Communications of the
Korean Mathematical Society CKMS

Suppose that $X$ is a real Banach space, $K$ is a non\-em\-pty closed convex subset of $X$ and $T:K\to K$ is a uniformly continuous $\phi$-hemicontractive operator or a Lipschitz $\phi $-hemicontractive operator. In this paper we prove that under certain conditions the three-step iteration methods with errors converge strongly to the unique fixed point of $T$. Our results extend the corresponding results of Chang [1], Chang et al. [2], Chidume [3]-[7], Chidume and Osilike [9], Deng [10], Liu and Kang [13], [14], Osilike [15], [16] and Tan and Xu [17].

Keywords: $\phi $-pseudocontractive operator, $\phi $-hemicontractive operators, the three-step iteration method with errors, fixed point, Banach spaces

MSC numbers: 47H05, 47H06, 47H10, 47H14