Non-existence results for a system of wave inequalities on locally finite graphs

Let $V$ be a locally finite, connected and weighted graph. We study non-existence results of non-trivial, non-negative solutions of the system $$\{\begin{array}{lllll}{u}_{tt}-\mathrm{\Delta }u\ge h_1\mathrm{\mid }v{\mathrm{\mid }}^p& \text{ in }V\times (0,\mathrm{\infty }),v_{tt}-\mathrm{\Delta }v\ge h_2\mathrm{\mid }u{\mathrm{\mid }}^q& \text{ in }V\times (0,\mathrm{\infty }),u=u_0;v=v_0& \text{ in }V\times \{0\},u_t=u_1;v_t=v_1& \text{ in }V\times \{0\},\end{array}$$ where $p,q>1$, $h_1,h_2$ are positive potentials. Under some volume growth condition of a ball, we prove that the system has no non-trivial non-negative solutions. In particular, our result is a natural extension of that in [$\mathit{\text{D.}}\mathit{\text{D.}}\mathit{\text{Monticelli,}}\mathit{\text{F.}}\mathit{\text{Punzo,}}\mathit{\text{and}}\mathit{\text{J.}}\mathit{\text{Somaglia.}}\mathit{\text{Nonexistence}}\mathit{\text{results}}\mathit{\text{for}}\mathit{\text{the}}\mathit{\text{semilinear}}\mathit{\text{wave}}\mathit{\text{equation}}\mathit{\text{on}}\mathit{\text{graphs.}}\mathit{\text{arXiv.2506.08697,}}\mathit{\text{2025.}}$] from a single inequality to a system.