CONVERGENCE THEOREMS OF THE ITERATIVE SEQUENCES FOR NONEXPANSIVE MAPPINGS
Jung-Im Kang, Yeol-Je Cho, Haiyun Zhou · Communications of the Korean Mathematical Society · 2004
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 ${x_n}$ be a sequence in D and ${t_n}$ , ${s_n}$ be real sequences such that (i) $0\;{\leq}\;t_n\;{\leq}\;t\;\;0\;as\;n\;->\;{\infty}\;and\;{\sum_{n=1}}^{\infty}\;t_ns_n\;\infty}\; $\mid$ $\mid$ x_n-Tx_n $\mid$ $\mid$ \;=\;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.