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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$.

References [Enhancements On Off] (What's this?)

  • [1] Ming-Po Chen and Mau-Hsiang Shih, Fixed point theorems for point-to-point and point-to-set maps, J. Math. Anal. Appl. 71 (1979), 516-524. MR 548780 (80i:54053)
  • [2] S. C. Chu and J. B. Diaz, Remarks on a generalization of Banach's principle of contractive mappings, J. Math. Anal. Appl. 11 (1965), 440-446. MR 0184218 (32:1691)
  • [3] Lj. B. Ćirić, A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267-273.
  • [4] M. Edelstein, On fixed points and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79. MR 0133102 (24:A2936)
  • [5] G. Hardy and T. Rogers, A generalization of a fixed point theorem of Reich, Canad. Math. Bull. 16 (1973), 201-206. MR 0324495 (48:2847)
  • [6] L. Janos, A converse of Banach's contraction theorem, Proc. Amer. Math. Soc. 16 (1967), 287-289. MR 0208589 (34:8398)
  • [7] A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Metric and Normed Spaces, Vol. 1, transl, by L. Baron, Graylock Press, Rochester, N.Y., 1957. MR 0085462 (19:44d)
  • [8] P. R. Meyer, A converse to Banach's contraction theorem, J. Res. Nat. Bur. Standards 71B (1967), 73-76. MR 0221469 (36:4521)

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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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