# 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.

Hint:

Let $$g(x)=\dfrac{x}{2}-f(x)$$ ,

Then $$f(x)=\dfrac{x}{2}-g(x)$$

$$f\left(\dfrac{x}{2}-f(x)\right)=\dfrac{\dfrac{x}{2}-f(x)}{2}-g\left(\dfrac{x}{2}-f(x)\right)=\dfrac{g(x)}{2}-g(g(x))$$

$$f'\left(\dfrac{x}{2}-f(x)\right)=\dfrac{g'(x)}{2}-g'(x)g'(g(x))$$

$$\therefore2g(x)-4g(g(x))+4\epsilon\left(\dfrac{x}{2}-g(x)\right)\left(\dfrac{g'(x)}{2}-g'(x)g'(g(x))\right)=\dfrac{x}{2}-g(x)$$

• I do not understand the fourth line of your hint. Shouldn't it be $f'(x)=(1/2)-g'(x)$? But more importantly, I do not see how your hint helps me to answer my question. – Gerhard S. Jul 12 at 9:26