zbMATH — the first resource for mathematics

Strong convergence and certain control conditions for modified Mann iteration. (English) Zbl 1189.47071
In this paper, the authors propose a new modified Mann iteration for computing fixed points of nonexpansive mappings in a Banach space setting. This new iterative scheme combines the modified Mann iteration introduced by T. H. Kim and H. K. Xu [“Strong convergence of modified Mann iterations”, Nonlinear Anal., Theory Methods Appl. 61, No. 1–2 (A), 51–60 (2005; Zbl 1091.47055)] and the viscosity approximation method introduced by A. Moudafi [“Viscosity approximation methods for fixed-points problems”, J. Math. Anal. Appl. 241, No. 1, 46–55 (2000; Zbl 0957.47039)]. The main results extend and improve some due to Xu and Kim in the aforementioned paper.

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
Full Text: DOI
[1] Browder, F.E.; Petryshyn, W.V., Construction of fixed points of nonlinear mappings, J. math. anal. appl., 20, 197-228, (1967) · Zbl 0153.45701
[2] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems, 20, 103-120, (2004) · Zbl 1051.65067
[3] Podilchuk, C.I.; Mammone, R.J., Image recovery by convex projections using a least-squares constraint, J. opt. soc. amer., 7, 517-521, (1990)
[4] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026
[5] Genel, A.; Lindenstrass, J., An example concerning fixed points Israel, J. math., 22, 81-86, (1975) · Zbl 0314.47031
[6] Halpern, B., Fixed points of nonexpansive maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[7] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[8] Moudafi, A., Viscosity approximation methods for fixed-points problems, J. math. anal. appl., 241, 46-55, (2000) · Zbl 0957.47039
[9] Kim, T.H.; Xu, H.K., Strong convergence of modified Mann iterations, Nonlinear anal., 61, 51-60, (2005) · Zbl 1091.47055
[10] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 2, 240-256, (2002) · Zbl 1013.47032
[11] Suzuki, T., Strong convergence of Krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. math. anal. appl., 305, 227-239, (2005) · Zbl 1068.47085
[12] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060
[13] Zeng, L.C.; Wong, N.C.; Yao, J.C., Strong convergence theorems for strictly pseudocontractive mappings of browder – petryshyn type, Taiwanese, J. math., 10, 837-850, (2006) · Zbl 1159.47054
[14] Lin, Y.C.; Wong, N.C.; Yao, J.C., Strong convergence theorems of Ishikawa iteration process with errors for fixed points of Lipschitz continuous mappings in Banach spaces, Taiwanese, J. math., 10, 543-552, (2006) · Zbl 1118.47053
[15] Zeng, L.C.; Yao, J.C., Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear anal., 64, 2507-2515, (2006) · Zbl 1105.47061
[16] Lin, Y.C., Three-step iterative convergence theorems with errors in Banach spaces, Taiwanese J. math., 10, 75-86, (2006) · Zbl 1106.47058
[17] Shioji, N.; Takahashi, W., Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. amer. math. soc., 125, 3641-3645, (1997) · Zbl 0888.47034
[18] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publishers Yokohama
[19] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036
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.