The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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3
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0answers
56 views

Could we compute $P(t^2)$?

Let $P$ be an operator such that $P(kx)=kP(x)$, $k \in \mathbb{C}$, $x$ is a variable, $P(xy)=P(xP(y))+P(P(x)y)-P(x)P(y)$, $x, y$ are variables. All variables commute. Let $P(t)=t$. Then ...
31
votes
1answer
4k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
2
votes
0answers
29 views

A question about scaling

One wants the function $\Delta ^2$ to be such that, $\Delta^2(k,\tau) = \Delta^2(\frac{k}{\lambda ^{\frac{4}{n+3}}}, \lambda \tau )$. Now from this how does this follow that, the following holds, ...
4
votes
2answers
231 views

Is a function $f$ satisfying $f(x+1) = f(x) + 1$ and $f(x^2) = (f(x))^2$ odd or even?

The problem statement, all variables and given/known data 1) $f(x+1)=f(x)+1$ 2) $f(x^2) =(f(x))^2$ Let a function $f \colon \mathbb{R} \to \mathbb{R}$ satisfy the above statements. Then prove ...
1
vote
2answers
72 views

Playing with a functional equation

I was playing with a functional equation and proved the result below: Let $f$ be such that $$f(f(z))=z$$ If $f^{-1}$ exists then $$f(z)=f^{-1}(z)$$ If $f'$ exists then as ...
3
votes
3answers
112 views

Given $f(x+y)=f(x)f(y), f'(0)=11,f(3)=3$, what is $f'(3)$?

The question is this: Given \begin{align} f(x+y)&=f(x)f(y)\\ f'(0)&=11\\ f(3)&=3 \end{align} What is $f'(3)$? And my solution: On differentiating the equation $f(x+y)=f(x)f(y)$ ...
7
votes
2answers
98 views

A functional equation with inequality

Find all (at least one) functions $f\colon \mathbb{R}\to \mathbb{R}$ (or show there is none), such that $$ f(x^3+x)≤x≤f(x^3)+f(x), \quad \text{for all $x\in \mathbb{R}$}. $$ This is a problem asked ...
9
votes
1answer
262 views

The value of the trilogarithm at $\frac{1}{2}$

From the functional equation $$\text{Li}_{3}(z) + \text{Li}_{3}(1-z)+ \text{Li}_{3} \Big( 1 - \frac{1}{z} \Big) = \zeta(3) + \frac{\ln^{3} (z)}{6}+ \frac{\pi^{2} \ln (z) }{6}- \frac{\ln^{2} (z) ...
6
votes
2answers
452 views

Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?

To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that ...
12
votes
1answer
399 views

Differentiable functions satisfying $f'(f(x))=f(f'(x))$

I am wondering whether or not there is a reasonable characterization of differentiable functions $f: \mathbb{R}\to \mathbb{R}$ such that $f'(f(x))=f(f'(x))$ for each $x\in\mathbb{R}$. (Or, if you like ...
10
votes
2answers
867 views

find functions f such that $f(f(x))=xf(x)+1$,

let $f:R\longrightarrow R$, and $f$ is continous,and such that $f(f(x))=xf(x)+1$, find all this $f$? follow is my some idea:(but I don't have solution) We have $f(f(0)) = 1$, so there is your $c = ...
66
votes
7answers
2k views

Does a non-trivial solution exist for $f'(x)=f(f(x))$?

Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind ...
-2
votes
2answers
140 views

Integral equation that's cant solve… Need a hand [closed]

Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations. $$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
3
votes
1answer
227 views

Find all polynomials $P(x)$ satisfying this functional equation

Find all polynomials $P(x)$ which have the property $$P[F(x)]=F[P(x)], \quad P(0) = 0$$ where $F(x)$ is a given function with the property $F(x)>x$ for all $x\geq 0$. This is an exercise ...
2
votes
2answers
181 views

Equation $f(x,y) f(y,z) = f(x,z)$

How to solve the functional equation $f(x,y) f(y,z) = f(x,z)$?
3
votes
1answer
127 views

Wealth indicator function for bidder agent logic

I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others). Given: Total number of competitors $n$ Amount of all ...
2
votes
1answer
93 views

