On the boundary value problem $u”+u=\alpha u^{-}+p(t),$ $u(0)=0=u(\pi )$

Abstract:

In this note we consider the boundary value problem $u'' + u = a{u^ - } + p(t),\;u(0) = 0 = u(\pi ),\;\alpha > 0$, and show that a necessary and sufficient condition for the problem to be solvable is that $\smallint _0^\pi {p(s)\sin s\;ds\; \leqslant 0}$. We thus answer in the affirmative a question posed by S. Fučik.