Let's assume that the you have :
\begin{align} \mathcal{L}_1 y &= \mathcal{F}_1(s) \quad s \in [0,1] \\ \mathcal{L}_2 y &= \mathcal{F}_2(s) \quad s \in ]1,2] \end{align}
where $\mathcal{L}_i$ are linear order 2 differential operators and $y$ the unknown.
Let's further assume that you have continuity conditions such that : \begin{align} (1) :& y(1^-)=y(1^+) \\ (2) :& y'(1^-)=y'(1^+) \end{align}
Does anyone knows how to adapt solvability conditions on that problem ?
Even if both problems are linear, when you stick them thanks to continuity the problem is no longer linear so I could not derive Fredholm alternative to get my solvability conditions.
Moreover, does an equivalent of "kernel" exist for the whole problem defined in [0,2] ?