argument.

For ${{\Delta t}}$ sufficiently small, has a solution.

First observe that, by the assumption on $m_0$, we have that $M^0\in{\mathcal{K}}_h$. We denote $$\label{eq:def_K_N}
 {\mathcal{K}}_h^N=\{M=(M^n)_{0\le n\le N} :\, M^0= \frac 1 {h^2} \int_{|x-x_{i,j}|_{\infty}\le h/2} m_0(x) dx,\,  M^n\in{\mathcal{K}}_h,\,\forall n=0,\dots,N\}$$ and we define a map $\Phi$ in the following way: Let $\Psi_1:{\mathcal{K}}^N_h\to ({\mathbb{R}}^{N_h^{2}})^N$ be the map which associates to $M=(M^n)_{0\le n\le N}$ the solution $U=(U^n)_{0\le n\le N}$ of and let $\Psi_2: ({\mathbb{R}}^{N_h^2})^N\to ({\mathbb{R}}^{N_h^2})^N$ the map that to $U=(U^n)_{0\le n\le N}$ associates the solution of . Then we show that the map $\Phi=\Psi_2\circ \Psi_1$, is well defined on ${\mathcal{K}}^N_h$ and maps this set into itself.\
We show that the map $\Psi_1$ is bounded by means of a barrier argument. By , the function $$V^n=\|U^N\|_\infty + \frac{M_0\Gamma (2-\alpha)}{\alpha(1-\alpha)}((N-n){{\Delta t}})^\alpha,$$ where $M_0=\sup_{i,j}|H(x_{i,j},0)-(f_h(M^{n+1}))_{i,j}|$, satisfies $V^N\ge U^N$ and $$\bar D_{\Delta t}^{\alpha} V_{i,j}^{n} - (\Delta_h V^{n})_{i,j} + g(x_{i,j},[D_hV^{n}]_{i,j}) \ge  f_h[M^{n+1}]_{i,j},\quad n=0,\dots,N-1.$$ By Proposition \[prop:ex\_HJ\], we get $U^n\le V^n$, $n=0,\dots,N$. In a similar way, we prove that $U^n\ge- V^n$, $n=0,\dots,N$, and therefore $$\|U^n\|_\infty \le \|U^N\|_\infty +CT^\alpha.$$ Hence $\Psi_1$ maps ${\mathcal{K}}_h^N$ in a bounded subset of $({\mathbb{R}}^{N_h^2})^N$.\
To prove that the map $\Psi_1$ is continuous, consider a sequence $M_{\kappa}=(M_{\kappa}^n)_{0\le n\le N}\in {\mathcal{K}}_h^N$ which tends to $M=(M^n)_{0\le n\le N}\in {\mathcal{K}}_h^N$ for 