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Iterative approaches to common fixed points of asymptotically nonexpansive mappings. (English) Zbl 1163.47055

Let D be a nonempty closed convex subset of a Banach space X. A mapping T:DD is said to be asymptotically nonexpansive if there exists a sequence k n of reals with k n 1 such that T h x-T hy k n x-y for all x,yD. A continuous strictly increasing function φ defined on + :=[0,) such that φ(0)=0 and lim r φ(r)= is called a gauge. A duality mapping J φ :xx * associated with gauge φ is defined as

J φ (x)={x * X * :x * (x)=xφ(x)andx * =φ(x)}·

X is said to have a weakly continuous duality mapping if a gauge φ exists such that the duality mapping J φ is single-valued and continuous from X with the weak topology to X * with the weak * topology.

In this paper, the authors design an iterative algorithm that converges strongly to common fixed points of a sequence of asymptotically nonexpansive mappings in a reflexive Banach space with weakly continuous duality mapping. The results proved in the paper show that the uniform smoothness requirement imposed on the space in the main results 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)], J. S. Jung [J. Math. Anal. Appl. 302, No. 2, 509–520 (2005; Zbl 1062.47069)] and R. Wittmann [Arch. Math. 58, No. 5, 486–491 (1992; Zbl 0797.47036)] is not required.

47J25Iterative procedures (nonlinear operator equations)
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces