Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces
Let \(E\) be a real Banach space with norm \(\|\cdot\|\), \(E^*\) its dual, \(\langle x,f\rangle\) denote the value of \(f\in E^*\) at \(x\in E\) and \(x_n\to x\) (respectively \(x_n\rightharpoonup x\), \(x_n\overset{*}\rightharpoonup x\)) denote strong (respectively weak, weak\(^*\)) convergence of the sequence \(\{x_n\}\) to \(x\). The norm of \(E\) is said to be smooth if \(\lim_{t\to 0}{\| x+ ty\|=\| x\|\over t}\) exists for each \(x\), \(y\) in its unit spere \(U= \{x\in E:\| x\|= 1\}\) 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 \(I(x)= \{f\in E^*:\langle x,f\rangle= \| x\|^2=\| f\|^2\}\) 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 \(x_n\rightharpoonup x\), \(I(x_n)\overset{*}\rightharpoonup I(x)\). In this paper, the author establishes the strong convergence of the iteration scheme \(\{x_n\}\) defined by \(x_{n+1}= \lambda_{n+1} a+(1- \lambda_{n+1}) T_{n+1} x_n\), \(n\geq 0\), \(a\), \(x_0\) 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 \textit{J. G. O'Hara}, \textit{P. Pillay} and \textit{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 \textit{J. S. Jung} and \textit{T. H. Kim} [Bull. Korean Math. Soc. 34, No. 1, 83--102 (1997; Zbl 0885.47020)] (see also \textit{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 [\textit{P.-L. Lions}, C. R. Acad. Sci., Paris, Sér. A 284, 1357--1359 (1977; Zbl 0349.47046); \textit{T. Shimizu} and \textit{W. Takahashi}, J. Math. Anal. Appl. 211, No. 1, 71--83 (1997; Zbl 0883.47075); \textit{R. Wittmann}, Arch. Math. 58, No. 5, 486--491 (1992; Zbl 0797.47036)] in the framework of a Hilbert space.
scientific article; zbMATH DE number 2136665
scientific article