  In this paper, we propose quasilinearization methods that convert nonlocal
fully-nonlinear parabolic systems into the nonlocal quasilinear parabolic
systems. The nonlocal parabolic systems serve as important mathematical tools
for modelling the subgame perfect equilibrium solutions to time-inconsistent
dynamic choice problems, which are motivated by the study of behavioral
economics. Different types of nonlocal parabolic systems were studied but left
behind the fully-nonlinear case and the connections among them. This paper
shows the equivalence in solvability between nonlocal fully-nonlinear and the
associated quasilinear systems, given their solutions are regular enough.
Moreover, we establish the well-posedness results for the nonlocal quasilinear
parabolic systems, so do that for the nonlocal fully-nonlinear parabolic
systems. The quasilinear case alone is interesting in its own right from
mathematical and modelling perspectives.
