  In this paper, we consider mean field games (MFGs) with a major and $N$ minor
agents. We first consider the limiting problem and allow the state coefficients
to vary with the conditional distribution in a nonlinear way. We use the
sufficient Pontryagin principle for optimality to transform the limiting
control problem into a system of two coupled conditional distribution dependent
forward backward stochastic differential equations (FBSDEs), and prove the
existence and uniqueness of solutions of the FBSDEs when the dependence between
major agent and minor agents is sufficiently weak and the convexity parameter
of the running cost of minor agents on the control is sufficiently large. A
weak monotonicity property is required for minor agents' cost functions and the
proof is based on a continuation method in coefficients. We then consider the
equilibrium property of MFG with major and minor agents and use the solution of
the limit problem to construct an $\mathcal{O}(N^{-\frac{1}{2}})$-Nash
equilibrium for MFG with a major and $N$ minor agents.
