  In this paper we study the continuous coagulation and multiple fragmentation
equation for the mean-field description of a system of particles taking into
account the combined effect of the coagulation and the fragmentation processes
in which a system of particles growing by successive mergers to form a bigger
one and a larger particle splits into a finite number of smaller pieces. We
demonstrate the global existence of mass-conserving weak solutions for a wide
class of coagulation rate, selection rate and breakage function. Here, both the
breakage function and the coagulation rate may have algebraic singularity on
both the coordinate axes. The proof of the existence result is based on a weak
L^1 compactness method for two different suitable approximations to the
original problem, i.e. the conservative and non-conservative approximations.
Moreover, the mass-conservation property of solutions is established for both
approximations.
