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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On fixed point theorems of contractive type


Authors: Mau Hsiang Shih and Cheh Chih Yeh
Journal: Proc. Amer. Math. Soc. 85 (1982), 465-468
MSC: Primary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1982-0656125-8
MathSciNet review: 656125
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Abstract: Let $ G$ be a continuous map of a nonempty compact metric space $ (X,d)$ into itself, such that for some positive integer $ m$, the iterated map $ {G^m}$ satisfying

$\displaystyle d({G^m}(x),{G^m}(y)) < \max \left\{ {d(x,y),d(x,{G^m}(x)),d(y,{G^m}(y)),d(x,{G^m}(y)),d(y,{G^m}(x))} \right\} $

for all $ x$, $ y \in X$ with $ x \ne y$. It is shown that (i) $ G$ has a unique fixed point $ {x^ * } \in X$; (ii) the sequence of iterates $ \left\{ {{G^k}(x)} \right\}$ converges to $ {x^ * }$ for any $ x \in X$; (iii) given $ \lambda $, $ 0 < \lambda < 1$, there exists a metric $ {d_\lambda }$, topologically equivalent to $ d$, such that $ {d_\lambda }(G(x)$, $ G(y)) \leqslant \lambda {d_\lambda }(x,y)$ for all $ x$, $ y \in X$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0656125-8
Keywords: Contraction, fixed point, remetrization
Article copyright: © Copyright 1982 American Mathematical Society

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