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 ...
3
votes
1answer
76 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 ...
1
vote
1answer
40 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
82 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
2answers
58 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
41 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
44 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
37 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
43 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
94 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
47 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
52 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
50 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
1answer
22 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 ...
1
vote
0answers
51 views
2
votes
1answer
62 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
94 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 ...
0
votes
0answers
20 views
Formulating math problem with rounding / discrete step
I have this problem, that can easily be solved by simulation or numerical optimization, but I wonder how to write it as a mathematical problem? It's two pricing schemes, one cost is evaluated at ...
3
votes
0answers
37 views
Order of Recursion?
Define an extended algebraic function f(a) as a function on a that utilizes any combination of recursive extensions and inversions of sequentiation. Example:
a + 1 = sequentiation. a + a = addition ...
5
votes
1answer
72 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 $n$.
...
4
votes
2answers
92 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
67 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
36 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
70 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
64 views
Solution to a functional equation
Let $n,i$ be positive integers and $C$ a strictly positive real value.
Consider the equation for $f$ :
$$1*\ln(f(n)) = C * \sum_{3<i* \ln(i) < \sqrt{n}} \left(\ln[f( i* \ln(i) )-1)] - \ln[(f( i* ...
6
votes
2answers
116 views
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
1
vote
1answer
71 views
Solution of Cauchy functional equation which has an antiderivative
Let $f\colon\mathbb R\to\mathbb R$ be a function such that
$$f(x+y)=f(x)+f(y)$$
for any $x,y\in\mathbb R$
i.e., it fulfills Cauchy functional equation.
Additionally, suppose that $F'=f$ for some ...
5
votes
2answers
150 views
Finding $f(x)$ of functional equation
I would appreciate if somebody could help me with the following problem:
Q: Find all conti-function $f(x)~ (x>0)$
$$xf(x^2)=f(x)$$
2
votes
3answers
77 views
Functional equation $f(xy)=f(x)+f(y)$
I want to prove the following claim:
If $f:(0,\infty)\to\mathbb{R}$ satisfying $f(xy)=f(x)+f(y)$, and if $f$ differentiable on $x_0=1$, then $f$ differentiable for all $x_0>0$.
Thank you.
5
votes
2answers
64 views
Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .
Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that
$$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$
I've tried subbing in heaps of values but I keep getting things like ...
8
votes
2answers
211 views
Factoring x + y
I'm trying to find functions $f(x)$ and $g(y)$ such that $$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
86 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
143 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
33 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)=?$
11
votes
2answers
351 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?
1
vote
2answers
51 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$ ?
5
votes
2answers
134 views
$f: \Bbb N→ \Bbb N$ , $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$
How to find all functions $f: \Bbb N→ \Bbb N$ which satisfy $f\big(f(m) +f(n)\big)=m+n$ $\space$ for all $m,n∈\Bbb N$
($\Bbb N$ is the set of all natural numbers, i.e. positive integers) ?
0
votes
1answer
26 views
Cauchy's Problem
I am looking at the Cauchy's functional equation here: http://en.wikipedia.org/wiki/Cauchy's_functional_equation.
Could someone help me on how to generalize the Cauchy's equation to $x \in ...
11
votes
4answers
227 views
If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x)\neq x$ for all $x$, must it be true that $f(f(x))\neq x$ for all $x$?
Let $f: \Bbb R → \Bbb R$ be a continuous function such that $f(x)=x$ has no real solution . Then is it true that $f(f(x))=x$ also has no real solution ?
1
vote
1answer
40 views
Cauchy functional equation with non choice
Assume ZF+ not AC. Then how many solutions are there for Cauchy functional equation?
Thank you
4
votes
2answers
87 views
Functional Equation $f(x+y)=f(x)+f(y)+f(x)f(y)$
I need to find all the continuous functions from $\mathbb R\rightarrow \mathbb R$ such that $f(x+y)=f(x)+f(y)+f(x)f(y)$. I know, what I assume to be, the general way to attempt these problems, but I ...
13
votes
3answers
264 views
Solving Differential Functional Equation $f(2x)=2f(x)f'(x)$
Find all functions satisfying $f(2x)=2f'(x)f(x)$
Under given condition, can't we find explicit solutions?
2
votes
2answers
58 views
A non-zero function satisfying $g(x+y)=g(x)g(y)$ must be exponential function
Let $g$ be a non-zero function satisfying $g(x+y)=g(x)g(y)$. Show that the function must be exponential function.
4
votes
1answer
89 views
Functional equation $(1-z)f(x)=f(\frac{1-z}{z}f(xz))$
May you could help me with the following functional equation: $(1-z)f(x)=f(\frac{1-z}{z}f(xz))$. I want to find all function $f:[0,\infty)\rightarrow[0,\infty)$ for $x>0,0<z<1$
Its an ...
7
votes
1answer
105 views
Find all differentiable functions $f$ such that $f\circ f=f$
I want to find all differentiable functions $f:\mathbb R \to \mathbb R$ such that $f\circ f=f$,
My attempt since $f$ is differentiable, $f'(f(x))f'(x)=f'(x)$ Now if $f'(x)\neq0$($f'=0$ means ...
1
vote
2answers
44 views
help with my hw its a quadratic equation [closed]
Write the quadratic equation using the following factor
$(7x-3) (-2x+1)$
$(R+9) (R-9)$
Factor: $x^2+6x+9$
Please help the hw is due tommorow
0
votes
0answers
30 views
What is the complete general solution to $s(z)-s(z-1)=f(z)$?
$$
\displaystyle s(z)=\sum _{k=-\infty }^{\infty } c_h(k) e^{i 2 \pi k z}+\sum _{k=-\infty }^{\infty } c_p(k) \left(\zeta \left(-k,a_0-z_0\right)-\zeta \left(-k,z-z_0+1\right)\right)
$$
I developed ...
1
vote
0answers
77 views
Is there a function $f$ such that $\Gamma (c+x)=\Gamma (c-f\left( x \right) )$?
I was just looking at Euler's reflection formula for Gamma function which states $$\Gamma (1-z)\Gamma (z)=\frac { \pi }{ \sin { (z\pi ) } } $$ but it seems to me that one more reflection formula ...
4
votes
1answer
47 views
Follow on from previous question: Functional Equation - a little tricky
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f[f(x)^2+f(y)]=xf(x)+y$ for all real numbers $x$ and $y$.
The answer to this has already been posted, but it doesn't explain why ...
1
vote
1answer
65 views
Injectivity of a Function [closed]
Sorry for confusion.
I am in the process of solving a functional equation, I need to show injectivity. (By the way i know that it is injective, I'm trying to prove it to myself).
Putting $f(x)=f(y)$ ...
7
votes
1answer
123 views
How to find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$?
Can someone please show me how to:
Find all polynomials $P(x)$ such that $P(x^2+2x+3)=[P(x+3)]^2$?
I've tried substitiuting $x=0,1$. Can't seem to figure it out. The square on the RHS is confusing ...
