Find all surjective functions $f(x+f(y))+f(y+f(x))=f(f(2x)+2y)$

Question

Find all surjective functions: $f:R_0^+→R_0^+$ such that: $$f(x+f(y))+f(y+f(x))=f(f(2x)+2y)$$

Note that $0$ is not in the domain and codomain.

This is what i found: $2f(x+f(x))=f(f(2x)+2x)$ And for some $a\in{\mathbb{R}}$ such that $f(a)=1$ : $f(x+1)+f(a+f(x))=f(f(2x)+2a)$ is that good? I don't know how to continue. Also: i noticed that $f(x)=x$ satisfy the conditions.

Answer 1

Idea:

Since swapping $x$ and $y$ we get the same expression on LHS we see that $$f(f(2x)+2y)= f(f(2y)+2x)$$ so, if we prove $f$ is injective we get $$f(2x)-2x= f(2y)-2y$$ which implies $f(x)= x$ for all $x$.

So we are left to prove $f$ is injective...