Let $f(x):\mathbb{R}\rightarrow\mathbb{R}$, be a strictly decreasing, convex and continuous function in $x$ with $f(x)>0, \forall x>0$, with $\underset{x\rightarrow\infty}{\textrm{Lim}} f(x)=0$.
Let $g(y,x):\mathbb{R}^2\rightarrow\mathbb{R}$, with $g(y,x)>0, \forall x,y>0$, be a continuous and strictly decreasing function in $x$, with $\underset{x\rightarrow\infty}{\textrm{Lim }} g(y,x)=0$ and single-peaked function in $y$, with $\underset{y\rightarrow\infty}{\textrm{Lim }} g(y,x)=0$.
Under which conditions (on $f$ and $g$) can I prove the existence of a point $x>0$ that satisfies $$f(x)=g(f(x),x).$$ I suppose I should use Brouwers fixed point theorem, but am unsure about the approach.