Bound sets for first order differential equations with general linear two-points boundary conditions

We consider differential equations with linear two-points boundary conditions. We present some existence results for bound sets defined as the intersection of sublevel sets of particular scalar functions, called bounding functions. All the three cases, namely continuous, locally lipschitzian and C-1-class bounding functions, are analized. Comparisons with previous results are given. Finally we apply the existence theorems to the homogeneous Cauchy problem and to the Picard problem.