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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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