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# Existence of bounded trajectories via upper and lower solutions

• The paper deals with the boundary value problem on the whole line

$u'' - f(u,u') + g(u) = 0$

$u(\infty,1) = 0$,   $u(+\infty) = 1$     $(P)$

where $g : R \to R$ is a continuous non-negative function with support $[0,1]$, and $f : R^2\to R$ is a continuous function. By means of a new approach, based on a combination of lower and upper-solutions methods and phase-plane techniques, 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 application, 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 application in the field of front-type solutions for reaction diffusion equations.

Mathematics Subject Classification: Primary: 34B15, 34A40.

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