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Solutions in weighted spaces of singular boundary value problems on the half-line. (English) Zbl 1124.34008

The paper is concerned with the following singular boundary value problem on the half-line \[ y''(t)+\Phi(t)f(t,y(t))=0,\qquad t\in (a,+\infty), \]
\[ y(a)=0, \qquad \lim_{t\to +\infty} y'(t)=0, \] where \(f:[a,+\infty)\times \mathbb{R}_0^+ \to \mathbb{R}^+\) is continuous and satisfies \[ \lim_{y\to 0^+}f(t,y)=+\infty \qquad \text{for each \(t\in (a,+\infty)\),} \] and \(\Phi: (a,+\infty)\to \mathbb{R}_0^+\) is continuous. Existence of at least one solution under appropriate conditions is proved by the method of upper and lower solutions. Furthermore, existence of at least two solutions is proved by the fixed point index theory on cones.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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