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Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces. (English) Zbl 1100.65049

The authors generalize and extend iterative methods to obtain a common fixed point of a finite number of nonexpansive mappings, those methods being connected, a.o., to the names of 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)], H. H. Bauschke [J. Math. Anal. Appl. 202, No. 1, 150–159 (1996; Zbl 0956.47024)], W. Takahashi, T. Tamura and M. Toyoda [Sci. Math. Jpn. 56, No. 3, 475–480 (2002; Zbl 1026.47042)]. Essentially, the generalization and extension has to do, on the one hand, with the kind of space (a Banach space with suitable properties) and, on the other hand, with the control conditions and the resulting sequence. To put it shortly, let \(T(1)\), \(T(2),\dots, T(r)\) be the given nonexpansive mappings from a suitable subset \(C\) to itself; given points \(u\) and \(x(0)\) in \(C\), a sequence \(\{x(n)\}\) is generated by \[ x(n+1)= \alpha(n)u+ (1-\alpha(n)) T(n+1) x(n),\quad n\geq 0,\tag{B} \] where \(T(n)= T(n\text{mod\,}r)\) and where \(\{a(n)\}\) is a real sequence in \([0,1]\) which satisfies the following two conditions:
(C1): \(\alpha(n)\to 0\) when \(n\to\infty\),
(C2): \(\Sigma\alpha(n)= \infty\).
Then, in Theorem 6, the authors state conditions about the Banach space and the sequence \(\{u(n)\}\) under which the sequence defined by (B) converges strongly to a common fixed point, assuming only the control conditions (C1) and (C2). As an application, they introduce strong convergence theorems connected with the feasibility problem.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
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