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Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems

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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:

$$\begin{array}{@{}rll}x_{n+1}&=&\alpha_{n}Qx_{n}+(1-\alpha_{n})Ty_{n},\\[1.5mm]y_{n}&=&\beta_{n}Sx_{n}+(1-\beta_{n})x_{n},\quad n\geq0.\end{array}$$

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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Authors and Affiliations

  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, China

    Yonghong Yao

  2. Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan

    Yeong-Cheng Liou

  3. Dipartimento di Matematica, Universitá della Calabria, 87036, Arcavacata di Rende, (CS), Italy

    Giuseppe Marino

Authors
  1. Yonghong Yao
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  2. Yeong-Cheng Liou
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  3. Giuseppe Marino
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Correspondence to Yonghong Yao.

Additional information

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

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