On Fixed Point Theory and Its Applications to Equilibrium Models

For a given set and a given (generally speaking, multivalued) mapping of this set into itself, we study the problem on the existence of fixed points of this mapping, i.e., of points contained in their images. We assume that the given set is nonempty and the given mapping is defined on the entire set. In these conditions, we give the description (redefinition) of the set of fixed points in the set-theoretic terms. This general idea is concretized for cases where the set is endowed with a topological structure and the mapping has additional properties associated with this structure. In particular, we provide necessary and sufficient conditions for the existence of fixed points of mappings with closed graph in Hausdorff topological spaces as well as in metric spaces. An example illustrating the possibilities and advantages of the proposed approach is given. The immediate applications of these results to the search of equilibrium states in game problems are also given: we describe the sets of saddle points in the minimax problem (an analogue of the Fan theorem) and of Nash equilibrium points in the game with many participants in cases where the sets of strategies of players are Hausdorff spaces or metrizable topological spaces.

Keywords

multivalued mapping; fixed point; saddle point; Nash equilibrium.

References

1. Kakutani S. A Generalization of Brouwer's Fixed Point Theorem. Duke Mathematical Journal, 1941, vol. 8, pp. 457-459. DOI: 10.1215/S0012-7094-41-00838-4
2. Park Sehie. Recent Results in Analytical Fixed Point Theory. Nonlinear Analysis, 2005, vol. 63, pp. 977-986. DOI: 10.1016/j.na.2005.02.026
3. Tarski A.A Lattice-Theoretical Fixpoint Theorem and Its Applications. Pacific Journal of Mathematics, 1955, vol. 5, no. 2, pp. 285-309. DOI: 10.2140/pjm.1955.5.285
4. Kantorovitch L. The Method of Successive Approximation for Functional Equations. Acta Mathematica, 1939, December, vol. 71, no. 1, pp. 63-97. DOI: 10.1007/BF02547750
5. Barendregt H. P. Lambda Calculus. Its Syntax and Semantics. North-Holland Publishing Company, 1981.
6. Li Jinlu Several Extensions of the Abian-Brown Fixed Point Theorem and Their Applications to Extended and Generalized Nash Equilibria on Chain-Complete Posets. Journal of Mathematical Analysis and Applications, 2014, vol. 409, no. 2, pp. 1084-1092. DOI: 10.1016/j.jmaa.2013.07.070
7. Fan K. Minimax Theorems. Proceedings of the National Academy of Sciences U.S.A., 1953, vol. 39, pp. 42-47. DOI: 10.1073/pnas.39.1.42
8. Nadler S.B.Jr. Multi-Valued Contraction Mappings. Pacific J. Math, 1969, vol. 30, pp. 475-488. DOI: 10.2140/pjm.1969.30.475
9. Arutyunov A.V. Covering Mappings in Metric Spaces and Fixed Points. Doklady Mathematics, 2007, vol. 76, no. 2, pp. 665-668. DOI:10.1134/S1064562407050079
10. Kuratowski K. Topology. Volume II. Academic Press, New York, 1968.
11. Borisovich Yu.G., Gel'man B.D., Myshkis A.D. etс. Vvedeniye v teoriyu mnogoznachnykh otobrazheniy i differentsial'nykh vklyucheniy [Introduction to the Theory of Multi-Valued Mappings and Differential Inclusions]. Moscow, Librokom, 2011. 224 p.