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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.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
Full Text: DOI
[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