Ykhjuy

Solve the following functional equation :
$f:Bbb Z rightarrow Bbb Z$, $f(f(x)+y)=x+f(y+2017)$

I have no prior experience with solving functional equation but still tried a bit. I set $x=y=0$ to get $f(f(0))=f(2017)$. Can we apply $f^{-1}$ both sides to get $f(0)=2017$? I am unable to carry this on.

You could take $y=0$ there and get $f(f(x))=x+f(2017)=x+n$ where $n$ is a constant. Thus $f(f(x))$ is a linear function. You can observe (guess?) what $f(x)$ could be like here. If it is so, then you have no problem taking $f^{-1}$ and you can easily find $n$.

The point is how to prove the nature of $f(x)$ if you know $f(f(x))$ is linear. This is an interesting problem. What is the square root (with integer values) of a linear function?

Now here I'm not sure $f(x)$ will end up being a simple function involving only arithmetic operations, or a function also involving the modulo.

Got it! I probably complicated it more than I had to so if anyone sees any way to simply this I would love to hear feedback.

To solve this, let $y=0$ so that
$f(f(x)) = x+f(2017)$.
Let $c=f(2017)$. Then for all $xinmathbb{Z}$,
$$f(f(x)) = x+c$$

Now plugging $x=y=0$ into the original equation we get
$f(f(0)) = f(2017)$. Taking $f$ of both sides yields
$f(f(f(0))) = f(f(2017))$ which is $f(0)+c=2017+c$ so
$f(0) = 2017$.

Now take $f$ of both sides of the original equation to get
$f(f(f(x)+y)) = f(x+f(y+2017))$
which is
$f(x)+y + c = f(x+f(y+2017))$
Setting $y=-2017$ gives
$$f(x)-2017 + c = f(x+f(0)) = f(x+2017)$$.

Now we return again to the original equation with $y=1$. This gives
$f(f(x)+1) = x+f(2018)$ which by the above formula is
$x+f(1)-2017 + c$. So
$$f(f(x)+1)-f(f(x)) = x+f(1)-2017+c - (x+c) = f(1)-2017$$

Now, as $f(f(x)) = x+c$, $f(f(x))$, and thus $f(x)$, can attain all values in $mathbb{Z}$. So for any $kinmathbb{Z}$, there is an $xinmathbb{Z}$ such that $f(x)=k$, and so the above formula becomes
$$f(k+1)-f(k) = f(1)-2017$$
for all $kinmathbb{Z}$. So for all $kinmathbb{Z}$,
$$f(k) = k+c_2$$
for some $c_2$. So the original equation becomes
$$x+y+2c_2=x+y+2017+c_2$$
so $c_2=2017$. Thus the only solution is
$$f(x) = x+2017$$

Let $n$ be an integer. Then at $f(x)=n$ and $y=0$, the equation $f(f(x)+y)=x+f(y+n)$ becomes $$f(n+y)=f(y+n)+x|_{f(x)=n}$$ so $f(0)=n$. Now at $f(x)=n+1$, we can derive similar relations, at $y=0$ and $y=-1$ respectively $$x|_{f(x)=n+1}=f(n+1)-f(n)=f(n)-f(n-1)$$ so $2f(n)=f(n-1)+f(n+1)$. By induction it can be proven that $2f(n)=f(n-x)+f(n+x)$ or $$2f(n)=f(x)+f(2n-x)$$ for any integer $x$. Assuming differentiability, $f'(x)-f'(2n-x)=0$ so $f$ is either a linear or periodic function. It cannot be periodic as $f(0)=n$ and $x|_{f(x)=0}=f(n)-f(0)$. Thus $f$ is a linear function and putting in $f(x)=ax+b$ gives $b=n$ and $a=1$ since $f(f(x))=x+f(n)$.