on ${\mathbb{R}}^{N_h}$) and, by standard monotonicity argument, $$\label{eq:lip0}
\|\Psi(V)-\Psi(W)\|_\infty\le  \|V-W\|_\infty.$$ for any $V, W\in {\mathbb{R}}^{N_h}$. For fixed $l,\kappa \in {\mathbb{Z}}^2$, call $\tau U$ the discrete function defined by $$(\tau U)^n_{i,j} = U^n_{l+i,\kappa+j}.$$ We have that $$\bar D_{\Delta t}^{\alpha} (\tau  U)_{i,j}^{n} - (\Delta_h (\tau U)^{n})_{i,j} + g(x_{i,j},[D_h(\tau U)^{n}]_{i,j})= f_h[M^{n+1}]_{i,j}+ E_{i,j}$$ where $$\begin{aligned}
E_{i,j}& =   f_h[M^{n+1}]_{i+l,j+\kappa} -f_h[M^{n+1}]_{i,j}  - g(x_{i+l,j+\kappa},[D_h(\tau U)^{n}]_{i,j}) + g(x_{i,j},[D_h(\tau U)^{n}]_{i,j}).\end{aligned}$$ Hence $ (\tau U)^n=  \Psi\left(\sum_{k=n+1}^{N} {\overline c}_{n}^k (\tau U)^k+\rho_\alpha E\right)$. Moreover, since $$\begin{aligned}
&f_h[M^{n+1}]_{i+l,j+\kappa} -f_h[M^{n+1}]_{i,j} \le C h \sqrt{l^2+\kappa^2},\\
&g(x_{i,j},[D_h(\tau U)^{n}]_{i,j}) - g(x_{i+l,j+\kappa},[D_h(\tau U)^{n}]_{i,j}) \le C(1+\|D_hU^n\|_\infty)h\sqrt{l^2+\kappa^2}\end{aligned}$$ it follows that $$\|E\|_\infty \le C(1+\|D_hU^n\|_\infty)h\sqrt{l^2+\kappa^2}.$$ By , we have $$\begin{aligned}
&\|(\tau U)^n-U^n\|_\infty= \left\|\Psi\left(\sum_{k=n+1}^{N} {\overline c}_{n}^k (\tau U)^k+\rho_\alpha E\right)
-\Psi\left(\sum_{k=n+1}^{N} {\overline c}_{n}^k  U^m\right)\right\|_\infty\\
&\le \left\|\sum_{k=n+1}^{N} {\overline c}_{n}^k( (\tau U)^k - U^k)\right\|_\infty+{\rho_\alpha}\|E\|_\infty.\end{aligned}$$ Hence $$\left\|\frac{(\tau U)^n - U^n}{h \sqrt{l^2+\kappa^2} }\right\|_\infty \le   \sum_{k=n+1}^{N}{\overline c}_{n}^k\left\| \frac{(\tau U)^k - U^k}{h \sqrt{l^2+\kappa^2}}\right\|_\infty + C \rho_\alpha (1+ \|D_hU^{n}\|_\infty).$$ For the arbitrariness of $l,\kappa$, the previous inequality implies that $$\bar D_{\Delta t}^{\alpha}(\|D_hU^n\|_\infty)\le C\|D_hU^n\|_\infty+C$$ and therefore, applying a backward version of the discrete Gronwall’s inequality with ${\omega}^n=\|D_hU^n\|_\infty$, ${\lambda}_1=C$, ${\lambda}_2={\lambda}_3=0$ and $g^n=C$, we obtain for some positive constant $L$.

\[prop:ex\_FP\] For ${{\Delta t}}$ sufficiently small, there exists a unique solution $(M^n)_{n=0}^N$ to . Moreover, $M^n\ge 0$ and $(M^n,\bar 1)_2=1$ for any $n=0,\dots,N$.

Equation 