Determine all functions $\Bbb R _{\ge 0} \rightarrow \Bbb R_{\ge 0}$ such that $f(x)+f(y)+2xy=f(x+y) $

Question

Determine all functions $\Bbb R _{\ge 0} \rightarrow \Bbb R_{\ge 0}$ such that $f(x)+f(y)+2xy=f(x+y) $

The only nice progress I could make was $f(x)=x^2$ is a solution and $f(0)=0 $

when we take $f(x)=x^2$ , we get $f(x)+f(y)+2xy=x^2+y^2+2xy =f(x+y) $

and for $(x,y)=(0,0)$, we get $2f(0)=f(0)$ $\implies f(0)=0$

I tried making substitution , but couldn't make a nice one , also I think Cauchy might help since we have the FE similar to $f(x)+f(y)=f(x+y) $

If possible, can someone post hints rather than solution, it helps me a lot.

Thanks in advance!

Without any additional assumptions we cannot say what $f$ is. If $f$ is continuous (or at least measurable) then additivity implies $g(t)=ct$ for some constant $c$. — Kavi Rama Murthy, Sep 06 '20 at 06:19

score 2 · Answer 1 · answered Sep 06 '20 at 05:58

2

Hint: If $f$ and $g$ are two solutions, what can you say about $f-g$?

answered Sep 06 '20 at 05:58

Misha Lavrov

What can you say? – nonuser Dec 05 '21 at 07:05
I'm not sure why you're asking me now what my hint meant a year ago. You can show that $f-g$ satisfies Cauchy's functional equation; therefore once one solution is found, you can find all others by adding a Cauchy solution. – Misha Lavrov Dec 05 '21 at 07:11
Because I didn't understand it. Thanks. – nonuser Dec 05 '21 at 07:55

score 1 · Answer 2 · answered Sep 06 '20 at 07:00

1

Hint. As you noted out that $f(x)=x^2$ satisfies the conditions, its natural to define the function $g(x):=f(x)-x^2$ or simply $f(x)=g(x)+x^2$.

Is this an additive Cauchy in $g$?

Can you show $g$ is bounded in a non-trivial interval?

Can you use the fact that $f$ is bounded?

Note that $g(x)> -10$ in the interval $[0,3]$ and so $g$ is linear but as $f(0)=0$ we get, $f(x)=kx+x^2$ for some $k$.

answered Sep 06 '20 at 07:00

Anand

I just had a tiny doubt , well it was stated in 4.1 ( evan's handout ) that "f is bounded (either above or below) in any nontrivial interval." here does any mean "any one non-trivial interval " or "every nontrivial interval " ? – Sunaina Pati Sep 06 '20 at 07:34
1

@Shubhangi It means in any one interval... For instance, see https://math.stackexchange.com/a/423494/687233 OR https://artofproblemsolving.com/community/c985173h1969998_cauchy_equations :P – Anand Sep 06 '20 at 08:15
1

Oh.. I got it ..Thanks :) Actually Evan's language was a bit confusing in that handout :D But yeah, now it's clear :) . Thank you so much , BTW is the MDP group active rn ? – Sunaina Pati Sep 06 '20 at 08:43
@Shubhangi Yeah yeah, its very active!! Not even a single message is posted there after the day your left :P – Anand Sep 06 '20 at 11:25
1

Aww... I see :rotfl: . Anyways Thanks, again for helping me :) – Sunaina Pati Sep 06 '20 at 12:13

2 Answers2