Let be a real Banach space with norm , its dual, denote the value of at and (respectively , ) denote strong (respectively weak, weak) convergence of the sequence to . The norm of is said to be smooth if exists for each , in its unit spere and is said to be uniformly smooth if the limit is attained uniformly for . The duality mapping from into the family of nonempty weak-star compact subsets of its dual is defined by for each . is single-valued if and only if is smooth. The single-valued is said to be weakly sequentially continuous if for each with , .
In this paper, the author establishes the strong convergence of the iteration scheme defined by , , , in a closed convex subset of for infinitely many nonexpansive mappings in a uniformly smooth Banach space 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.