Strong convergence on iterative methods of Cesàro means for nonexpansive mapping in Banach space. (English) Zbl 1472.47090

Summary: Two new iterations with Cesàro’s means for nonexpansive mappings are proposed and the strong convergence is obtained as \(n \rightarrow \infty\). Our main results extend and improve the corresponding results of H.-K. Xu [J. Math. Anal. Appl. 298, No. 1, 279–291 (2004; Zbl 1061.47060)], Y.-S. Song and the second author [Appl. Math. Comput. 186, No. 2, 1120–1128 (2007; Zbl 1121.65063)], and Y.-H. Yao et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 6, 2332–2336 (2009; Zbl 1223.47107)].


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


[1] Baillon, J. B., Un theoreme de type ergodique pour les contractions non lineairs dans un espaces de Hilbert, Comptes Rendus de l’Académie des Sciences A, 280, 22, 1511-1514 (1975) · Zbl 0307.47006
[2] Bruck, R. E., A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel Journal of Mathematics, 32, 2-3, 107-116 (1979) · Zbl 0423.47024 · doi:10.1007/BF02764907
[3] Xu, H.-K., Viscosity approximation methods for nonexpansive mappings, Journal of Mathematical Analysis and Applications, 298, 1, 279-291 (2004) · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[4] Song, Y.; Chen, R., Viscosity approximate methods to Cesàro means for non-expansive mappings, Applied Mathematics and Computation, 186, 2, 1120-1128 (2007) · Zbl 1121.65063 · doi:10.1016/j.amc.2006.08.054
[5] Yao, Y.; Liou, Y.-C.; Zhou, H., Strong convergence of an iterative method for nonexpansive mappings with new control conditions, Nonlinear Analysis: Theory, Methods & Applications, 70, 6, 2332-2336 (2009) · Zbl 1223.47107 · doi:10.1016/j.na.2008.03.014
[6] Takahashi, W., Nonlinear Functional Analysis. Nonlinear Functional Analysis, Fixed Point Theory and its Applications, Yokohama, Japan: Yokohama Publishers, Yokohama, Japan · Zbl 1223.47107 · doi:10.1016/j.na.2008.03.014
[7] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, Journal of Mathematical Analysis and Applications, 75, 1, 287-292 (1980) · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6
[8] Xu, H.-K., Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66, 1, 240-256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
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