  The paper discusses the conditions for the existence of fixed points of
multivalued mappings that are not based on the linear structure of the set. The
descriptions for the sets of fixed points for mappings with closed graph in
compact Hausdorff spaces and in metrizable spaces, as well as for continuous
functions in spaces with convergence (Frechet topology) and in spaces with
Scott topology are provided. Applications to the problem of the equilibrium are
given: the sets of saddle points and of Nash equilibria for compact Hausdorff
and metrizable spaces of strategies of players are described.
