; \(0\in (I-S)\tilde{x}+N_{\mathrm{Fix}(T)}\tilde{x}\) , where N Fix(T) denotes the normal cone to the set of fixed points of T.
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The first author was partially supposed by National Natural Science Foundation of China Grant 10771050.
The second author was partially supposed by the grant NSC 97-2221-E-230-017.
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Yao, Y., Liou, YC. & Marino, G. Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems. J. Appl. Math. Comput. 31, 433–445 (2009). https://doi.org/10.1007/s12190-008-0222-5
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DOI: https://doi.org/10.1007/s12190-008-0222-5
Keywords
- Nonexpansive mapping
- Two-step iterative algorithm
- Hierarchical fixed point problem
- Variational inequality