Second order nonlinear differential equations with linear impulse and periodic boundary conditions. (Q664413)

The paper deals with the existence of solutions to the problem \[ -(p\,(t)\,y')\,'+q(t)\,y=f(t,y),~y(c-)=d_1\,y(c+), \] \[ (p\,y')'(c-)=d_2\,(p\,y')'(c+),~y(a)=y(b),~(p\,y')'(a)=(p\,y')'(b), \] where \(-\infty <a<c<b<\infty ,\) \(p,\,q:[a,b]\to \mathbb C\) are continuous on \([a,c)\cup (c,b],\) regulated on \([a,b],\) \(p\neq 0\) on \([a,b],\) \(d_1,\,d_2\in \mathbb C,\) \(f\:[a,c)\cup (c,b]\times \mathbb C^2\to \mathbb C\) is continuous and for each \(\xi _0\in \mathbb C\) there exist the finite limits \[ \lim \limits {_{\substack{ (t,\xi )\to (t,\xi _0)\\t<\,c}} }\,f(t,\xi )\quad \text{ and }\quad \lim \limits {_{\substack{ (t,\xi )\to (t,\xi _0)\\t>\,c}} }f(t,\xi ). \] The paper provides two existence results for the non-resonance case. Both of them utilize the operator representation of the problem and the properties of the Green's function derived in the paper. The former one is based on the contraction mapping principle and, in addition, it ensures also the uniqueness of the solution. The latter one is based on the Schauder fixed point theorem.