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Strong convergence of viscosity iteration methods for nonexpansive mappings. (English) Zbl 1168.47053

Summary: We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for the viscosity iterative scheme are given and the strong convergence of the viscosity iterative scheme to a solution of a certain variational inequality is established.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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[1] F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, no. 1, pp. 82-90, 1967. · Zbl 0148.13601
[2] S. S. Chang, “On Halpern/s open question,” Acta Mathematica Sinica, vol. 48, no. 5, pp. 979-984, 2005. · Zbl 1125.47315
[3] Y. J. Cho, S. M. Kang, and H. Zhou, “Some control conditions on iterative methods,” Communications on Applied Nonlinear Analysis, vol. 12, no. 2, pp. 27-34, 2005. · Zbl 1088.47053
[4] B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, no. 6, pp. 957-961, 1967. · Zbl 0177.19101
[5] P.-L. Lions, “Approximation de points fixes de contractions,” Comptes Rendus de l/Académie des Sciences. Série A-B, vol. 284, no. 21, pp. 1357-1359, 1977. · Zbl 0349.47046
[6] S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287-292, 1980. · Zbl 0437.47047
[7] S. Reich, “Approximating fixed points of nonexpansive mappings,” PanAmerican Mathematical Journal, vol. 4, no. 2, pp. 23-28, 1994. · Zbl 0856.47032
[8] N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641-3645, 1997. · Zbl 0888.47034
[9] R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486-491, 1992. · Zbl 0797.47036
[10] H.-K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240-256, 2002. · Zbl 1013.47032
[11] 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
[12] 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
[13] N. C. Wong, D. R. Sahu, and J. C. Yao, “Solving variational inequalities involving nonexpansive type mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 12, pp. 4732-4753, 2008. · Zbl 1182.47050
[14] T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 61, no. 1-2, pp. 51-60, 2005. · Zbl 1091.47055
[15] 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
[16] X. Qin, Y. Su, and M. Shang, “Strong convergence of the composite Halpern iteration,” Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 996-1002, 2008. · Zbl 1135.47052
[17] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. · Zbl 0712.47043
[18] K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984. · Zbl 0537.46001
[19] J. S. Jung and C. H. Morales, “The Mann process for perturbed m-accretive operators in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 2, pp. 231-243, 2001. · Zbl 0997.47042
[20] T. Suzuki, “Strong convergence of Krasnoselskii and Mann/s type sequences for one-parameter nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227-239, 2005. · Zbl 1068.47085
[21] L. S. Liu, “Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114-125, 1995. · Zbl 0872.47031
[22] B. E. Rhoades, “Some theorems on weakly contractive maps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 4, pp. 2683-2693, 2001. · Zbl 1042.47521
[23] Ya. I. Alber and A. N. Iusem, “Extension of subgradient techniques for nonsmooth optimization in Banach spaces,” Set-Valued Analysis, vol. 9, no. 4, pp. 315-335, 2001. · Zbl 1049.90123
[24] Ya. I. Alber, S. Reich, and J.-C. Yao, “Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2003, no. 4, pp. 193-216, 2003. · Zbl 1028.47049
[25] J. S. Jung, “Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2536-2552, 2006. · Zbl 1101.47053
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