Li, Xiao Nan; Gu, J. S. Strong convergence of modified Ishikawa iteration for a nonexpansive semigroup in Banach spaces. (English) Zbl 1226.47081 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 4, 1085-1092 (2010). Summary: We give certain control conditions for a modified Ishikawa iteration to compute common fixed points of a kind of nonexpansive semigroup in Banach spaces. These results improve and extend those in [S. Plubtieng and R. Wangkeeree, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, A, 3110–3118 (2009; Zbl 1203.47072)] and [Y.-S. Song and S.-M. Xu, J. Math. Anal. Appl. 338, No. 1, 152–161 (2008; Zbl 1138.47040)]. Cited in 1 ReviewCited in 11 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H10 Fixed-point theorems 47H20 Semigroups of nonlinear operators Keywords:normalized duality mapping; nonexpansive semigroup; strong convergence; common fixed point; modified Ishikawa iteration Citations:Zbl 1203.47072; Zbl 1138.47040 PDFBibTeX XMLCite \textit{X. N. Li} and \textit{J. S. Gu}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 4, 1085--1092 (2010; Zbl 1226.47081) Full Text: DOI References: [1] Browder, F. E., Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Natl. Acad. Sci. USA, 53, 1272-1276 (1965) · Zbl 0125.35801 [2] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75, 287-292 (1980) · Zbl 0437.47047 [3] Halpern, B., Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101 [4] Moudafi, A., Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241, 46-55 (2000) · Zbl 0957.47039 [5] Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060 [6] Kim, T. H.; Xu, H. K., Strong convergence of modified Mann iterations, Nonlinear Anal., 61, 51-60 (2005) · Zbl 1091.47055 [7] Somyot, P.; Rattanaporn, W., Strong convergence of modified Mann iterations for a countable family of nonexpansive mappings, Nonlinear Anal. (2008) [8] Takahashi, W., Nonlinear Functional Analysis—Fixed Point Theory and Its Applications (2000), Yokohama Publishers Inc.: Yokohama Publishers Inc. Yokohama, (in Japanese) · Zbl 0997.47002 [9] Megginson, R. E., An Introduction to Banach Space Theory (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0910.46008 [10] Chen, R.; Song, Y., Convergence to common fixed point of nonexpansive semigroup, J. Comput. Appl. Math., 200, 566-575 (2007) · Zbl 1204.47076 [11] Song, Y.; Xu, S., Strong convergence theorems for nonexpansive semigroup in Banach spaces, J. Math. Anal. Appl., 338, 152-161 (2008) · Zbl 1138.47040 [12] Chang, S. S., On Chidume’s open questions and approximation solutions of multi-valued strongly accretive mapping equation in Banach spaces, J. Math. Anal. Appl., 216, 94-111 (1997) · Zbl 0909.47049 [13] Xu, H. K., Iterative algorithms for nonlinear operators, J. London Math. Soc., 2, 240-256 (2002) · Zbl 1013.47032 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.