  In this paper, we study a class of fully nonlinear metric flow on K\"ahler
manifolds, which includes the J-flow as a special case. We provide a sufficient
and necessary condition for the long time convergence of the flow, generalizing
the result of Song-Weinkove. As a consequence, under the given condition, we
solved the corresponding Euler equation, which is fully nonlinear of
Monge-Amp\`ere type. As an application, we also discuss a complex
Monge-Amp\`ere type equation including terms of mixed degrees, which was first
posed by Chen.
