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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 mappings


Author: C. E. Chidume
Journal: Proc. Amer. Math. Soc. 120 (1994), 545-551
MSC: Primary 47H10; Secondary 47H09, 47H15
MathSciNet review: 1165050
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Abstract: 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}\} _{n = 1}^\infty $ be a real sequence satisfying: (i) $ 0 < {c_n} < 1$ for all $ n \geqslant 1$; (ii) $ \sum\nolimits_{n = 1}^\infty {{c_n} = \infty } $; and (iii) $ \sum\nolimits_{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) \leqslant cb(t)$ for all $ c \geqslant 1$. Then the sequence $ \{ {x_n}\} _{n = 1}^\infty $ generated by $ {x_1} \in K$,

$\displaystyle {x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1,$

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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1165050-6
PII: S 0002-9939(1994)1165050-6
Article copyright: © Copyright 1994 American Mathematical Society