  This article presents the variant of the approach introduced in the recent
work of Bensoussan, Wong, Yam and Yuan [13] to the generic first-order mean
field game problem. A major contribution here is the provision of new crucial a
priori estimates, whose establishment is fundamentally different from the
mentioned work since the associated forward-backward ordinary differential
equation (FBODE) system is notably different. In addition, we require
monotonicity conditions intimately on the coefficient functions but not on the
Hamiltonians to handle their non-separable nature and nonlinear dynamics; as
tackling Hamiltonians directly, it potentially dissolves much useful
information. Compared with the assumptions used in [13], we introduce an
additional requirement that the first-order derivative of the drift function in
the measure variable cannot be too large relative to the convexity of the
running cost function; this requirement only arises when the Hamiltonian is
non-separable, and this phenomenon can also be seen in the existing literature.
On the other hand, we require less here for the second-order differentiability
of the coefficient functions in comparison to that in [13]. Our approach
involves first demonstrating the local existence of a solution over small time
interval, followed by the provision of new crucial a priori estimates for the
sensitivity of the backward equation with respect to the initial condition of
forward dynamics; and finally, smoothly gluing the local solutions together to
form a global solution. In addition, we establish the local and global
existence and uniqueness of classical solutions for the mean field game and its
master equation.
