I want to prove that the following claims are equivalent:

1) The boundary value problem $Ly = f, By = c$ has a unique solution for all $f \in \mathcal{C}[a,b]$ and $c \in \mathbb{C}^2$,

2) $Ly = 0, By = c$ has a unique solution for all $c \in \mathbb{C}^2$.

I'm not sure where to start really. One idea I had was to assume that the claim in (1) is not true, so that there are two solutions $y$ and $\tilde{y}$, and then $L(y-\tilde{y}) = Ly - L\tilde{y} = f-f = 0$, but $y \neq \tilde{y}$, and since a homogeneous system $Ly = 0$ always have at least the trivial solution $y = 0$, this would imply that $Ly = 0, By=c$ has at least two distinct solutions. However, I'm not sure if it is even true that the trivial solution $y=0$ is a solution to $Ly=0, By=c$.

Any help?


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