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Approximation for fixed points of asymptotically nonexpansive mappings. (English) Zbl 0734.47037
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)].

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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##### 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
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