Two point boundary value problems for second order ordinary differential equations across many resonant points (Q1312725)

This article deals with the existence and uniqueness of solution to the two point boundary value problem across many resonant points: \(y''+f(x,y)=0\), \(y(0)=a\), \(y(1)=b\), mainly under the following conditions: (1) \(f\) and \(f_ y\) are continuous on \([0,1] \times \mathbb{R}\); (2) \(A \leq f_ y(x,y) \leq \beta(x) \leq B\), \(\int^ 1_ 0 \beta (x)dx<B \alpha_ 1+(1-\alpha_ 1)A\), where \(A\), \(B>0\) with \(0<A<k^ 2 \pi^ 2<B\), \(k\) is the minimal positive integer suiting the inequality and \(\alpha_ 1\) is the minimal positive root of \(\sqrt A \text{ctg} (\sqrt A(1-\alpha)/(2k))=\sqrt B \text{tg} (\sqrt B \alpha/(2k))\). The results are proved by the optimal control theory method.