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Zbl 1153.65340
Wu, Dingping; Chang, Shih-Sen; Yuan, George X.
Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 63, No. 5-7, A, 987-999 (2005). ISSN 0362-546X

Summary: Some sufficient and necessary conditions for the iterative sequence converging to a common fixed points for a family of nonexpansive mappings in Banach spaces are obtained. The results presented in this paper not only give an affirmative answer to Halpern's open question and a partial answer to the Reich's open question but also extend and improve some recent results of {\it H. H. Bauschke} [J. Math. Anal. Appl. 202, No. 1, 150--159 (1996; Zbl 0956.47024)], {\it B. Halpern} [Bull. Am. Math. Soc. 73, 957--961 (1967; Zbl 0177.19101)], {\it P. L. Lions} [C. R. Acad. Sci., Paris, Sér. A 284, 1357--1359 (1977; Zbl 0349.47046)], {\it R. Wittmann} [Arch. Math. 58, No.~5, 486--491 (1992; Zbl 0797.47036)], {\it S. Reich} [J. Math. Anal. Appl. 75, 287-292 (1980; Zbl 0437.47047); Panam. Math. J. 4, No. 2, 23--28 (1994; Zbl 0856.47032)], {\it N. Shioji} and {\it W. Takahashi} [Proc. Am. Math. Soc. 125, No. 12, 3641--3645 (1997; Zbl 0888.47034)], {\it W. Takahashi} et al. [J. Approximation Theory 91, No. 3, 386--397 (1997; Zbl 0904.47045)], and {\it H.-K. Xu} [Bull. Aust. Math. Soc. 65, 109--113 (2002; Zbl 1030.47036)]. As applications, at the end of the paper, we utilize our results to study the feasibility problem.
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
49M05 Methods of successive approximation based on necessary conditions

Keywords: nonexpansive mapping; iterative sequence; fixed point; uniformly smooth Banach space; normalized duality mapping

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