Numerically solve an integral equation of the form $f(x)=f(0) - \frac{1}{2} \int_0^x f(x-y) \cdot (1-H(y)^2)\, dy.$

Question

I have an integral equation of the form :

$$f(x)=f(0) - \frac{1}{2} \int_0^x f(x-y) \cdot (1-H(y)^2)\, dy.$$

I am looking for an efficient method to solve this integral equation, in particular I am interested in obtaining $\lim_{x \rightarrow \infty} f(x)$?

The function $H(y)$ is defined by: $$ H’(y)=1-H(y)-f(y),\qquad H(0) = 0. $$