On coincidence points of mappings in generalized metric spaces

Let $X$ be a space with $\infty$-metric $\rho$ (with possibly infinite value) and $Y$ a space with $\infty$-distance $d$ satisfying the identity axiom. We consider the problem of the coincidence point for the mappings $F,G:X \to Y$ of the existence of the solution for the equation $F(x)=G(x).$ We provide conditions of the existence of the coincidence points in terms of the covering set for the mapping $F$ and the Lipschitz set for the mapping $G$ in the space $X\times Y.$ The $\alpha$-covering set ($\alpha > 0$) of the mapping $F$ — is the set of such $(x,y),$ that
$$\exists u\in X \ F(u)=y, \ \ \rho(x,u)\leq \alpha^{-1}d(F(x),y), \ \ \rho(x,u)<\infty,$$
and the $\beta$- Lipschitz set ($\beta\geq 0$) for the mapping $G$ — is the set of such $(x,y),$ that
$$ \forall u\in X\,\, G(u)=y \, \Rightarrow \, d(y,G(x))\leq \beta \rho(u,x).$$
The new results are compared with the known theorems about the coincidence points.