Neumann problems for second order ordinary differential equations across resonance (Q1894288)

The boundary value problem \((*)\) \(y''+ f(t, y)= 0\), \((t, y)\in W= [0, 1]\times \mathbb{R}\), \(y'(0)= a\), \(y'(1)= b\) is considered. Some results concerning the unique solvability of \((*)\) are established. Here \(f\), \(f_ y\in C(W)\) and \(f\) belongs to the nearly-resonance domain: \(A\leq f_ y\leq \beta(t)\leq B\) for some \(k^ 2\pi^ 2< A< (k+ 1)^ 2 \pi^ 2\leq B< \infty\), \(\beta\in L[0, 1]\). By means of Pontryagin's maximum principle, the exact estimates, which guarantee the existence of a unique solution to the corresponding linearized BVP, are also obtained.