Convergence of the Ishikawa iteration process for nonexpansive mappings
Let \(D\) be a subset of a normed space \(X\) and \(T: D\to X\) be a nonexpansive mapping. Given a sequence \(\{x_n\}\) in \(D\) and two real sequences \(\{t_n\}\) and \(\{s_n\}\) satisfying \hskip17mm (i) \(0\leq t_n\leq t< 1\) and \(\sum^\infty_{n= 1} t_n= \infty\), \hskip17mm (ii) \(0\leq s_n\leq 1\) and \(\sum^\infty_{n= 1} s_n< \infty\), \hskip17mm (iii) \(x_{n+ 1}= t_n T(s_n Tx_n+ (1- s_n) x_n)+ (1- t_n) x_n\), \(n= 1,2,3,\dots\), the author proves that if \(\{x_n\}\) is bounded, then \(\lim_{n\to \infty} |Tx_n- x_n|= 0\). The conditions on \(D\), \(X\), and \(T\) are shown which guarantee the weak and strong convergence of the Ishikawa iteration process to a fixed point of \(T\).
scientific article; zbMATH DE number 902573