  In this article, we consider generalized Nash games where the associated
constraint map is not necessarily self. The classical Nash equilibrium may not
exist for such games and therefore we introduce the notion of best approximate
solution for such games. We investigate the occurrence of best approximate
solutions for such generalized Nash games consisting of infinitely many players
in which each player regulates the strategy variable lying in a topological
vector space. Based on the maximum theorem and a fixed point result for
Kakutani factorizable maps, we derive the existence of best approximate
solutions under the quasi-concavity and weak continuity assumption on players'
objective functions. Furthermore, we demonstrate the occurrence of best
approximate solutions for quasi-optimization problems.
