Contraction of Fredholm equation

Question

I am supposed to find fitting conditions for two fuctions $g(x)\in C([0,1])$ and $K(x,t)\in C([0,1]^2)$ such that the function $$y(x)=g(x)+\int_0^1K(x,t)y(t)^2dt$$ has a unique solution. For this I wanted to show that the function is a contraction mapping, then by Banach's fixed point theorem the iteration would converge towards a unique solution. So far I was able to show, that $$\|T(y_1)-T(y_2)\|_{\infty} \le \|y_1-y_2\|_{\infty}*\int_0^1\|K(x,t)*(y_1(t)+y_2(t))\|_{\infty}dt$$ Where $T(y_i)$ denotes the definition at the top. Here is where I run into trouble: for the mapping to be a contraction the integral term has to be $<1$, which would be fine if it were not for $(y_1(t)+y_2(t))$. My question would be how can I "remove" the dependency on $y_1, y_2$? I tried playing around with the mean value theorem to no avail. Any help is much appreciated. Thanks in advance