  We consider a general class of finite-player stochastic games with mean-field
interaction, in which the linear-quadratic cost functional includes linear
operators acting on controls in $L^2$. We propose a novel approach for deriving
the Nash equilibrium of the game semi-explicitly in terms of operator
resolvents, by reducing the associated first order conditions to a system of
stochastic Fredholm equations of the second kind and deriving their solution in
semi-explicit form. Furthermore, by proving stability results for the system of
stochastic Fredholm equations, we derive the convergence of the equilibrium of
the $N$-player game to the corresponding mean-field equilibrium. As a
by-product, we also derive an $\varepsilon$-Nash equilibrium for the mean-field
game, which is valuable in this setting as we show that the conditions for
existence of an equilibrium in the mean-field limit are less restrictive than
in the finite-player game. Finally, we apply our general framework to solve
various examples, such as stochastic Volterra linear-quadratic games, models of
systemic risk and advertising with delay, and optimal liquidation games with
transient price impact.
