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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:
65J15Equations with nonlinear operators (numerical methods)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
49M05Numerical methods in calculus of variations based on necessary conditions