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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 $T:K\to K$ be a nonexpansive mapping with nonempty set of all fixed points $F\left(T\right)$. For a sequence $\left\{{\alpha }_{n}\right\}$ of real numbers in $\left[0,1\right]$ and an arbitrary $u\in K$, the sequence $\left\{{x}_{n}\right\}$ in $K$ defined by ${x}_{0}\in K$ and

${x}_{n+1}={\alpha }_{n}u+\left(1-{\alpha }_{n}\right)T{x}_{n},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}n\ge 0,$

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 $Sx:=\left(1-\delta \right)x+\delta Tx,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\delta \in \left(0,1\right)$, instead of $T$, and prove a strong convergence theorem for approximating the fixed points of $T$.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces