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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 ·, E * its dual, x,f denote the value of fE * at xE and x n x (respectively x n x, x n *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 t0 x+ty=x t exists for each x, y in its unit spere U={xE:x=1} and is said to be uniformly smooth if the limit is attained uniformly for (x,y)U×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)={fE * :x,f=x 2 =f 2 } for each xE. 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 }E with x n x, I(x n ) *I(x).

In this paper, the author establishes the strong convergence of the iteration scheme {x n } defined by x n+1 =λ n+1 a+(1-λ n+1 )T n+1 x n , n0, a, x 0 in a closed convex subset of E 2 for infinitely many nonexpansive mappings T n :CC 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.

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