A functional-differential equation

Consider the functional differential equation
$$4f\left(\frac x2-f(x)\right)+4~\epsilon~ f(x)f'\left(\frac x2-f(x)\right)=f(x)$$
for all $x\geq0$ together with the initial condition $f(0)=0$ and the additional constraint $f(x)\geq0$ for all $x$. This equation arises as a first-order optimality condition in a problem that I am studying. $~\epsilon~$ is a parameter in an interval around $0$.

For all $~\epsilon~$ the equation has the trivial solution $~f(x)=0~$ for all $~x~$, which is not of any interest for me.

The equation also has a linear solution
$$f(x)=\frac x{4(1-\epsilon)},$$
which is valid for all $\epsilon<1$.

Finally, for $\epsilon=0$, there exists the non-linear but smooth solution
$$f_0(x)=\frac{1+2x-\sqrt{1+4x}}8.$$

My question is whether you have any idea how I can embed the solution $f_0$ for the singular case $\epsilon=0$ into a whole family of solutions $f_\epsilon$ for values of $\epsilon$ close to $0$.

I understand that this is a singular perturbation problem. Although I am primarily interested in an analytical solution, I would already be happy about an existence proof.

A numerical solution is not what I am looking for but it may be useful to get an idea about how the solution family could look like.

I would appreciate any hints or references.