Second-Order Boundary Value Problem With Set of the Associated Green's Function May Have Zeros

Abstract

We consider the nonlinear second-order boundary value problem $$u''+\kappa^{2}u=f(t, u(t)), \  \  \  \  t\in (0, T),T>0,$$ $$u(0)=u(T),\  \ u'(0)= u'(T),$$  where $0<\kappa<\frac{\pi}{T}$,  $f:[0, T]\times[0,\infty)\to [0,\infty]$ is continuous. We use the sets of the associated Green's function may have zeros at some interior points. In particular, we study the problems where the associated Green's function may have zeros. The proof is based on the fixed point theorem in cones.