  We investigate the production and freeze-out of dark matter with a constant
thermally averaged cross-section in a generic bouncing universe framework. Our
result shows that, there is a novel avenue that dark matter is produced
thermally and take a weakly freezing-out process, besides two previously known
cases, the thermally production & strongly freezing-out case and the
non-thermally production & weakly freezing-out case, in which the relic
abundance of dark matter are inverse and proportional to its cross-section
respectively. We calculated the relic abundance of dark matter for this new
case, and find its relic abundance is independent of its cross-section. We also
present the cosmological constraints on the cross-section and mass relation of
dark matter for this new case.
