Journal of the
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J. Korean Math. Soc. 1999; 36(6): 1061-1073

Printed November 1, 1999

Copyright © The Korean Mathematical Society.

Ishikawa and Mann iterative processes with errors for nonlinear $\Phi$-strongly quasi-accretive mappings in normed linear spaces

H. Y. Zhou and Y. J. Cho

Abstract

Let $X$ be a real normed linear space. Let $T: D(T)\subset X\rightarrow X$
be a uniformly continuous and $\phi$-strongly quasi-accretive mapping. Let
$\{\alpha_n\}_{n=0}^\infty$, $\{\beta_n\}_{n=0}^\infty$ be two real sequences in
$[0,1]$ satisfying the following conditions:
\roster
\item"{\rm (i)}" $\alpha_n\rightarrow 0,\;\beta_n\rightarrow 0$ as $n\rightarrow \infty$,
\item"{\rm (ii)}" $\sum_{n=0}^{\infty}\alpha_n=\infty$.
\endroster
Set $Sx=x-Tx$ for all $x\in D(T)$.
Assume that $\{u_n\}_{n=0}^{\infty}$ and $\{v_n\}_{n=0}^{\infty}$
are two sequences in $D(T)$ satisfying $\sum_{n=0}^{\infty}\|u_n\|<\infty$
and $v_n\rightarrow 0$ as $n\rightarrow\infty$.
Suppose that, for any given $x_0\in X$, the Ishikawa type iteration sequence
$\{x_n\}_{n=0}^\infty$ with errors defined by
$$
\left\{\eqalign{
&x_{n+1}=(1-\alpha_n)x_n+\alpha_nSy_n+u_n,\cr
&y_n=(1-\beta_n)x_n+\beta_nSx_n+v_n}\right.\leqno \text{\rm (IS)$_1$}
$$
for all $n=0,1,2,\cdots$
is well-defined. We prove that $\{x_n\}_{n=0}^\infty$ converges strongly to the unique zero of $T$ if and only if $\{Sy_n\}_{n=0}^\infty$ is bounded. Several related results deal with iterative approximations of fixed points of
$\phi$-hemicontractions by the Ishikawa iteration with errors in a normed linear
space. Certain conditions on the iterative parameters $\{\alpha_n\}_{n=0}^\infty$, $\{\beta_n\}_{n=0}^\infty$
and $T$ are also given which guarantee the strong convergence of the iteration processes.

Keywords: Ishikawa iteration with errors, $\phi$-strongly quasi-accretive mapping, $\phi$-hemicontraction, normed linear space

MSC numbers: 47H10, 47H17, 40A05

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