Existence of Solutions of a Nonlinear Differential Equation
L. Cesari, R. Kannan · Proceedings of the American Mathematical Society · 1983
A criterion is proved for the existence of at least one solution to the equation $u'' + u = g(u) + h$ with $u(0) = u(\pi ) = 0$, where $h \in {L_2}[0,\pi ]$ and $g$ is continuous monotone nonincreasing.