The standard way to write $ \text{ y is a function of x} $ is
$ y = f(x) $
This is taken to mean that $y$ is the value of function $f$ evaluated at $x$. For simplicity let's take $f$ to be some mapping, $ f:\Bbb R\to\Bbb R$.
I cannot understand if mathematics authors are justified in using the notation
$y = y(x)$
to declare that $y$ is a function of $x$. The reason is a type mismatch, it cannot be be possible for $y$ to be a binary relation, as well as some element in the codomain of the binary relation.
Is the notation above commonly accepted? I have seen it in a few published papers, and am not sure whether it is an abuse or has some sound mathematical reasoning.
Without knowing the context to which you are referring I would say the author doesn't wish to a lot of $(x)$'s. If, for example, there is integration or differentiation with respect to $x$ then you have to take the fact that $y$ is a function of $x$ into account. On the other hand if there is integration or differentiation with respect to $t$ then perhaps you can consider $y$ to be a constant. This answer is speculative because we are not given a complete example.