STRONG CONVERGENCE OF ISHIKAWA ITERATIONS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS (Nonlinear Analysis and Convex Analysis)
Gang Eun Kim · Kyoto University Research Information Repository (Kyoto University) · 2004
Let $C$ be a nonempty bounded closed convex subset of a uniformly convex Banach space.We prove that if $T:Carrow C$ is both compact iterates and asymptotically nonexpansive, the lshlhwa iteration process with errors defined by $x1$ $\in C,$ $x_{n+1}=$ $cx_{n}z_{h}+\beta_{n}T^{n}y_{\tau\iota}+\gamma_{n}u_{\mathfrak{n}}$ , and $y_{n}=\alpha_{n}'x_{n}+\beta_{\mathrm{r}\iota}' T'*x_{n}\mathit{1}-$ $\sqrt{n}v_{n}$ converges strongly to some fixed point of $T$ .This generalizes the recent theorems due to Rhoades [5], Schu[ 6] and Schu [7].$\mathrm{K}\mathrm{e}\mathrm{y}\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{s}arrow \mathrm{t}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{g}$ convergence, fixed point, Mann and Ishikawa iteration process, asymptotically nonexpansive mapping.