  We provide a thorough study of a general class of linear-quadratic extended
mean field games and control problems in any dimensions where the mean field
terms are allowed to be unbounded and there are also presence of cross terms in
the objective functionals. Our investigation focuses on the unique existence of
equilibrium strategies for the extended mean field problems by employing the
stochastic maximum principle approach and the appropriate fixed point argument.
We provide two distinct proofs, accompanied by two sufficient conditions, that
establish the unique existence of the equilibrium strategy over a global time
horizon. Both conditions emphasize the importance of sufficiently small
coefficients of sensitivity for the cross term, of state and control, and mean
field term. To determine the required magnitude of these coefficients, we
utilize the singular values of appropriate matrices and Weyl's inequalities.
The present proposed theory is consistent with the classical one, namely, our
theoretical framework encompasses classical linear-quadratic stochastic control
problems as particular cases. Additionally, we establish sufficient conditions
for the unique existence of solutions to a particular class of non-symmetric
Riccati equations, and we illustrate a counterexample to the existence of
equilibrium strategies. Furthermore, we also apply the stochastic maximum
principle approach to examine linear-quadratic extended mean field type
stochastic control problems. Finally, we conduct a comparative analysis between
our method and the alternative master equation approach, specifically
addressing the efficacy of the present proposed approach in solving common
practical problems, for which the explicit forms of the equilibrium strategies
can be obtained directly, even over any global time horizon.
