Fixed points theorems in the non-compact case and application to Mathematical Economics[Théorèmes de points fixes dans le cas non compact et application à l'Economie Mathématique]

The starting point of the fixed point theory is Brower's theorem (1912), which could be extended to multivariate functions (also called correspondences) defined on a convex and compact subset, with non-empty convex values and verifying a continuity condition, following the generalization, by Fan (1961), of the Knaster-Kuratowski-Mazurkiewicz lemma (known as the KKM lemma) to the topological vectorial spaces. This result can be interpreted as a result of the existence of maximal elements frequently used in game theory and mathematical economics (Monique Florenzano (1980) - Gerard Debreu (1982)). Several works then followed to extend these results to correspondences not necessarily defined on a compact set. Brower's theorem has been extended to multi-shell functions (also called correspondences) defined on a convex and compact subset, with non-empty convex values and verifying a continuity condition, following the generalisation, by Fan (1961), of the Knaster-Kuratowski-Mazurkiewicz lemma (known as the KKM lemma) to topological vector spaces. This result can be interpreted as a result of the existence of maximal elements frequently used in game theory and mathematical economics (Monique Florenzano (1980) - Gerard Debreu (1982)). Several works then followed to extend these results to correspondences not necessarily defined on a compact set. In a first step, I propose to put in order the results of the existence of fixed points and maximal elements, which have been advanced in the literature, in the non-compact case. For this purpose, the notion of covering a correspondence is introduced to give a generalized version of the Knaster-Kuratowski-Mazurkiewicz (1984) lemma which is used to prove the existence of fixed points for correspondences defined on a convex non-empty subset of a separate topological vector space. An existence result of maximal elements for irreflexive preference matches defined on a not necessarily compact set is deduced. As a corollary, we obtain the corresponding results of Browder-Fan (1968), Shafer-Sonnschien (1975), Borglin-Keiding (1976), Tarafdar (1977), Border (1985), Mehta (1987), Tan-Yuan (1993) and Ding-Tan (1993). In a second step, I used the maximum element existence result obtained in the first step to prove the existence of equilibrium in qualitative games with an infinite number of agents for preference matches defined on a convex subset of a topologically separated vector space and satisfying an overlap condition. This allowed establishing the existence of equilibrium in generalized games (or abstract economy) with non-compact choice sets. The results obtained generalise the corresponding results of Gales-Mas Collel (1975), Yannelis and Prabakar (1983), Toussaint (1984), Tulcea (1988) and Ding-Tan (1993).