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Zbl 0871.47045
Zhou, Haiyun; Jia, Yuting
Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption. (English)
[J] Proc. Am. Math. Soc. 125, No.6, 1705-1709 (1997). ISSN 0002-9939; ISSN 1088-6826/e

Summary: 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) $\alpha_n\to 0$, $\beta_n\to 0$ as $n\to\infty$. Then the Ishikawa iterative sequence $\{x_n\}_{n=1}^{\infty}$ generated by $$(I)\qquad x_1\in K,\quad x_{n+1}=(1-\alpha_n)x_n+\alpha_nTy_n,\quad y_n=(1-\beta_n)x_n+\beta_nTx_n,\quad n\geq 1,$$ converges strongly to the unique fixed point of $T$.

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)

Keywords: Ishikawa iteration; strong pseudocontraction; strictly convex Banach space; unique fixed point

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