Densely Defined Equilibrium Problems
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In the present work, we deal with set-valued equilibrium problems, for which we provide sufficient conditions for the existence of a solution. The conditions, that we consider, are imposed not on the whole domain, but rather on a self-segment-dense subset of it, a special type of dense subset. As an application, we obtain a generalized Debreu–Gale–Nikaïdo-type theorem, with a considerably weakened Walras law in its hypothesis. Furthermore, we consider a noncooperative \(n\)-person game and prove the existence of a Nash equilibrium, under assumptions that are less restrictive than the classical ones.
KeywordsSelf-segment-dense set Set-valued equilibrium problem Debreu–Gale–Nikaïdo-type theorem Nash equilibrium
Mathematics Subject Classification47H04 47H05 26B25 26E25 90C33
The authors are grateful to two anonymous referees for their helpful comments and suggestions which led to improvement of the original submitted version of this work. This work was supported by a Grant of the Romanian Ministry of Education, CNCS—UEFISCDI, Project number PN-II-RU-PD-2012-3-0166. This paper is a result of a research made possible by the financial support of the Sectoral Operational Programme for Human Resources Development 2007–2013, co-financed by the European Social Fund, under the project POSDRU/159/1.5/S/132400-“Young successful researchers—professional development in an international and interdisciplinary environment”.
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