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Proceedings of the American Mathematical Society

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

 

 

Quasicontractions on metric spaces


Author: Brian Fisher
Journal: Proc. Amer. Math. Soc. 75 (1979), 321-325
MSC: Primary 54H25
DOI: https://doi.org/10.1090/S0002-9939-1979-0532159-9
MathSciNet review: 532159
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Abstract: It is proved that if T is a continuous mapping on the complete metric space X into itself satisfying the inequality

\begin{displaymath}\begin{array}{*{20}{c}} \hfill {d({T^p}x,{T^q}y) \leqslant c\... ...;{\text{and}}\;0 \leqslant s,s' \leqslant q\} } \\ \end{array} \end{displaymath}

for all x,y in X, where $ 0 \leqslant c < 1$, for some fixed positive integers p and q, then T has a unique fixed point. Further, it is shown that the condition that T be continuous is unnecessary if $ q\;({\text{or}}\;p) = 1$.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0532159-9
Keywords: Metric space, completeness, fixed point, quasi-contraction
Article copyright: © Copyright 1979 American Mathematical Society