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# Questions tagged [functional-equations]

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation and possibly other conditions. Solving the equation means finding all functions satisfying the equation. For basic questions about functions use more suitable tags like (functions) or (elementary-set-theory).

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### How to extract associative binary operations from a class of binary operations on $\mathbb{R}$

I would like to find (all) associative binary operations of the form $$u_{1}*u_{2}=\ln{\left[e^{u_{1}}+e^{u_{2}}\right]}+Q\left(u_{2}-u_{1}\right),$$ where $Q$ is an arbitrary function. My effort: ...
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### Prove that $\forall x \in \mathbb{R}, f(x)=0$ [duplicate]

Suppose $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable function and satisfies $$\forall x \in \mathbb{R}, \vert f'(x)\vert \leq \vert f(x)\vert, \quad f(0)=0.$$ Prove or disprove $f(x)=0$ How ...
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### Romanian Master of Mathematics 2019 Day 2 Prob. 5. (Still lack of solution 2019 Feb. 28th)

I found this problem is interesting, but I do not know how to do it. I want to know in general, how can one deal with such a functional problem. Are there any recommend books, lecture notes and etc. ...
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### Find all function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$

If $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and $a,b,c>0$, then find all function such that : $$f(ax)f(by)=f(ax+by)+cxy,\quad \text{where } a,b,c>0 \text{ for all } x,y\in \Bbb{R}.$$ My attempt ...
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Given $a<0$ such that $k:=2^{a}+3^{a}\in (0,1)$ and define $f(x):=x^{a}$ for all $x>0$. Then, $f(2x)+f(3x)=x^{a}(2^{a}+3^{a})=kf(x)$. Somebody know another example of a function $f:(0,+\infty)\... 0answers 33 views ### Additive and Bijective function on the real line I was trying to solve the following functional equation:$f(f(x-y))=f(x)-f(y)$. And I concluded that$f$must be additive and bijective. The question is: Let$f:\mathbb{R} \to \mathbb{R}$be an ... 2answers 466 views ###$2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if$f(2)=5$find$f(5)$This question has been asked before, however I am interested in seeing why my approach to finding a solution does not work.$2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if$f(2)=5$find$f(5)$What I have ... 1answer 82 views ### If$f(x)f(y)+f(xy)\le -\frac{1}{4},\forall x,y\in[0,1)$, show that$f(x)=-\frac{1}{2}$Let$f:[0,1) \to \mathbb{R}$be a function such that $$f(x)f(y)+f(xy)\le -\dfrac{1}{4} \quad \forall\, x,y\in[0,1).$$ Show that $$f(x)=-\dfrac{1}{2} \quad \forall\, x \in[0,1).$$ I have proved that ... 2answers 61 views ### Find$x$where$f(e^{(x+1)})=x-\ln(x)$approaches one. Given that $$x \in [1,\infty) \quad f(e^{(x+1)})=x-\ln(x)$$ and $$\lim_{x \to a} f(x)=1$$ Find$a$. I got to the point: $$\ln(a)-\ln(\ln(a)-1)=2$$ But from there on I could not get to$a=e^...
I am trying to find all functions $$f : [0, 1] \rightarrow [0, 1]$$ continuous monotone strictly increasing such that $$f(v) = v - \frac{v}{2} \cdot f^{-1}(\frac{v}{2}).$$ However I have no idea how ...