Containers and methods are disclosed whereby portions of an available stand-by supply of flowable material within a container may be dispensed substantially immediately on-demand. Preferably, the containers include an open-ended container shell having a resilient, shape-retaining side wall defining an
interior space, a liner positioned within the
interior space of the container shell, and a cap closing the open-ended container shell to seal the liner against leakage of flowable material. The liner has a flaccid liner body for containing a flowable material to be dispensed. The cap has a
discharge opening to allow an amount of flowable material to be discharged therethrough from the liner body. Importantly, the container includes a valve
system which includes one-way
discharge and inlet valves. The one-way
discharge valve allows only flowable material to be discharged from the liner via the discharge opening, but prevents
ambient air from being introduced into the liner when the discharge of flowable material is finished. The one-way
inlet valve serves to trap air within the
interior space of the container shell in response to a collapsing force applied against the side wall of the container shell which, in turn, allows
ambient air to enter the interior space when such collapsing force is removed.