JOURNAL BROWSE
Search
Advanced SearchSearch Tips
CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS
Kang, Jung-Im; Cho, Yeol-Je; Zhou, Hai-Yun;
  PDF(new window)
 Abstract
In this paper, we will prove the following: Let D be a nonempty of a normed linear space X and T : D -> X be a nonexpansive mapping. Let be a sequence in D and , be real sequences such that (i) $0\;{\leq}\;t_n\;{\leq}\;t\;<\;1\;and\;{\sum_{n=1}}^{\infty}\;t_n\;=\;{\infty},\;(ii)\;(a)\;0\;{\leq}\;s_n\;{\leq}\;1,\;s_n\;->\;0\;as\;n\;->\;{\infty}\;and\;{\sum_{n=1}}^{\infty}\;t_ns_n\;<\;{\infty}\;or\;(b)\;s_n\;=\;s\;for\;all\;n\;{\geq}\;1\;and\;s\;{\in}\;[0,1),\;(iii)\;x_{n+1}\;=\;(1-t_n)x_n+t_nT(s_nTx_n+(1-s_n)x_n)\;for\;all\;n\;{\geq}\;1.$ Then, if the sequence {x_n} is bounded, then $lim_{n->\infty}\;x_n-Tx_n\;=\;0$. This result improves and complements a result of Deng [2]. Furthermore, we will show that certain conditions on D, X and T guarantee the weak and strong convergence of the Ishikawa iterative sequence to a fixed point of T.
 Keywords
nonexpansive mapping;Ishikawa iterative sequence;strong and weak convergence theorems.;
 Language
English
 Cited by
 References
1.
Iterative Methods for Nonlinear Operator Equations in Banach spaces, 2002.

2.
J. Math. Anal. Appl., 1996. vol.199. pp.769-775 crossref(new window)

3.
Contemporary Math., 1983. vol.21. pp.115-123 crossref(new window)

4.
Proc. Amer. Math. Soc., 1976. vol.59. pp.65-71 crossref(new window)

5.
Proc. Amer. Math. Soc., 1974. vol.44. pp.375-380 crossref(new window)

6.
J. Math. Anal. Appl., 1993. vol.178. pp.301-308 crossref(new window)