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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Approximation of fixed points
of strongly pseudocontractive maps
without Lipschitz assumption


Authors: Zhou Haiyun and Jia Yuting
Journal: Proc. Amer. Math. Soc. 125 (1997), 1705-1709
MSC (1991): Primary 47H17
MathSciNet review: 1389522
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Abstract: In the present paper, the following result is shown: Let $X$ be a real Banach space with a uniformly convex dual $X^*$, and let $K$ be a nonempty closed convex and bounded subset of $X$. Assume that $T:\,K\rightarrow K$ is a continuous strong pseudocontraction. Let $\{\alpha _n\}^{\infty }_{n=1}$ and $\{\beta _n\}^{\infty }_{n=1}$ be two real sequences satisfying (i) $0<\alpha _n,\,\beta _n<1$ for all $n\ge 1$; (ii) $\sum _{n=1}^{\infty }\alpha _n=\infty $; and (iii) $\alpha _n \rightarrow 0,\, \beta _n \rightarrow 0$ as $n\rightarrow \infty .$ Then the Ishikawa iterative sequence $\{x_n\}_{n=1}^{\infty }$ generated by

\begin{eqnarray*}\text {\rm (I)} \;\left \{ \begin {array}{l} x_1\in K,\\ x_{n+1}=(1-\alpha _n)x_n+\alpha _nTy_n,\\ y_n=(1-\beta _n)x_n+\beta _nTx_n,\,n\geq 1, \end {array} \right . \end{eqnarray*}

converges strongly to the unique fixed point of $T$.


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Additional Information

Zhou Haiyun
Affiliation: Department of Mathematics, Hebei Teachers University, Shijiazhuang 050016, People’s Republic of China

Jia Yuting
Affiliation: Department of Mathematics, Hebei Teachers University, Shijiazhuang 050016, People’s Republic of China

DOI: http://dx.doi.org/10.1090/S0002-9939-97-03850-1
PII: S 0002-9939(97)03850-1
Keywords: The Ishikawa iteration, strong pseudocontraction, strictly convex Banach space
Received by editor(s): December 5, 1995
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society