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Existence of bounded trajectories via upper and lower solutions. (English) Zbl 0979.34019
Summary: The authors deal with the boundary value problem on the whole line \[ u''- f(u, u')+ g(u)= 0,\quad u(-\infty)= 0,\quad u(+\infty)= 1,\tag{P} \] where \(g: \mathbb{R}\to \mathbb{R}\) is a continuous nonnegative function with support \([0,1]\), and \(f: \mathbb{R}^2\to \mathbb{R}\) is a continuous function. By means of a new approach, based on a combination of lower and upper solutions methods and phase-plane techniques, they prove an existence result for (P) when \(f\) is superlinear in \(u'\); by a similar technique, they get a nonexistence one. As an application, the authors investigate the attractivity of the singular point \((0,0)\) in the phase-plane \((u,u')\). They refer to a forthcoming paper for a further application in the field of front-type solutions to reaction diffusion equations.

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C99 Qualitative theory for ordinary differential equations
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