  The emergence of the graphon theory of large networks and their infinite
limits has enabled the formulation of a theory of the centralized control of
dynamical systems distributed on asymptotically infinite networks (Gao and
Caines, IEEE CDC 2017, 2018). Furthermore, the study of the decentralized
control of such systems was initiated in (Caines and Huang, IEEE CDC 2018,
2019), where Graphon Mean Field Games (GMFG) and the GMFG equations were
formulated for the analysis of non-cooperative dynamic games on unbounded
networks. In that work, existence and uniqueness results were introduced for
the GMFG equations, together with an epsilon-Nash theory for GMFG systems which
relates infinite population equilibria on infinite networks to finite
population equilibria on finite networks. Those results are rigorously
established in this paper.
