Knapsack problem for Baumslag–Solitar groups

In this work we investigate a decidability problem of group version of the knapsack problem for the Baumslag–Solitar group $BS(p,q)$. We proved, that the knapsack problem is decidable for the group $BS(p,q)$ for coprime integers $p > 1$, $q > 1$. In the case where $p=1$, $q\in \mathbb{N}$, we proved that the knapsack problem is decidable for the group $BS(1,q)$ with some restriction on the input of the problem. However, the problem of the decidability of the knapsack problem for the group $BS(1,q)$ on the whole set of inputs remains open.