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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Fixed point theorems for mappings with a contractive iterate at a point


Author: Janusz Matkowski
Journal: Proc. Amer. Math. Soc. 62 (1977), 344-348
MSC: Primary 54H25
MathSciNet review: 0436113
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Abstract: Let (X,d) be a complete metric space, $ T:X \to X$, and $ \alpha :[0,\infty )^5 \to [0,\infty )$ be nondecreasing with respect to each variable. Suppose that for the function $ \gamma (t) = \alpha (t,t,t,2t,2t)$, the sequence of iterates $ {\gamma ^n}$ tends to 0 in $ [0,\infty )$ and $ {\lim _{t \to \infty }}(t - \gamma (t)) = \infty $. Furthermore, suppose that for each $ x \in X$ there exists a positive integer $ n = n(x)$ such that for all $ y \in X$,

$\displaystyle d({T^n}x,{T^n}y) \leqslant \alpha (d(x,{T^n}x),d(x,{T^n}y),d(x,y),d({T^n}x,y),d({T^n}y,y)).$

Under these assumptions our main result states that T has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.

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

  • [1] L. F. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Amer. Math. Soc. 26 (1970), 615-618. MR 42 #919. MR 0266010 (42:919)
  • [2] K. Iseki, A generalization of Sehgal-Khazanchi's fixed point theorems, Math. Seminar Notes Kobe Univ. 2 (1974), 1-9. MR 0436108 (55:9058)
  • [3] L. Khazanchi, Results on fixed points in complete metric space, Math. Japonicae (to appear). MR 0400197 (53:4032)
  • [4] V. M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Amer. Math. Soc. 23 (1969), 631-634. MR 40 #3531. MR 0250292 (40:3531)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0436113-5
PII: S 0002-9939(1977)0436113-5
Keywords: Fixed point, orbit, complete metric, Cauchy sequence
Article copyright: © Copyright 1977 American Mathematical Society