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Gu, Gendai; Wang, Shenghua; Cho, Yeol Je

Strong convergence algorithms for hierarchical fixed points problems and variational inequalities. (English) Zbl 1227.49013

J. Appl. Math. 2011, Article ID 164978, 17 p. (2011).
Summary: We introduce a new iterative scheme that converges strongly to a common fixed point of a countable family of nonexpansive mappings in a Hilbert space such that the common fixed-point is a solution of a hierarchical fixed-point problem. Our results extend the ones of Moudafi, Xu, Cianciaruso et al., and Yao et al.

Cited in 13 Documents

MSC:

49J40 Variational inequalities
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47J25 Iterative procedures involving nonlinear operators

Keywords:

iterative scheme; nonexpansive mappings in a Hilbert space; hierarchical fixed- point problem
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\textit{G. Gu} et al., J. Appl. Math. 2011, Article ID 164978, 17 p. (2011; Zbl 1227.49013)
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References:

[1] Y. H. Yao, Y. J. Cho, and Y. C. Liou, “Iterative algorithms for hierarchical fixed points problems and variational inequalities,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1697-1705, 2010. · Zbl 1205.65192
[2] A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,” Inverse Problems, vol. 23, no. 4, pp. 1635-1640, 2007. · Zbl 1128.47060
[3] P. E. Mainge and A. Moudafi, “Strong convergence of an iterative method for hierarchical fixed-point problems,” Pacific Journal of Optimization, vol. 3, no. 3, pp. 529-538, 2007. · Zbl 1158.47057
[4] G. Marino and H.-K. Xu, “Explicit hierarchical fixed point approach to variational inequalities,” Journal of Optimization Theory and Applications, vol. 149, no. 1, pp. 61-78, 2011. · Zbl 1221.49012
[5] F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, vol. 2009, Article ID 208692, 13 pages, 2009. · Zbl 1180.47040
[6] A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46-55, 2000. · Zbl 0957.47039
[7] H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279-291, 2004. · Zbl 1061.47060
[8] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0708.47031
[9] G. Marino and H.-K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 43-52, 2006. · Zbl 1095.47038
[10] P.-E. Maingé, “Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 469-479, 2007. · Zbl 1111.47058
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