Consider two sequences $(X_0, X_1, \dots)$ and $(Y_0, Y_1, \dots)$, where $X_k\in \mathbb{R}^n$ and $Y_k \in\mathbb{R}^n$. Suppose two continuous nonlinear functions $f$ and $g$, so that the following recursions hold:
\begin{align*} X_{k} &= \arg\max f(X_k, Y_{k+1})\\ Y_{k+1} &= g(Y_k, X_k) \end{align*} with $Y_0$ given. Here, $\arg\max$ returns the maximizer $X_k$ of function $f(X_k, Y_{k+1})$.
It is known that for fixed $X_k=X, \forall k$, $Y_{k}$ converges to $Y$ as $k$ goes to infinity. Besides, all $X_k$ and $Y_k$ are nonnegative.
How can I show that both $X_k$ and $Y_k$ converge and under what conditions?