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Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. (English) Zbl 1062.47069
Let $E$ be a real Banach space with norm $\Vert\cdot\Vert$, $E^*$ its dual, $\langle x,f\rangle$ denote the value of $f\in E^*$ at $x\in E$ and $x_n\to x$ (respectively $x_n\rightharpoonup x$, $x_n\overset*\to\rightharpoonup x$) denote strong (respectively weak, weak$^*$) convergence of the sequence $\{x_n\}$ to $x$. The norm of $E$ is said to be smooth if $\lim_{t\to 0}{\Vert x+ ty\Vert=\Vert x\Vert\over t}$ exists for each $x$, $y$ in its unit spere $U= \{x\in E:\Vert x\Vert= 1\}$ 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 $I(x)= \{f\in E^*:\langle x,f\rangle= \Vert x\Vert^2=\Vert f\Vert^2\}$ 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 $\{x_n\}\in E$ with $x_n\rightharpoonup x$, $I(x_n)\overset*\to\rightharpoonup I(x)$. In this paper, the author establishes the strong convergence of the iteration scheme $\{x_n\}$ 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 {\it J. G. O’Hara}, {\it P. Pillay} and {\it 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 {\it J. S. Jung} and {\it T. H. Kim} [Bull. Korean Math. Soc. 34, No. 1, 83--102 (1997; Zbl 0885.47020)] (see also {\it 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 [{\it P.-L. Lions}, C. R. Acad. Sci., Paris, Sér. A 284, 1357--1359 (1977; Zbl 0349.47046); {\it T. Shimizu} and {\it W. Takahashi}, J. Math. Anal. Appl. 211, No. 1, 71--83 (1997; Zbl 0883.47075); {\it R. Wittmann}, Arch. Math. 58, No. 5, 486--491 (1992; Zbl 0797.47036)] in the framework of a Hilbert space.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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