Approximation of common fixed points of nonexpansive semigroups in Hilbert spaces. (English) Zbl 1250.47079

The authors prove the strong convergence of an iterative sequence generated by a nonexpansive semigroup and a contraction mapping.


47J25 Iterative procedures involving nonlinear operators
47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
Full Text: DOI


[1] F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041-1044, 1965. · Zbl 0128.35801
[2] 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, pp. 82-90, 1967. · Zbl 0148.13601
[3] 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
[4] B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957-961, 1967. · Zbl 0177.19101
[5] P.-L. Lions, “Approximation de points fixes de contractions,” Comptes Rendus de l’AcadĂ©mie des Sciences, vol. 284, no. 21, pp. A1357-A1359, 1977. · Zbl 0349.47046
[6] R. Wittmann, “Approximation of fixed points of nonexpansive mappings,” Archiv der Mathematik, vol. 58, no. 5, pp. 486-491, 1992. · Zbl 0797.47036
[7] F. Deutsch and I. Yamada, “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 33-56, 1998. · Zbl 0913.47048
[8] S. Li, L. Li, and Y. Su, “General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 9, pp. 3065-3071, 2009. · Zbl 1177.47075
[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
[10] G. Marino, V. Colao, X. Qin, and S. M. Kang, “Strong convergence of the modified Mann iterative method for strict pseudo-contractions,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 455-465, 2009. · Zbl 1165.65353
[11] H. K. Xu, “An iterative approach to quadratic optimization,” Journal of Optimization Theory and Applications, vol. 116, no. 3, pp. 659-678, 2003. · Zbl 1043.90063
[12] 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
[13] I. Yamada, N. Ogura, Y. Yamashita, and K. Sakaniwa, “Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space,” Numerical Functional Analysis and Optimization, vol. 19, no. 1-2, pp. 165-190, 1998. · Zbl 0911.47051
[14] S. Plubtieng and R. Punpaeng, “Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces,” Mathematical and Computer Modelling, vol. 48, no. 1-2, pp. 279-286, 2008. · Zbl 1169.47055
[15] N. Shioji and W. Takahashi, “Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 34, no. 1, pp. 87-99, 1998. · Zbl 0935.47039
[16] T. Shimizu and W. Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 211, no. 1, pp. 71-83, 1997. · Zbl 0883.47075
[17] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28, Cambridge University Press, Cambridge, UK, 1990. · Zbl 0708.47031
[18] S.-S. Chang, Y. J. Cho, and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science Publishers, Huntington, NY, USA, 2002. · Zbl 1070.47054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.