CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS

Title & Authors
CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS
Kang, Jung-Im; Cho, Yeol-Je; Zhou, Hai-Yun;

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 $\small{{x_n}}$ be a sequence in D and $\small{{t_n}}$, $\small{{s_n}}$ 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 $lim_{n->\infty}\;$\small{\mid}$$\small{\mid}$x_n-Tx_n$\small{\mid}$$\small{\mid}$\;=\;0$. 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
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