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Viscosity method for hierarchical fixed point problems with an infinite family of nonexpansive nonself-mappings. (English) Zbl 1266.47099

Summary: A viscosity method for hierarchical fixed point problems is presented to solve variational inequalities, where the involved mappings are nonexpansive nonself-mappings. Solutions are sought in the set of the common fixed points of an infinite family of nonexpansive nonself-mappings. The results generalize and improve the recent results announced by many other authors.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000. · Zbl 1184.76244 · doi:10.1063/1.1314337
[2] H.-K. Xu, “Approximating curves of nonexpansive nonself-mappings in Banach spaces,” Comptes Rendus de l’Académie des Sciences. Série I. Mathématique, vol. 325, no. 2, pp. 151-156, 1997. · Zbl 0888.47036 · doi:10.1016/S0764-4442(97)84590-9
[3] W. Takahashi and G.-E. Kim, “Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 32, no. 3, pp. 447-454, 1998. · Zbl 0947.47049 · doi:10.1016/S0362-546X(97)00482-3
[4] L. C. Ceng and A. Petru\csel, “Krasnoselski-Mann iterations for hierarchical fixed point problems for a finite family of nonself mappings in Banach spaces,” Journal of Optimization Theory and Applications, vol. 146, no. 3, pp. 617-639, 2010. · Zbl 1210.47094 · doi:10.1007/s10957-010-9679-0
[5] S.-A. Zhang, X.-R. Wang, H. W. J. Lee, and C.-K. Chan, “Viscosity method for hierarchical fixed point and variational inequalities with applications,” Applied Mathematics and Mechanics. English Edition, vol. 32, no. 2, pp. 241-250, 2011. · Zbl 1296.47105 · doi:10.1007/s10483-011-1410-8
[6] D. C. Youla, “Mathematical theory of image restoration by the method of convex projections,” in Image Recovery: Theory and Applications, H. Stark, Ed., pp. 29-78, Academic Press, New York, NY, USA, 1987.
[7] C. Byrne, “A unified treatment of some iterative algorithms in signal processing and image reconstruction,” Inverse Problems, vol. 20, no. 1, pp. 103-120, 2004. · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006
[8] I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), vol. 8 of Studies in Computational Mathematics, pp. 473-504, North-Holland, Amsterdam, The Netherlands, 2001. · Zbl 1013.49005 · doi:10.1016/S1570-579X(01)80028-8
[9] I. Yamada, N. Ogura, and N. Shirakawa, “A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems,” in Inverse Problems, Image Analysis, and Medical Imaging (New Orleans, LA, 2001), vol. 313 of Contemporary Mathematics, pp. 269-305, American Mathematical Society, Providence, RI, USA, 2002. · Zbl 1039.47051 · doi:10.1090/conm/313/05379
[10] 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
[11] Y. Yao and Y.-C. Liou, “Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems,” Inverse Problems, vol. 24, no. 1, Article ID 015015, 8 pages, 2008. · Zbl 1154.47055 · doi:10.1088/0266-5611/24/1/015015
[12] 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
[13] S.-Y. Matsushita and W. Takahashi, “Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 2, pp. 412-419, 2008. · Zbl 1169.47052 · doi:10.1016/j.na.2006.11.007
[14] P. L. Combettes, “Quasi-Fejérian analysis of some optimization algorithms,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 115-152, North-Holland, Amsterdam, The Netherlands, 2001. · Zbl 0992.65065 · doi:10.1016/S1570-579X(01)80010-0
[15] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society. Second Series, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[16] J.-P. Gossez and E. Lami Dozo, “Some geometric properties related to the fixed point theory for nonexpansive mappings,” Pacific Journal of Mathematics, vol. 40, no. 3, pp. 565-573, 1972. · Zbl 0223.47025 · doi:10.2140/pjm.1972.40.565
[17] J. S. Jung, “Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 509-520, 2005. · Zbl 1062.47069 · doi:10.1016/j.jmaa.2004.08.022
[18] Y. Song and R. Chen, “Viscosity approximation methods for nonexpansive nonself-mappings,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 316-326, 2006. · Zbl 1103.47053 · doi:10.1016/j.jmaa.2005.07.025
[19] J. G. O’Hara, P. Pillay, and H.-K. Xu, “Iterative approaches to convex feasibility problems in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 9, pp. 2022-2042, 2006. · Zbl 1139.47056 · doi:10.1016/j.na.2005.07.036
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