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Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption. (English) Zbl 0871.47045
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 $$n\geq 1$$; (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:
 47J25 Iterative procedures involving nonlinear operators
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##### References:
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