A Landesman–Lazer-type condition for asymptotically linear second-order equations with a singularity

We consider the T-periodic problem \begin{gather*}x''+g(t,x)=0,\\x(0)=x(T),\qq x'(0)=x'(T),\end{gather*} where g: [0,T]×]0,+∞[→ℝ exhibits a singularity of a repulsive type at the origin, and an asymptotically linear behaviour at infinity. In particular, for large x, g(t, x) is co...

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Bibliographic Details
Published in:Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2012-12, Vol.142 (6), p.1263-1277
Main Authors: Fonda, Alessandro, Garrione, Maurizio
Format: Article
Language:English
Subjects:
Asymptotes
Asymptotic methods
Asymptotic properties
Infinity
Mathematical analysis
Origins
Singularities
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Summary:We consider the T-periodic problem \begin{gather*}x''+g(t,x)=0,\\x(0)=x(T),\qq x'(0)=x'(T),\end{gather*} where g: [0,T]×]0,+∞[→ℝ exhibits a singularity of a repulsive type at the origin, and an asymptotically linear behaviour at infinity. In particular, for large x, g(t, x) is controlled from both sides by two consecutive asymptotes of the T-periodic Fučik spectrum, with possible equality on one side. Using a suitable Landesman–Lazer-type condition, we prove the existence of a solution.
ISSN:0308-2105
1473-7124
DOI:10.1017/S0308210511000151