Existence and multiplicity for an O.D.E. with nonlinear conditions (Q1355050)

The nonlinear problem \[ u''(x)+g(x,u)=0,\;u'(0)=-f(u(0)),\;u'(\pi)=f(u(\pi)) \] is considered, where \(f\) and \(g\) are continuous, \(f\) is increasing and \(g\) either satisfies a sign condition or is increasing with respect to \(u\). Under suitable additional assumptions on \(f\) and \(g\), which give among others that \(u=0\) is a solution, it is shown that the problem has multiple solutions, where a lower bound of the number of solutions is given. If it is not known that \(u=0\) is a solution, a different set of conditions on \(f\) and \(g\) is given such that the problem has at least one solution.