We investigate the existence of unique steady-state solution to the nonlinear aggregation and multiple fragmentation equation under non-equilibrium conditions induced due to external injection and removal mechanisms of particles in a weighted \(L^1\) space. These mechanisms force the system to deviate from the natural volume conservation property. We establish existence and uniqueness results using the fixed point theorem for a wide class of physically significant kinetic kernels, including singular cases for the fragmentation events. Strong linear stability of the steady-state is proved, and the asymptotic properties of the time-dependent solution are studied to exhibit exponential convergence rate using the semigroup theory of operators. The analysis is further executed to establish the existence results for the multi-component framework. Numerical simulations are also presented to support the theoretical findings.