Find all polynomial solutions of the functional equation given …

Let f be a polynomial a) $f(x)f(x+1)=f(x^2+x+1)$ b) $f(x)f(2x^2)=f(2x^3+x)$
3
votes
4answers
227 views

Prove that $f(f(x))=x$ has no roots … $f$ having a general form [duplicate]

This problem gave me some headache, especially because $f$ have its own general form : let $f(x) = ax^2 + bx + c$. Suppose that $f(x) = x$ has no real roots. Show that equation $f(f(x))=x$ has also ...
3
votes
3answers
211 views

If $f$ is even and $y'=f(y)$ then $y$ is odd

Let $f\in C^1(\mathbb{R}, \mathbb{R})$ be an even function. Consider the maximal solution $y\colon\left]\alpha ,\beta\right[\to \mathbb{R}$ of the IVP $$y'=f(y),\ y(0)=0$$ Prove that $y$ ...
1
vote
1answer
95 views

Second order nonlinear delay differential equation

I have to solve the following delay differential equation: $$\ddot{x}(t)+A\sin(\omega x(t-\tau))=0$$ Can someone give me a hint on how to solve this equation? Thanks
0
votes
0answers
84 views

Seeking a function which satisfies a given functional equation.

I wish to find a function $f: \mathbb{R} \rightarrow \mathbb{R}$ which satisfies: $f(u) \geq 0$ when $0 \leq u < 1$, $f(u)=0$ when $u<0$ or $u \geq 1$, $\int_0^1 f(u) du=1$, and ...
0
votes
1answer
323 views

Finding the Extremals of a Functional J.

The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by $$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$ I have found ...
1
vote
1answer
129 views

Cauchy’s functional equation for non-negative arguments

Function $f:[0,+\infty)\rightarrow\mathbb{R}$ satisfies $f(x+y)=f(x)+f(y)$ for every non-negative $x$ and $y$. It’s bounded from below with some non-positive constant $m$. Does it imply that $f$ has ...
6
votes
1answer
201 views

An elementary functional equation.

I am finding this functional equation from a past high school mathematics competition rather tricky. Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}^+$ such that: ...
1
vote
1answer
196 views

exponential additive functional equation

Let $S$ be a semigroup with no identity element and $m:S\to \Bbb C$ be given function($m\not\equiv 0$) satisfying the exponential functional equation $$ m(x+y)=m(x)m(y) $$ for all $x, y\in S$. Find ...
3
votes
1answer
105 views

Proving that a function is differentiable and equal to a constant value for all x

Let $f(x)$ denote a strictly positive continuous function defined on all real numbers with the property that $f(2012)=2012$ and $f(x)=f(x+f(x))$ for all $x$. Prove that $f(x)=2012$ for all $x$. I am ...
4
votes
0answers
89 views

$f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$

Consider the equation: $f(a+b,\lambda) = f (a,\lambda) \cdot f(b,\lambda)$, for $a \geq 0$ and $b \geq 0$. Is my understanding that this simple functional equation is important in analysis. Can ...
0
votes
2answers
89 views

Can D'Alembert's functional be derived from Cauchy functionas?

Is it possible to derive D'Alembert's functional equation from Cauchy's functional equations? If so, can somebody kindly point me to a reference? Edit: Can $f(x + y) + f(x − y) = 2f(x)f(y)$ be ...
2
votes
1answer
68 views

Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$

Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$. How to prove that: The equation above has a unique solution $U_n$ ...
4
votes
4answers
135 views

functional equation involving $ f(x/k) $

given the equation $$ 1= f(x)+f(x/2)+f(x/3)+f(x/4) $$ how could i solve it ?? or the most general equation $$ 1= f(x)+f(x/2)+f(x/3)+f(x/4)+....+f(x/N) $$ for a given 'N' number, where could i use ...
3
votes
0answers
103 views

Order of Recursion?

