International Journal of Mathematics and Mathematical Sciences 
Volume 2006 (2006), Article ID 69542, 12 pages
doi:10.1155/IJMMS/2006/69542

The equivalence between the convergences of Mann and Ishikawa iteration methods with errors for demicontinuous φ-strongly accretive operators in uniformly smooth Banach spaces

Zeqing Liu,1 Li Wang,2 Jeong Sheok Ume,3 and Shin Min Kang4

 1Department of Mathematics, Liaoning Normal University Dalian, Liaoning 116029, China
2Department of Science, Shenyang Institute of Aeronautical Engineering Shenyang, Liaoning 110034, China
3Department of Applied Mathematics, Changwon National University, Changwon 641-773, Korea
4Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, Korea
Received 22 October 2004; Revised 5 March 2006; Accepted 12 March 2006

Abstract

We investigate the equivalence between the convergences of the Mann iteration method and the Ishikawa iteration method with errors for demicontinuous φ-strongly accretive operators in uniformly smooth Banach spaces. A related result deals with the equivalence of the Mann iteration method and the Ishikawa iteration method for φ-pseudocontractive operators in nonempty closed convex subsets of uniformly smooth Banach spaces. The results presented in this paper extend and improve the corresponding results in the literature.