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Strong convergence of the iterative methods for hierarchical fixed point problems of an infinite family of strictly nonself pseudocontractions. (English) Zbl 1253.65091
Summary: We deals with a new iterative algorithm for solving hierarchical fixed point problems of an infinite family of pseudocontractions in Hilbert spaces by $y_n = \beta_n S x_n + (1 - \beta_n)x_n, x_{n+1} = P_C[\alpha_n f(x_n) + (1 - \alpha_n) \sum^\infty_{i=1} \mu^{(n)}_i T_i y_n]$, and $\forall n \geq 0$, where $T_i : C \mapsto H$ is a nonself $k_i$-strictly pseudocontraction. Under certain approximate conditions, the sequence $\{x_n\}$ converges strongly to $x^\ast \in \bigcap^\infty_{i=1} F(T_i)$, which solves some variational inequality. The results here improve and extend some recent results.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 49J40 Variational methods including variational inequalities 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
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##### References:
 [1] 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 [2] 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 [3] 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 [4] L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “A general iterative method with strongly positive operators for general variational inequalities,” Computers & Mathematics with Applications. An International Journal, vol. 59, no. 4, pp. 1441-1452, 2010. · Zbl 1189.49006 · doi:10.1016/j.camwa.2009.11.007 [5] M. Tian, “A general iterative algorithm for nonexpansive mappings in Hilbert spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 3, pp. 689-694, 2010. · Zbl 1192.47064 · doi:10.1016/j.na.2010.03.058 [6] L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “A general composite iterative algorithm for nonexpansive mappings in Hilbert spaces,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2447-2455, 2011. · Zbl 1292.47042 · doi:10.1016/j.camwa.2011.02.025 [7] Y. H. Yao and Y. C. Liou, “Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems,” Abstract and Applied Analysis, vol. 2010, Article ID 763506, 19 pages, 2010. · Zbl 1203.49048 · doi:10.1155/2010/763506 · eudml:229127 [8] L.-C. Ceng, Q. H. Ansari, and J.-C. Yao, “Relaxed extragradient iterative methods for variational inequalities,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112-1123, 2011. · Zbl 1229.65109 · doi:10.1016/j.amc.2011.01.061 [9] Y. H. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extragradient method to the minium-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID Article ID 817436, 9 pages, 2012. · Zbl 1232.49011 · doi:10.1155/2012/817436 [10] Y. Yao, R. Chen, and J.-C. Yao, “Strong convergence and certain control conditions for modified Mann iteration,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 6, pp. 1687-1693, 2008. · Zbl 1189.47071 · doi:10.1016/j.na.2007.01.009 [11] Y. L. Song, H. Y. Hu, Y. Q. Wang, et al., “Strong convergence of a new general iterative method for variational inequality problems in Hilbert spaces,” Fixed Point Theory and Applications, vol. 2012, 46 pages, 2012. · Zbl 06210364 · doi:10.1186/1687-1812-2012-46 [12] 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 [13] Y. H. Wang and Y. H. Xia, “Strong convergence for asymptotically pseudocontractions with the demiclosedness principle in Banach spaces,” Fixed Point Theory and Applications, vol. 2012, 45 pages, 2012. · Zbl 1273.47087 · doi:10.1186/1687-1812-2012-45 [14] S. S. Chang, Y. J. Cho, and H. Y. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science Publisher Inc., Huntington, NY, USA, 2002. · Zbl 1070.47054 [15] H. Y. Zhou, “Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 2, pp. 456-462, 2008. · Zbl 1220.47139 · doi:10.1016/j.na.2007.05.032 [16] F. E. Browder and W. V. Petryshyn, “Construction of fixed points of nonlinear mappings in Hilbert space,” Journal of Mathematical Analysis and Applications, vol. 20, pp. 197-228, 1967. · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6 [17] Y. H. Wang and L. Yang, “Modified relaxed extragradient method for a general system of variational inequalities and nonexpansive mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 818970, 14 pages, 2012. · Zbl 1246.65094 · doi:10.1155/2012/818970