Define an extended algebraic function $f(a)$ as a function on $a$ that utilizes any combination of recursive extensions and inverses of sequentiation. Example: $a + 1$ , sequentiation. $a + a$, ...
5
votes
2answers
189 views

Prove that if a particular function is measurable, then its image is a rect line

I´m really stuck with this problem of my homework. I don´t have any idea, how to begin. Let $f$ be a function, $f:\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x+y)=f(x)+f(y)$ , $\forall ...
5
votes
1answer
97 views

Functional Equation help

Came across this problem a little while ago but can't seem to get beyond a certain point. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ such that $f(n+1)>f(n)$ and $$f(f(n))=3n$$ for all $n$. ...
8
votes
2answers
677 views

Do there exist functions satisfying $f(x+y)=f(x)+f(y)$ that aren't linear?

Do there exist functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y),$ but which aren't linear? I bet you they exist, but I can't think of any examples. Furthermore, what ...
3
votes
3answers
127 views

Functional equations with 3 variables

What are the general solutions of the functional equations? $$ f(x,y)+f(y,z)=\frac{1}{f(x,z)} $$ $$ f(x,y)f(y,z)f(x,z)=1 $$
0
votes
2answers
56 views

Some functional equations in two variables

I have two questions. i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation $$ |f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|} $$ has a solution ...
0
votes
2answers
220 views

linear functional-equation $ \,2f\left(x+1\right)=f\left(x\right)+f\left(2x\right) $

I'm looking for all functions : $ \ R\rightarrow R\ $ satisfying: $ \,2f\left(x+1\right)=f\left(x\right)+f\left(2x\right) $
1
vote
1answer
85 views

Solution to a functional equation

Let $n,i$ be positive integers and $C$ a strictly positive real value. Consider the equation for $f$ : $$1*\ln(f(n)) = C * \sum_{3<i* \ln(i) < \sqrt{n}} \left(\ln[f( i* \ln(i) )-1)] - \ln[(f( i* ...
6
votes
2answers
312 views

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.

Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$. Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
1
vote
1answer
594 views

Solution of Cauchy functional equation which has an antiderivative

Let $f\colon\mathbb R\to\mathbb R$ be a function such that $$f(x+y)=f(x)+f(y)$$ for any $x,y\in\mathbb R$ i.e., it fulfills Cauchy functional equation. Additionally, suppose that $F'=f$ for some ...
5
votes
2answers
194 views

Finding $f(x)$ of functional equation

I would appreciate if somebody could help me with the following problem: Q: Find all conti-function $f(x)~ (x>0)$ $$xf(x^2)=f(x)$$
4
votes
3answers
2k views

Functional equation $f(xy)=f(x)+f(y)$ and differentiability

I want to prove the following claim: If $f:(0,\infty)\to\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ differentiable on $x_0=1$, then $f$ differentiable for all $x_0>0$. Thank you.
6
votes
2answers
142 views

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .

Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that $$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$ I've tried subbing in heaps of values but I keep getting things like ...
10
votes
2answers
292 views

Factoring x + y

I'm trying to find functions $f(x)$ and $g(y)$ such that $$f(x)\cdot g(y) = x + y$$ I can't seem to find a single solution to this problem. Anything I try becomes of the form $f(x,y) \cdot g(y) or ...
2
votes
1answer
1k views

Functional equation $f(y/x)=xf(y)-yf(x)$

Are there any solutions to $f:\mathbb{R} \backslash \{0\} \rightarrow \mathbb{R}$ with $f(\frac{y}{x})=x \cdot f(y)-y \cdot f(x)$ other than $f(x)=0$ for all $x \in \mathbb{R}$? I already have found ...
11
votes
1answer
170 views

How find this function $f(x)$

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous such that $\dfrac{f(x)}{x}=f'\left(\dfrac{x}{2}\right)$. Find $f(x)$. (2):if $f(x)$ on $x\in[a,b] $ be continuous,find all $f(x)$? I think this is an ...
0
votes
1answer
41 views

Finding the function [duplicate]

I would appreciate if somebody could help me with the following problem: Q: $f(x):$ conti-function and $f(2x)-f(x)=x^3$ find $f(x)=?$
14
votes
2answers
1k views

Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?

If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
2
votes
2answers
75 views

Finding Value, Related To Functional Equation

$f(x)$ is continuous for $\forall x \in R$ and $f(2x)-f(x)=x^{3}$ (1) $f(x)+f(-x)$ is constant ? (2) $f(0)=0$ ? I don't know how to use the continuity. especially for $f(0)=0$ ?