Consider the second-order ODE
with $y$ an $L^2$ complex-valued function on $[a,b]$ subject to the boundary conditions:
$\alpha_1y(a)-\alpha_2y'(a)=0$ and $\beta_1y(b)-\beta_2y'(b)=0$
where $f$ is also an arbitrary $L^2$ complex-valued function on $[a,b]$, $p$,$q$ and $w$ are real-valued $C^1$ functions on $[a,b]$ and $\lambda$ is a complex constant.
Suppose that we are able to construct a function which, when evaluated for a given $f$, returns a solution $y$ that meets the above constraints. Is this $y$ then a unique solution to $(1)$?
[Alternative characterisation: I have avoided functional analysis terminology above so as not to make the problem too generic and thus eliminate important information, but if we take $(1)$ to define an operator $D$ from a subset of $L^2$ onto $L^2$, I am wondering whether a second operator $E$ which determines a $y$ for any given $f$ is in fact the inverse. P.S. The methodology for constructing $E$ given $D$ and the circumstances under which this is possible are already clear to me.]