*(English)*Zbl 1062.47069

Let $E$ be a real Banach space with norm $\parallel \xb7\parallel $, ${E}^{*}$ its dual, $\langle x,f\rangle $ denote the value of $f\in {E}^{*}$ at $x\in E$ and ${x}_{n}\to x$ (respectively ${x}_{n}\rightharpoonup x$, ${x}_{n}\stackrel{*}{\rightharpoonup}x$) denote strong (respectively weak, weak${}^{*}$) convergence of the sequence $\left\{{x}_{n}\right\}$ to $x$. The norm of $E$ is said to be smooth if ${lim}_{t\to 0}\frac{\parallel x+ty\parallel =\parallel x\parallel}{t}$ exists for each $x$, $y$ in its unit spere $U=\{x\in E:\parallel x\parallel =1\}$ and is said to be uniformly smooth if the limit is attained uniformly for $(x,y)\in U\times U$. The duality mapping $I$ from $E$ into the family of nonempty weak-star compact subsets of its dual ${E}^{*}$ is defined by $I\left(x\right)=\{f\in {E}^{*}:\langle x,f\rangle =\parallel x{\parallel}^{2}=\parallel f{\parallel}^{2}\}$ for each $x\in E$. $I$ is single-valued if and only if $E$ is smooth. The single-valued $I$ is said to be weakly sequentially continuous if for each $\left\{{x}_{n}\right\}\in E$ with ${x}_{n}\rightharpoonup x$, $I\left({x}_{n}\right)\stackrel{*}{\rightharpoonup}I\left(x\right)$.

In this paper, the author establishes the strong convergence of the iteration scheme $\left\{{x}_{n}\right\}$ defined by ${x}_{n+1}={\lambda}_{n+1}a+(1-{\lambda}_{n+1}){T}_{n+1}{x}_{n}$, $n\ge 0$, $a$, ${x}_{0}$ in a closed convex subset of ${E}_{2}$ for infinitely many nonexpansive mappings ${T}_{n}:C\to C$ in a uniformly smooth Banach space $E$ with a weakly sequentially continuous duality maping.

The main theorem (Theorem 5) extends a recent result of *J. G. O’Hara*, *P. Pillay* and *H.-K. Xu* [Nonlinear Anal., Theory Methods Appl. 54A, No. 8, 1417–1426 (2003; Zbl 1052.47049)] to a Banach space setting. For the same iteration scheme, with finitely many mappings, a complementary result to a result of *J. S. Jung* and *T. H. Kim* [Bull. Korean Math. Soc. 34, No. 1, 83–102 (1997; Zbl 0885.47020)] (see also *H. H. Bauschke* [J. Math. Anal. Appl. 202, No. 1, 150–159 (1996; Zbl 0956.47024)]) is obtained by imposing some other condition on the sequence of parameters. The results proved in the present paper also improve results in [*P.-L. Lions*, C. R. Acad. Sci., Paris, Sér. A 284, 1357–1359 (1977; Zbl 0349.47046); *T. Shimizu* and *W. Takahashi*, J. Math. Anal. Appl. 211, No. 1, 71–83 (1997; Zbl 0883.47075); *R. Wittmann*, Arch. Math. 58, No. 5, 486–491 (1992; Zbl 0797.47036)] in the framework of a Hilbert space.

##### MSC:

47J25 | Iterative procedures (nonlinear operator equations) |

47H09 | Mappings defined by “shrinking” properties |

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |