Looking at this question, I wanted to use the hint in order to show that the r.h.s is locally lipschitz w.r.t $y$
So $$f(x,y)=\frac{1}{2}\left\vert \frac {x}{y}+\frac {y^3}{x^3}\right\vert$$
Hence, $$\vert f(x,y_1)-f(x,y_2)\vert = \frac{1}{2} \left\vert \frac{x^4(y_2-y_1)+y_1y_2(y_1^3-y_2^3)}{x^3y_1y_2} \right\vert$$
Now, the hint by Siminore suggests to use the triangle inequality and the Mean Value Theorem: therefore, I obtain
$$ \frac{1}{2} \left\vert \frac{x(y_2-y_1)}{y_1y_2} \right\vert + \frac{1}{2} \left\vert \frac{(y_1^3-y_2^3)}{x^3} \right\vert \leq \left\vert \frac{x(y_2-y_1)}{y_1y_2} \right\vert + \frac{3}{2} M^2\left\vert \frac{y_1-y_2}{x^3} \right\vert $$
where $M$ is the max of $y \mapsto 3y^2$ for $y$ in a neigbourhood of $y_0$.
I can't understand how to bound the first term in order to show the Lipschitz conditions. How can I move?