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Iterative approximation of fixed points of nonexpansive mappings. (English) Zbl 1095.47034

Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\) which has a uniformly Gâteaux differentiable norm and let \(T: K\rightarrow K\) be a nonexpansive mapping with nonempty set of all fixed points \(F(T)\). For a sequence \(\{\alpha_n\}\) of real numbers in \([0,1]\) and an arbitrary \(u\in K\), the sequence \(\{x_n\}\) in \(K\) defined by \(x_0\in K\) and \[ x_{n+1}=\alpha_n u+ (1-\alpha_n) T x_n,\,\,n\geq 0, \] was introduced by B. Halpern [Bull. Am. Math. Soc. 73, 957–961 (1967; Zbl 0177.19101)] and subsequently studied by several authors in order to approximate the fixed points of \(T\), see, e.g., the reviewer’s recent monograph [V. Berinde, “Iterative approximation of fixed points” (Efemeride, Baia Mare) (2002; Zbl 1036.47037)].
In the present paper, the authors use the same kind of iterative scheme but with the averaged map \(S\), given by \(Sx:=(1-\delta)x+\delta Tx,\,\,\delta\in (0,1)\), instead of \(T\), and prove a strong convergence theorem for approximating the fixed points of \(T\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
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References:

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