Schu, Jürgen Approximation for fixed points of asymptotically nonexpansive mappings. (English) Zbl 0734.47037 Proc. Am. Math. Soc. 112, No. 1, 143-151 (1991). The author studies the convergence of the iteration sequence \(z_{n+1}=\mu_{n+1}T^ n(z_ n),\) where T is an asymptotically nonexpansive self-mapping of a nonempty closed, bounded, and starshaped subset of a smooth reflexive Banach space. Related previous work is due to K. Goebel, B. Halpern, W. A. Kirk, and P. Vijayaraju [e.g. Bull. Calcutta Math. Soc. 80, No.2, 133-136 (1988; Zbl 0667.47032)]. Reviewer: J.Appell (Würzburg) Cited in 36 Documents MSC: 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. Keywords:convergence of the iteration sequence; asymptotically nonexpansive self- mapping of a nonempty closed, bounded, and starshaped subset of a smooth reflexive Banach space Citations:Zbl 0667.47032 PDFBibTeX XMLCite \textit{J. Schu}, Proc. Am. Math. Soc. 112, No. 1, 143--151 (1991; Zbl 0734.47037) Full Text: DOI References: [1] Bernard Beauzamy, Introduction to Banach spaces and their geometry, North-Holland Mathematics Studies, vol. 68, North-Holland Publishing Co., Amsterdam-New York, 1982. Notas de Matemática [Mathematical Notes], 86. · Zbl 0491.46014 [2] Joseph Diestel, Geometry of Banach spaces — selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. · Zbl 0307.46009 [3] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171 – 174. · Zbl 0256.47045 [4] Benjamin Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957 – 961. · Zbl 0177.19101 [5] S. K. Samanta, Fixed point theorems in a Banach space satisfying Opial’s condition, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1-4, 251 – 258 (1984). · Zbl 0636.47046 [6] Jürgen Schu, Iterative approximation of fixed points of nonexpansive mappings with starshaped domain, Comment. Math. Univ. Carolin. 31 (1990), no. 2, 277 – 282. · Zbl 0717.47022 [7] P. Vijayaraju, Fixed point theorems for asymptotically nonexpansive mappings, Bull. Calcutta Math. Soc. 80 (1988), no. 2, 133 – 136. · Zbl 0667.47032 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.