Bounded solutions in a given set of differential systems

The criteria for an entirely bounded solution in a given set of a differential system with a quasi-linear part are developed sequentially via an asymptotic boundary value problem. This enables us to prove several bounded solutions separated by given functions. So, the multiplicity results can be also obtained in this way. For possible applications in epidemics and population dynamics, see e.g., the references in Gaines and Santanilla (Rocky Mountain J. Math. 12 (1982) 669–678).