Location of fixed points in the presence of two cycles

Alfonso Ruiz-Herrera


Any orientation-preserving homeomorphism of the plane having a two
cycle has also a fixed point. This well known result does not
provide any hint on how to locate the fixed point, in principle it
can be anywhere. J. Campos and R. Ortega in {\it Location of fixed
points and periodic solutions in the plane} consider the class of
Lipschitz-continuous maps and locate a fixed point in the region
determined by the ellipse with foci at the two cycle and
eccentricity the inverse of the Lipschitz constant. It will be
shown that this region is not optimal and a sub-domain can be
removed from the interior. A curious fact is that the ellipse
mentioned above is relevant for the optimal location of fixed
point in a neighbourhood of the minor axis but it is of no
relevance around the major axis.


Ellipse; Lipschitz-continuous homeomorphism; two cycle; fixed point

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