We are asked to show that the Volaterra Equation $$ f(x)=\phi(x)+\lambda\int_a^x K(x,y)f(y)dy, x\in[a,b] $$ has a unique solution function $f\in C[a,b]$ for any $K\in C([a,b]\times[a,b]), \phi\in C[a,b]$, and $\lambda\in \mathbb{R}$.
My approach was to let $\tau$ be a transform and $\tau(f(x))=\phi(x)+\lambda\int_a^x K(x,y)f(y)dy$ for all x. Then I would like to show that $\tau$ is a contraction and then use the Fixed Point Theorem. But the difficulty comes because I cannot figure out a way to get rid of $K$ or $\lambda$. Please note that this is a problem appeared in my real analysis course and we have not really touched more general functional analysis theory. So an answer or hint without too much deep functional analysis detail will be better.