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Approximation of fixed points of strongly pseudocontractive mappings. (English) Zbl 0802.47058
Summary: Let \(E\) be a real Banach space with a uniformly convex dual, and let \(K\) be a nonempty closed convex and bounded subset of \(E\). Let \(T: K\to K\) be a continuous strongly pseudocontractive mapping of \(K\) into itself. Let \(\{c_ n\}^ \infty_{n=1}\) be a real sequence satisfying:
(i) \(0<c_ n <1\) for all \(n\geq 1\);
(ii) \(\sum_{n=1}^ \infty c_ n= \infty\); and
(iii) \(\sum_{n=1}^ \infty c_ n b(c_ n) <\infty\), where \(b: [0,\infty)\to [0,\infty)\) is some continuous nondecreasing function satisfying \(b(0)=0\), \(b(ct)\leq cb(t)\) for all \(c\geq 1\).
Then the sequence \(\{x_ n\}^ \infty_{n=1}\) generated by \(x_ 1\in K\), \[ x_{n+1}= (1- c_ n) x_ n+ c_ n Tx_ n, \qquad n\geq 1, \] converges strongly to the unique fixed point of \(T\). A related result deals with the Ishikawa iteration scheme when \(T\) is Lipschitzian and strongly pseudocontractive.

MSC:
47J25 Iterative procedures involving nonlinear operators
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