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Proceedings of the American Mathematical Society

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

 
 

 

A monotonicity theorem for the family $f_{a}(x)=a-x^{2}$


Author: Leo Jonker
Journal: Proc. Amer. Math. Soc. 85 (1982), 434-436
MSC: Primary 58F20; Secondary 34C25, 54C05
DOI: https://doi.org/10.1090/S0002-9939-1982-0656118-0
MathSciNet review: 656118
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Abstract: Let ${f_a}(x) = a - {x^2}$, $x \in [ - \tfrac {1} {2} - \tfrac {1} {2}\sqrt {1 + 4a}$, $\tfrac {1} {2} + \tfrac {1} {2}\sqrt {1 + 4a} ]$ and $a \in [0,2]$. It is proved that if ${f_a}$ has a periodic orbit of odd period $n$ and if $b > a$, then ${f_b}$ has a periodic orbit of period $n$. This is equivalent to the corresponding result for the function family ${g_\lambda }(x) = \lambda x(1 - x)$, $x \in [0,1]$, $\lambda \in [0,4]$.


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Keywords: Mapping on an interval, periodic points, monotonicity
Article copyright: © Copyright 1982 American Mathematical Society