*(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 $[0,1]$ and an arbitrary $u\in K$, the sequence $\left\{{x}_{n}\right\}$ in $K$ defined by ${x}_{0}\in K$ and

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 $Sx:=(1-\delta )x+\delta Tx,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\delta \in (0,1)$, 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 |