Show a solution to $y(x)=g(x)+\int\limits_{0}^{x}k(x,t,y(t))dt$ exists under certain assumptions on $k(x,t,z)$ and $g(x)$.

Question

I got this homework question that I am stuck on. Let $J = [0, a]$ (with $a > 0$ fixed). Let $g(x)$ be a function which is continuous at all $x \in J$ and let $k(x, t, z)$ be a function which is continuous and bounded in $\{(x, t, z) \in J\times J \times \mathbb{R} : t < x\}$. Prove that there exists at least one solution of the equation: $$y(x)=g(x)+\int\limits_{0}^{x}k(x,t,y(t))dt, \ \ \ x\in J$$ It can apparently be proven in a similar way as the Peano existence theorem http://en.wikipedia.org/wiki/Peano_existence_theorem I would show my workings so far but I just have no idea where to start on this one. I think that we need to differentiate the given equation to transform it into an equivalent differential equation and then show that a solution to the DE exists. Any help would be much appreciated, Thanks.