I am currently having trouble solving a difference equations problem where the initial value is given and the end value. I simplified the problem to the following:
\begin{align}
y^{k - 1} - 2y^{k} + y^{k+1} = 0, \qquad k=1, \dots, T-1, y^{i} \in \mathbb{R}^{d} \text{ for } i\in [0, \dots, T], 
\end{align}
where the values for $y^{0}$ and $y^{T}$ are given. I have tried googling but the only thing I found are examples where $y^{0}$ and $y^{1}$ are known, which is not case here sadly.

**Question:** Does anyone know how to solve this?

Thanks for any help or tips in advance!