Let be a real Banach space, the topological dual space of , the dual pair between and , , the domain of and the set of all fixed points of , respectively, and the normalized duality mapping defined by , . A mapping is said to be
(1) asymptotically nonexpansive if there exists a sequence in with such that for all and ,
(2) asymptotically pseudo-contractive if there exists a sequence in with and for any there exists such that for all .
In this paper some convergence theorems of modified Ishikawa and Mann iterative sequence with errors for asymptotically pseudo-contractive and asymptotically nonexpansive mappings in Banach spaces are obtained.
Let be a mapping, let be a nonempty convex subset of , let be a given point, and let , , and be four sequences in . Then the sequence defined by , for all is called the modifies Ishikawa iterative sequence with errors of , where and are two bounded sequences in .
If and , , then . The sequence , , is called the modified Mann iterative sequences with errors of . Main result is the following:
Theorem 1. Let be a real uniformly smooth Banach space, let be a non-empty bounded closed convex subset of , let be an asymptotically pseudo-contractive mapping with a sequence , , and let . Let , , , and be four sequences in satisfying the following conditions:
(i) , ;
(ii) , , ;
(iii) , .
Let be any given point and let , be the modified Ishikawa iterative sequence with errors. Then:
(1) If converges strongly to a fixed point of in , there exists a nondecreasing function , such that , for all .
(2) Conversely, if there exists a strictly increasing function , satisfying preceding inequality, then .
A similar result for asymptotically nonexpansive mappings is proved. If is the modified Mann iterative sequence with errors, a similar result for asymptotically pseudo-contractive mappings is obtained.
This paper generalizes the results by [K. Goebel and W. A. Kirk, Proc. Am. Math. Soc. 35, No. 1, 171-174 (1972; Zbl 0256.47045), W. A. Kirk, Am. Math. Mont. 72, 1004-1006 (1965; Zbl 0141.32402), Q. H. Liu, Nonlinear Anal. Theory Methods Appl. 26, No. 11, 1835-1842 (1996; Zbl 0861.47047), J. Schu, J. Math. Anal. Appl. 15, No. 2, 407-413 (1991; Zbl 0734.47036)].