On the Existence of Spatial Equilibrium and Generalized Fixed Point Theorems

The problem of immigration-proof division into countries of a finite-dimensional area is studied. This is a kind of Tiebout equilibrium, in which the principle of migration consistency assumes that the border residents do not have incentives for the change of jurisdiction, i.e., at the point of their location costs of citizens for border countries are equal. It is assumed that a measurable population density is given and it is required that the cross-country boundary be represented by a continuous surface (curve). The proof of the existence of required division is based on the reducing of the problem to the searching of a fixed point. However, the conditions of classical theorems on fixed points of Brouwer-Kakutani or the "conical" Krasnoselskii's theorem are violated. This led us to the development of their generalization to the case of a mapping, possibly acting outside the domain. It is proved that a continuous mapping defined on a convex compact has a fixed point if it satisfies one of two boundary conditions: "compression" or "expansion". This result also is extended to point-to-set "Kakutani maps": applying it, we prove the existence of an immigration-consistent division into countries of any compact finite-dimensional area in very general setting..