The Sturm-Liouville problem for the equation \(X''=P(T,X)Q(X')\) (Q917756)

Existence theorems are proved for Sturm-Liouville problems determined by equations of the form \(x''=f(t,x,x')\) and boundary conditions \(a_ 0x(0)-b_ 0x'(0)=0,\) \(a_ 1x(1)+b_ 1x'(1)=0,\) where \(a_ 0\), \(a_ 1\), \(b_ 0\), \(b_ 1\) are non-negative real numbers with \(a_ 0+b_ 0\), \(a_ 1+b_ 1\) and \(a_ 0+a_ 1\) positive. It is proved that there is a solution in the case \(x''=p(x)q(x')\), provided p and q are real- valued continuous functions on \({\mathbb{R}}\) and q has two zeros of opposite signs. This result remains true for \(x''=A(t)q(x')\) if A: [0,1]\(\to {\mathbb{R}}\) is a \(C^ 1\)-function.