Existence of bounded trajectories via upper and lower solutions

The paper deals with the boundary value problem on the whole line [GRAPHICS] where g: R --> R is a continuous non-negative function with support [0, 1], and f : R-2 --> R is a continuous function. By means of a new approach, ba sed on a combination of lower and upper-solutions methods and phase-plane t echniques, we prove an existence result for (P) when f is superlinear in u' ; by a similar technique, we also get a non-existence one. As an applicatio n, we investigate the attractivity of the singular point (0,0) in the phase -plane (u, u'). We refer to a forthcoming paper [13] for a further applicat ion in the field of front-type solutions for reaction diffusion equations.