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Strong convergence theorem for two commutative asymptotically nonexpansive mappings in Hilbert space. (English) Zbl 1219.47119
Summary: Let \(C\) be a bounded closed convex subset of a Hilbert space \(H\) and \(T,S:C\rightarrow C\) be two asymptotically nonexpansive mappings such that \(ST=TS\). We establish a strong convergence theorem for \(S\) and \(T\) in Hilbert space by a hybrid method. The results generalize and unify many corresponding results.
MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, no. 1, pp. 171-174, 1972. · Zbl 0256.47045
[2] H. Ishihara and W. Takahashi, “A nonlinear ergodic theorem for a reversible semigroup of Lipschitzian mappings in a Hilbert space,” Proceedings of the American Mathematical Society, vol. 104, no. 2, pp. 431-436, 1988. · Zbl 0692.47010
[3] Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,” Bulletin of the American Mathematical Society, vol. 73, no. 4, pp. 591-597, 1967. · Zbl 0179.19902
[4] S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, pp. 147-150, 1974. · Zbl 0286.47036
[5] W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506-510, 1953. · Zbl 0050.11603
[6] J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153-159, 1991. · Zbl 0709.47051
[7] 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
[8] T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 1140-1152, 2006. · Zbl 1090.47059
[9] 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
[10] Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, Ph.D. thesis, Université de Paris, Paris, France, 1968.
[11] S. Plubtieng and K. Ungchittrakool, “Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2306-2315, 2007. · Zbl 1133.47051
[12] C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400-2411, 2006. · Zbl 1105.47060
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