Existence of fixed points on compact epilipschitz sets without invariance conditions

• Main Authors: Quincampoix Marc, Kamenskii Mikhail
• Format: Article
• Language: English
• Published: SpringerOpen 2005-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2005/603074

<p/> <p>We provide a new result of existence of equilibria of a single-valued Lipschitz function <inline-formula><graphic file="1687-1812-2005-603074-i1.gif"/></inline-formula> on a compact set <inline-formula><graphic file="1687-1812-2005-603074-i2.gif"/></inline-formula> of <inline-formula><graphic file="1687-1812-2005-603074-i3.gif"/></inline-formula> which is locally the epigraph of a Lipschitz functions (such a set is called epilipschitz set). Equivalently this provides existence of fixed points of the map <inline-formula><graphic file="1687-1812-2005-603074-i4.gif"/></inline-formula>. The main point of our result lies in the fact that we do not impose that <inline-formula><graphic file="1687-1812-2005-603074-i5.gif"/></inline-formula> is an "inward vector" for all point <inline-formula><graphic file="1687-1812-2005-603074-i6.gif"/></inline-formula> of the boundary of <inline-formula><graphic file="1687-1812-2005-603074-i7.gif"/></inline-formula>. Some extensions of the existence of equilibria result are also discussed for continuous functions and set-valued maps.</p>