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Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space. (English) Zbl 1153.65340

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 H. H. Bauschke [J. Math. Anal. Appl. 202, No. 1, 150–159 (1996; Zbl 0956.47024)], B. Halpern [Bull. Am. Math. Soc. 73, 957–961 (1967; Zbl 0177.19101)], P. L. Lions [C. R. Acad. Sci., Paris, Sér. A 284, 1357–1359 (1977; Zbl 0349.47046)], R. Wittmann [Arch. Math. 58, No. 5, 486–491 (1992; Zbl 0797.47036)], 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)], N. Shioji and W. Takahashi [Proc. Am. Math. Soc. 125, No. 12, 3641–3645 (1997; Zbl 0888.47034)], W. Takahashi et al. [J. Approximation Theory 91, No. 3, 386–397 (1997; Zbl 0904.47045)], and 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:

65J15 Numerical solutions to equations with nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
49M05 Numerical methods based on necessary conditions
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References:

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