Problem 4

Let

c>0

be a given positive real and

\mathbb{R}_{>0}

be the set of all positive reals. Find all functions

f:\mathbb{R}_{>0}\to\mathbb{R}_{>0}

such that

f((c+1)x+f(y))=f(x+2y)+2cx\quad\text{for all }x,y\in\mathbb{R}_{>0}.

Step 2 of 3: Rewrite in terms of the non-negative excess g(x)=f(x)-2x

Previous step Next step

g(x)=f(x)-2x\ge0\ \Longrightarrow\ g((c+1)x+2y+g(y))+2g(y)=g(x+2y)

Detailed analysis

Set $g(x)=f(x)-2x\ge0$ . Substituting $f(t)=g(t)+2t$ into the functional equation and cancelling $2(c+1)x+4y$ from both sides yields $g((c+1)x+2y+g(y))+2g(y)=g(x+2y)$ for all $x,y>0$ .