Weak and strong convergence theorems for nonexpansive mappings. (English) Zbl 1011.47040

The Mann and Ishikawa iterative processes with errors for a self operator \(T\) on a Banach space were introduced by L. S. Liu [Indian J. Pure Appl. 26, 649-659 (1995; Zbl 0853.47029) and J. Math. Anal. Appl. 194, 114-125 (1995; Zbl 0872.47031)] in order to study the random occurrence of errors in concrete numerical computations, where strong convergence theorems for different classes of operators (Lipschitzian and strongly accretive; Lipschitzian and strictly pseudocontractive; demicontinuous and strongly accretive) were obtained. Afterwards, many other authors have proved convergence theorems for the Mann and Ishikawa iterations with errors associated to an operator satisfying several contractive type conditions: Z.-Y. Huang [Comput. Math. Appl. 36, 13-21 (1998; Zbl 0938.47042); Comput. Math. Appl 37, 1-7 (1999; Zbl 0942.47046) and Comput. Math. Appl 39, 137-317 (2000; Zbl 0956.47025)]; L.-S. Liu [Nonlinear Anal., Theory Methods Appl. 34, 307-317 (1998; Zbl 0931.47055)]; M. O. Osilike [J. Math. Anal. Appl. 213, 91-105 (1998; Zbl 0904.47056)], and others.
Later, Y.-G. Xu [J. Math. Anal. Appl. 224, 91-101 (1995; Zbl 0936.47041)] argued that Liu’s definition is not fully appropriate and proposed an improved version of it by inserting the error terms using linear convex combinations. Convergence theorems for the Mann and Ishikawa iterations with errors in this new form have been also obtained by C. E. Chidume and C. Moore [Proc. Am. Math. Soc. 127, 1163-1170 (1999; Zbl 0913.47052)] for a continuous hemicontractive self map.
In the paper under review, the authors prove various weak and strong convergence theorems for Ishikawa type iterative processes with errors associated to one operator \(T\) or to a pair \(S\), \(T\) of self-operators, defined on a closed convex subset of a real Banach space. Some of the obtained results are generalizations of the previous results of the last author and his collaborators [W. Takahashi and G. E. Kim, Math. Jap. 48, 1-9 (1998; Zbl 0913.47056) and W. Takahashi and T. Tamura, J. Convex. Anal. 5, 45-56 (1998; Zbl 0916.47042)].


47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems