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We provide a new result of existence of equilibria of a single-valued Lipschitz function on a compact set of 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 . The main point of our result lies in the fact that we do not impose that is an "inward vector" for all point of the boundary of . Some extensions of the existence of equilibria result are also discussed for continuous functions and set-valued maps.