A famous functional equation


That is following problem: Find all functions f:\mathbb{R}\to\mathbb{R} such that
f(x^2 + y + f(y)) = (f(x))^2 + 2\cdot y\; \forall x,y\in\mathbb{R}.(*)
It is famous! Why? Because, it is from AMM(problem 10908, posted by Wu Wei Chao) and it is one in problems from Bulgarian TST 2003, Vietnam TST 2004 and Iran TST 2007. However, in Vietnam TST 2004 it has form:


Find all real values of a, for which there exists one and only one function f: \mathbb{R} \to \mathbb{R} and satisfying the equation
f(x^2 + y + f(y)) = (f(x))^2 + a\cdot y\;\forall x,y\in\mathbb{R}.(**)
We easy see that, if a is an answer then a=2. In fact,
If a= 0. We have at least two functions satisfying, therefore a= 0 not satisfy.
If a\not = 0. Suppose f is function satisfy, because \{f^{2}(x)+ay|x,y\in\mathbb{R}\} = \mathbb{R}(note that : a\not = 0) we have f is surjection (1).
By (1) exist b such that f(b) = 0, from (**) we have f(x^{2} + b) = f^{2}(x) + a b\forall x\in\mathbb{R} and  f(x^{2} - b) = f^{2}(x) - ab\forall x\in\mathbb{R} , therefore 2ab = f(x^{2} + b) - f(x^{2} - b)\forall x\in\mathbb{R} , choose x = 0 here we have b = 0(note that :f( - b) = 0). So f(x) = 0 iff x = 0 (2).
In (**) choose y = 0 we have f(x^{2}) = f^{2}(x)\forall x\in\mathbb{R} (3).
In (3) choose x = 1 and by (2) we have f(1) = 1 (4).
In (**) choose y = 1 we have f(x^{2} + 2) = f(x^{2}) + a\forall x\in\mathbb{R} (5). In (5) choose x = 0 then a= f(2). Therefore a^{2} = f^{2}(2) = f(4) = f(2) + a= 2a, so a= 2.
So, we’ll back to equation (*). These are solutions of this one:
1. Solution of harazi, a member of the site mathlinks.ro.
In (*) put y=-\dfrac{f^2(x)}{2} we have f\left(x^2-\dfrac{f^2(x)}{2} +f\left( -\dfrac{f^2(x)}{2}\right)\right)=0\forall x\in\mathbb{R}, from here and by above we have f\left( -\dfrac{f^2(x)}{2}\right )=

-x^2+\dfrac{f^2(x)}{2}\forall x\in\mathbb{R} (1)
From (1), in (*) Put y by -\dfrac{f^2(y)}{2} we have f(x^2-y^2)=f^2(x)-f^2(y)\forall x,y\in\mathbb{R}
From here we have f(x+y)=f(x)+f(y)\; \forall x,y\in\mathbb{R} (2)
Now, by f^2(x)=f(x^2)\forall x\in\mathbb{R} and (2) we see that f^2(x+y)=f((x+y)^2)\forall x,y\in\mathbb{R}

\Rightarrow (f(x)+f(y))^2=f(x^2+2xy+y^2)\forall x,y\in\mathbb{R}

\Rightarrow f^2(x)+2f(x)f(y)+f^2(y)=f(x^2)+2f(xy)+f(y^2)\forall x,y\in\mathbb{R}

\Rightarrow f(xy)=f(x)f(y)\forall x,y\in\mathbb{R} \; (3)
Final, by (2),(3) and Proposition 2.7 ([1]) we have f(x)=x\forall x\in\mathbb{R} or f(x)=0\forall x\in\mathbb{R}. Checking condition (*) we obtain

f(x)=x\forall x\in\mathbb{R}.

2. Solution of N.T.TUAN, a member of the site mathlinks.ro.
In (*) choose x=0 we have f(x+f(x))=2x\forall x\in\mathbb{R}\; (1) .
By above: |f(x)|=|f(-x)|\forall x\in\mathbb{R}, if b=f(a)=f(-a)(a>0) then b>0 (because f(x)>0\forall x>0). Setting y=a and y=-a in (*) we have f(x^2+a+b)=f^2(x)+2a

\forall x\in\mathbb{R}\Rightarrow f(a+b)=2a

f(x^2-a+b)=f^2(x)-2a\forall x\in\mathbb{R}

\Rightarrow f(2b)=0(\text{by above})
Therefore 2b=0\Rightarrow b=0, contradiction! So, f(x)=-f(-x)\forall x\in\mathbb{R}, form that, see

f(x)>0\forall x>0(2) and f(x)<0\forall x<0(3).
Now, for all y<0, in (*) setting x=\sqrt{-y} we have f(f(y))+f(y)-2y=0\forall y<0 (4)
Final,from (3),(4) and idea of Example 1.10 ([1]) we see that f(x)=x\forall x<0, but

f(x)=-f(-x)\forall x\in\mathbb{R}, we have
f(x)=x\forall x\in\mathbb{R}.

Link download http://tuan.nguyentrung.googlepages.com/A_famous_functional_equation.pdf
Bibliography
[1] Functional equations and How to solve them, Christopher G.Small.
[2] http://www.mathlinks.ro/viewtopic.php?p=18289

1 thought on “A famous functional equation”

  1. tai sao ơ dòng thư 8 trên xuông lai có f(b)=0 thoa man ca 2 pt
    f(x^2+b)= (f(x))^2 +b và …(như trên)
    Em nghĩ nó chi thoa man đươc môt pt thôi chư nhi. Tai sao pt dâu trư lai co đươc.

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