Consider the following integral equation

$\phi(x) = f(x) + \frac{1}{x}\int_0^x N(x,y)\phi(y)\;dy$,

where $f$ and $N$ are continuous and bounded functions. Are solutions $\phi$ of the above equation unique? If so, can one get an estimate of the form

$\sup_{(0,x)} |\phi| \leq C \sup_{(0,x)}|f|$ ?

Additional info: Assume that $\phi(0)=\phi(1)=f(0)=0$ and that $\|N\|_\infty\geq 1$. I am only interested in a solution (or lack thereof) on the interval $[0,1]$.

As a note, without the $\frac{1}{x}$ multiplier, this is a Volterra equation of the second kind and existence/uniqueness of a solution $\phi$ is well-known and the above estimate is indeed satisfied.