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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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