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.



A Volterra-type equation
Not in general, since if $N(x,y)=a>1$ then the equation with $f=0$ has the solution $\phi(x)=x^{a-1}$. If, however, $N(0,0)<1$ (assuming as in the question that $N$ is continuous) then one can use the Banach fixed-point theorem (on short intervals) to get a unique solution.


