barrier arguments.

\[le:barrier\_argument\] For ${{\Delta t}}>0$, the following inequality holds $$\label{eq:barrier1}
    \bar D_{\Delta t}^\alpha ((N-n){{\Delta t}})^\alpha\ge \frac{\alpha(1-\alpha)}{\Gamma (2-\alpha)},\qquad n=0,\dots,N-1.$$

Well posedness of the discrete problem {#sec:well_posed}
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In this section, we study the well posedness of the discrete problem .

Uniqueness
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The uniqueness argument is classical in Mean Field Games theory and relies on the monotonicity of the coupling cost $f$ and convexity of the discrete Hamiltonian $g$.

\[thm:uniqueness\] The problem has at most one solution.

We suppose by contradiction that there are two solutions $(U,M)$ and $(\tilde{U},\tilde{M})$ such that $U^N = \tilde{U}^N$ and $M^{0} = \tilde{M}^{0}$. Then we have $$\begin{aligned}
\bar D_{\Delta t}^{\alpha} (U^{n}& -\tilde{U}^{n})_{i,j} - (\Delta_h (U^{n}-\tilde{U}^{n}))_{i,j} \\
& + g(x_{i,j},[D_hU^{n}]_{i,j}) - g(x_{i,j},[D_hU^{n}]_{i,j}) -  (f_h[M^{n+1}] - f_h[\tilde{M}^{n+1}] )_{i,j}=0.
\end{aligned}$$ Multiplying by $Z_{i,j}$ and summing over all $i,j$ leads to $$\label{eq:u_diff}
\begin{aligned}
(\bar D_{\Delta t}^{\alpha} & (U^{n} -\tilde{U}^{n}),Z)_2 - ((\Delta_h (U^{n}-\tilde{U}^{n})),Z)_2 \\
 & + \sum_{i,j} (g(x_{i,j},[D_hU^{n}]_{i,j}) - g(x_{i,j},[D_h\tilde{U}^{n}]_{i,j}))Z_{i,j} - (( f_h[M^{n+1}] - f_h[\tilde{M}^{n+1}]),Z )_2=0.
\end{aligned}$$ Multiplying the discrete equation by $W_{i,j}$ and summing over all $i,j$ we obtain $$(D_{\Delta t}^\alpha M^{n+1},W)_2 - (M^{n+1}, \Delta_h W)_2 + \sum_{i,j} M_{i,j}^{n+1} [D_hW]_{i,j} \cdot \nabla_q g  (x_{i,j},[D_hU^{n}]) = 0.$$ From this and the similar equation satisfied by $\tilde{M}^{n+1}$, we obtain $$\label{eq:m_diff}
\begin{aligned}
