Consider the boundary value problem $u^{\prime \prime }+u=\alpha u^{-}+p\left(t\right)$, $u\left(0\right)=0=u\left(\pi \right)$, $\alpha \le 0$. We show that a necessary and sufficient condition for the problem to be solvable is that ${\int }_0^{\pi }p\left(t\right)sintdt\ge 0$. We thus answer positively a counter part of a long-standing open problem posed by S. Fučík. A more general problem is also considered and the sufficient condition for the existence of solution is obtained.

中文翻译：

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关于S.Fučík的一个开放问题

考虑边值问题 ${ü}^{\prime \prime }+ü=\alpha {ü}^{-}+p（Ť）$， $ü（0）=0=ü（\pi ）$， $\alpha \le 0$。我们证明要解决该问题的必要和充分条件是${\int }_0^{\pi }p（Ť）罪ŤdŤ\ge 0$。因此，我们肯定地回答了S.Fučík提出的长期存在的开放问题。还考虑了一个更一般的问题，并获得了解决方案存在的充分条件。