  In this paper, we study a class of infinite horizon fully coupled
McKean-Vlasov forward-backward stochastic differential equations (FBSDEs). We
propose a generalized monotonicity condition involving two flexible functions.
Under this condition, we establish the well-posedness results for infinite
horizon McKean-Vlasov FBSDEs by the method of continuation, including the
unique solvability, an estimate of the solution, and the related continuous
dependence property of the solution on the coefficients. Based on the
solvability result, we study an infinite horizon mean field control problem.
Moreover, by choosing appropriate form of the flexible functions, we can
eliminate the different phenomenon between the linear-quadratic (LQ) problems
on infinite horizon and finite horizon proposed in Wei and Yu (SIAM J. Control
Optim. 59: 2594-2623, 2021).
