Abuse of notation in declaring a variable is a function of another? 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.
 A: There is nothing odd or abusive in this notation. The heart of the problems seems to lay in the fact that the notation
$$
=
$$
is understood in mathematics in two different senses. 
First, it is the equality operator, meaning that expression of the form
$$
A=B
$$
is true if $A$ is equal to $B$.
Second, it is an assignment operator, meaning that the expression of the form
$$
A=B
$$
amounts to "$A$ is defined to be the same as $B$, which we already know". It is this second meaning is usually supposed in the expressions of the form $y=y(x)$. For instance, I often write in my papers something along the line: "Consider a system of ODE
$$
\dot x=f(x),
$$where $x=x(t)$" emphasizing that letter $x$ in my text denotes function $x(t)$.
A: This bothers me too and I never do it in my private life. However, when teaching calculus it's unavoidable. To keep my inner-stickler happy, I secretly think to myself that $x=\DeclareMathOperator{id}{id}\id_X$ where $X$ is the domain of $y$. Thus we have
$$
y=y\circ\id_X=y\circ x
$$
Defining $y(x)$ as $y\circ x$ then justifies the notation $y=y(x)$ and I don't have to think about elements.
A: 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.
