Location of fixed points in the presence of two cycles

Abstract

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.