Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 

 

A monotonicity theorem for the family $ f\sb{a}(x)=a-x\sp{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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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]$.


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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58F20, 34C25, 54C05

Retrieve articles in all journals with MSC: 58F20, 34C25, 54C05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0656118-0
Keywords: Mapping on an interval, periodic points, monotonicity
Article copyright: © Copyright 1982 American Mathematical Society