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

MSC:
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
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Full Text: DOI
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 · doi:10.1016/j.mcm.2010.06.038
[2] A. Moudafi, “Krasnoselski-Mann iteration for hierarchical fixed-point problems,” Inverse Problems, vol. 23, no. 4, pp. 1635-1640, 2007. · Zbl 1128.47060 · doi:10.1088/0266-5611/23/4/015
[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 · http://www.ybook.co.jp/online/pjoe/vol3/pjov3n3p529.html
[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 · doi:10.1007/s10957-010-9775-1
[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 · doi:10.1155/2009/208692 · eudml:117785
[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 · doi:10.1006/jmaa.1999.6615
[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 · doi:10.1016/j.jmaa.2004.04.059
[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 · doi:10.1017/CBO9780511526152
[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 · doi:10.1016/j.jmaa.2005.05.028
[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 · doi:10.1016/j.jmaa.2005.12.066