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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 points of certain self maps on an interval


Author: Chi Song Wong
Journal: Proc. Amer. Math. Soc. 42 (1974), 234-235
MSC: Primary 26A18
MathSciNet review: 0325869
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Abstract: Let T be a self map on a bounded interval [a, b] with $ a,b \in T([a,b])$. Suppose that for any x, y in [a, b],

$\displaystyle \vert T(x) - T(y)\vert \leqq \tfrac{1}{2}(\vert x - T(x)\vert + \vert y - T(y)\vert).$

It is shown without the continuity of T that the midpoint of [a, b] is a fixed point of T. A nontrivial example is given.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0325869-5
PII: S 0002-9939(1974)0325869-5
Article copyright: © Copyright 1974 American Mathematical Society