  We investigate the properties of a class of mode-coupling equations for the
glass transition where the density mode decays into multiple relaxation
channels. We prove the existence and uniqueness of the solutions for Newtonian
as well as Brownian dynamics and demonstrate that they fulfill the requirements
of correlation functions, in the latter case the solutions are purely
relaxational. Furthermore, we construct an effective mode-coupling functional
which allows to map the theory to the case of a single decay channel, such that
the covariance principle found for the mode-coupling theory for simple liquids
is properly generalized. This in turn allows establishing the maximum theorem
stating that long-time limits of mode-coupling solutions can be calculated as
maximal solutions of a fixed-point equation without relying on the dynamic
solutions.

Soft Condensed Matter;Statistical Mechanics