On Single-Valuedness of Quasimonotone Set-Valued Operators

Abstract

A Nash problem is a noncooperative game in which the objective function of each player also depends on the decision variable of the other player. In order to solve such difficult problem, a classical approach is to write the optimality conditions of each of the problems obtaining thus a variational inequality. If the objective functions are nondifferentiable, the variational inequality can be set-valued, that is defined by a point-to-set map. Indeed, the derivatives are replaced by subdifferentials which are monotone if the objective functions are convex in the player’s variable. And if the objective functions are quasiconvex in terms of the player’s variable, the normal operator will advantageously replace the derivatives [3]. But solving a set-valued variational inequality is clearly more difficult than solving a single-valued variational inequality. It is thus very important to know sufficient conditions ensuring that a set-valued map, in particular a normal operator, is single-valued. Any monotone set-valued map that is also lower semi-continuous at a given point of the interior of its domain is actually single-valued at this point. This famous result is due to Kenderov [18] in 1975. Such a pointwise property of monotone maps has its local and dense counterparts (see, e.g. [14] and [10], respectively). The aim of this chapter is to answer to the somehow natural question: “what can one expect as a similar single-valuedness result for the more general class of quasimonotone set-valued maps”. The three points of view, that is pointwise, local and dense aspects, are treated, and a central role is played by the concept of directional single-valuedness. For the sake of simplicity, in Sects. 6.3 and 6.4, we only consider finite dimensional setting even if all of the results of Sect. 6.3, respectively 6.4, hold true in Banach spaces, respectively in Hilbert spaces.