of the problem.\
The boundedness and continuity of the map $\Phi_1$ and the continuity of the map $\Psi_2$ implies that $\Phi=\Psi_2\circ\Psi_1$ is a continuous map on the compact, convex set ${\mathcal{K}}_h^N$. Moreover by Prop. \[prop:ex\_FP\], $\Phi$ maps ${\mathcal{K}}_h^N$ into itself. Hence, by Brouwer’s Theorem, $\Phi$ admits a fixed point $M$, which clearly a solution of system .

Convergence {#sec:convergence}
===========

In this section we prove a convergence result for the scheme . We relies on the analysis performed in [@acc Section 3] for the Mean Field Games system and in [@lx Theorem 3.2] for computation related to fractional derivatives.\
A preliminary result for the analysis is a fundamental identity which relies on the duality structure of the discrete system.

Let $(U,M)$ be a solution of and $({\tilde U},{\tilde M})$ a solution of the perturbed system $$\label{eq:conv_scheme_pert}
\left\{
\begin{array}{ll}
\bar D_{\Delta t}^{\alpha} {\tilde U}_{i,j}^{n} - (\Delta_h {\tilde U}^{n})_{i,j} + g(x_{i,j},[D_h{\tilde U}^{n}]_{i,j}) = f_h[{\tilde M}^{n+1}]_{i,j}+a^n_{i,j}, \\[6pt]
D_{\Delta t}^\alpha {\tilde M}_{i,j}^{n+1} - (\Delta_h {\tilde M}^{n+1})_{i,j} - {\mathcal{B}}_{i,j}({\tilde U}^{n},{\tilde M}^{n+1}) = b^{n+1}_{i,j}, \\[6pt]
{\tilde U}^{N}_{i,j} = u_T(x_{i,j}), \,   {\tilde M}_{i,j}^{0}=  \frac 1 {h^2} \int_{|x-x_{i,j}|_{\infty}\le h/2} m_0(x) dx,
\end{array}
\right.$$ for $n=0,\dots,N-1$ and $0\le i,j\le N_h$. Then $$\label{eq:conv_fund_id}
\begin{aligned}
&  \sum_{n=0}^{N-1}   \bigg(f_h[M^{n+1}] - f_h[\tilde{M}^{n+1}],M^{n+1} - 