Abstract and Applied Analysis
Volume 2012 (2012), Article ID 909187, 15 pages
Research Article

The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

1Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
2Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China

Received 15 August 2012; Accepted 27 August 2012

Academic Editor: Yongfu Su

Copyright © 2012 Zhiqun Xue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let be an arbitrary uniformly smooth real Banach space, let be a nonempty closed convex subset of , and let be a uniformly generalized Lipschitz generalized asymptotically -strongly pseudocontractive mapping with . Let be four real sequences in and satisfy the conditions: (i) , ; (ii) as and ; (iii) . For some , let be any bounded sequences in , and let be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of is equivalent to that of .