Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On fixed point theorems of contractive type
HTML articles powered by AMS MathViewer

by Mau Hsiang Shih and Cheh Chih Yeh PDF
Proc. Amer. Math. Soc. 85 (1982), 465-468 Request permission

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 \[ d({G^m}(x),{G^m}(y)) < \max \left \{ {d(x,y),d(x,{G^m}(x)),d(y,{G^m}(y)),d(x,{G^m}(y)),d(y,{G^m}(x))} \right \} \] 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 $x$, $y \in X$.
References
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54H25
  • Retrieve articles in all journals with MSC: 54H25
Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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