Convergence of the solutions of the discounted equation: the discrete case

We derive a discrete version of the results of our previous work. If $M$ is a compact metric space, $c:M\times M\to \mathbb{R}$ a continuous cost function and $\lambda \in (0,1)$, the unique solution to the discrete $\lambda$-discounted equation is the only function $u_{\lambda }:M\to \mathbb{R}$ such that $$\mathrm{\forall }x\in M,u_{\lambda }(x)=\underset{y\in M}{\min }\lambda u_{\lambda }(y)+c(y,x)\mathrm{.}$$ We prove that there exists a unique constant $\alpha \in \mathbb{R}$ such that the family of $u_{\lambda }+\alpha \mathrm{/}(1-\lambda )$ is bounded as $\lambda \to 1$ and that for this $\alpha$, the family uniformly converges to a function $u_0:M\to \mathbb{R}$ which then verifies $$\mathrm{\forall }x\in X,u_0(x)=\underset{y\in X}{\min }{u}_0(y)+c(y,x)+\alpha \mathrm{.}$$ The proofs make use of Discrete Weak KAM theory. We also characterize $u_0$ in terms of Peierls barrier and projected Mather measures.