  Generalized mode-coupling theory (GMCT) is a first-principles-based and
systematically correctable framework to predict the complex relaxation dynamics
of glass-forming materials. The formal theory amounts to a hierarchy of
infinitely many coupled integro-differential equations, which may be
approximated using a suitable finite-order closure relation. Although previous
studies have suggested that finite-order GMCT leads to well-defined solutions,
and that the hierarchy converges as the closure level increases, no rigorous
and general result in this direction is known. Here we unambiguously establish
the existence and uniqueness of solutions to generic, schematic GMCT
hierarchies that are closed at arbitrary order. We consider two types of
commonly invoked closure approximations, namely mean-field and exponential
closures. We also distinguish explicitly between overdamped and underdamped
glassy dynamics, corresponding to hierarchies of first-order and second-order
integro-differential equations, respectively. We find that truncated GMCT
hierarchies closed under an exponential closure conform to previously developed
mathematical theories, such that the existence of a unique solution can be
readily inferred. Self-consistent mean-field closures, however, of which the
well-known standard-MCT closure approximation is a special case, warrant
additional arguments for mathematical rigour. We demonstrate that the existence
of a priori bounds on the solution is sufficient to also prove that unique
solutions exist for such self-consistent hierarchies. To complete our analysis,
we present simple arguments to show that these a priori bounds must exist,
motivated by the physical interpretation of the GMCT solutions as density
correlation functions. Overall, our work contributes to the theoretical
justification of GMCT for studies of the glass transition, placing GMCT on a
firmer mathematical footing.
