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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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Does the functional square root of the sine adhere to a "Sum = Integral" identity?

Background There are a number of functions that are subject to "Sum = Integral" identities. Some of them are listed in this question. For instance, when we set $ {\rm sinc}\, x := \sin(x)/x \...
Max Muller's user avatar
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4 votes
1 answer
112 views

Solutions to $(f(x)-f(y))^3=f\left(x^3\right)-f\left(y^3\right)$

I was wondering, if there are more solutions to the functional equations, than $f(x) = const$. Maybe someone has an idea of how to find all solutions (or all continuous solutions)? Find all the ...
Vlad Boiko's user avatar
0 votes
0 answers
9 views

Functional equation proof [duplicate]

I'm stuck with this problem: Knowing that for $x,y\in\mathbb{R}$ $$f(xy+x+y)=f(xy)+f(x)+f(y)$$ prove that $$f(x+y)=f(x)+f(y)$$ What I got so far: f(0)=0 (if x=y=0 then $f(0)=3f(0)$). I tried letting $...
Anon's user avatar
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1 vote
1 answer
60 views

General form of Jacobi Theta Transformation $\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} e^{n^2 \pi x} $

I was looking into the functional equation of $\zeta(s)$ and at one point the proof uses the Jacobi Theta Transformation: $$\sum_{n \in \mathbb{Z}} e^{n^2 \pi / x} = \sqrt{x} \sum_{n \in \mathbb{Z}} ...
Kashif's user avatar
  • 728
3 votes
1 answer
133 views

The Uniqueness of the Logarithm as a Group Isomorphism between the Positive Reals and Reals

My Group Theory textbook asks of me that I prove the following statement: If $\varphi: \mathbb{R}^{+} \rightarrow \mathbb{R}$ be any isomorphism between the groups $(\mathbb{R}^{+}, \cdot)$, $(\...
Barbatulka's user avatar
2 votes
3 answers
105 views

The functional equation $f(x+1) = \dfrac{z x + f(2x)}{x + 1}$

Let $x$ be real. Consider this equation. $f(\exp(x) +1) = \dfrac{f(2) \exp(x) + f(2 \exp(x))}{\exp(x) + 1}$ Apart from the trivial solutions $f(\exp(x)) = C$ for a constant $C$ independant of $x$, are ...
mick's user avatar
  • 16.4k
1 vote
0 answers
66 views

Solving $f(x) = f(\frac{a + b x}{c + d x}) = f(\frac{a' + b' x}{c' + d' x})$?

How to solve the equation $$f(x) = f(\frac{a x + b}{c x + d}) = f(\frac{a'x + b'}{c'x + d'})$$ For given real $a,a',b,b',c,c',d,d'$ ? Maybe this system of equations is a bit overdetermined in its ...
mick's user avatar
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0 votes
0 answers
46 views

How to find period of an arbitrary periodic function?

Is there any known algorithm to find the period of a generic function which is known to be periodic? The most direct approach would be to solve for $T$ the functional equation $$f(x+T)=f(x)$$ which is ...
Sanjana's user avatar
  • 265
5 votes
1 answer
133 views

Solving the functional equation $f(x)=\frac4{(x+4)^2}f(\frac x{x+4})+2f(2x+1)$

Consider the functional equation $$f(x)=\frac4{(x+4)^2}\,f\!\left(\frac x{x+4}\right)+2f(2x+1),\quad x\geq0.$$ I believe that there should be a unique non-identically-zero solution up to scaling. ...
LegionMammal978's user avatar
2 votes
0 answers
35 views

About a property of solutions to the delay equation $\frac{dx}{dt} = Ax(t) + F(t,\,x_t)$ in a Banach space $E$

I'm dealing with the equation \begin{equation*} \tag{1} \frac{dx}{dt} = Ax(t) + F(t, \, x_t),\end{equation*} in which $A$ is the generator of a $C_0$-semigroup $(T(t))_{t \, \geqslant \, 0}$ on a ...
user405919's user avatar
1 vote
1 answer
30 views

Lipschitz Continuity of inf mappings

For simplicity, let $\mathcal{B} = \mathcal{C}[[-\tau, 0], \mathbb{R}]$ for some $\tau \geq 0$. I am working with time delay differential equations of the form $$ \dot{x}(t) = F(x_t) = \inf_{\theta \...
Olayo's user avatar
  • 87
5 votes
0 answers
75 views

Finding twice-differentiable $f$ such that $f(x) = x^{-a} - 1 + f(1) + f'(1)(x-1)$

$a$ is a constant, $x \in (0,\infty)$. I encountered this differential equation playing around with some material, unsure if it hides something interesting. In fact, plugging $f(1)$ returns an ...
Lele's user avatar
  • 51
0 votes
1 answer
67 views

Solving for $f$ when $f(x+y)=f(x)+f(y)+axy$ where $a$ is a real number

This question has been asked before here but I have followed a different approach We have: $$f(0) = 0$$ $$f(x) = f(\frac{x}{2}+\frac{x}{2}) = 2f(\frac{x}{2})+a\frac{x^2}{4}$$ $$f(\frac{x}{2}) = 2f(\...
s_a94248's user avatar
3 votes
0 answers
76 views

A very interesting function equation $f(a,b)=f(a,c)+f(c,b)$ implies $f(a,b)=g(a)-g(b)$.

Function equation: $f(a,b)=f(a,c)+f(c,b)$ for all positive reals $a>c>b\geq 0$. My solution: $f(a,c)=f(a,b)-f(c,b)$, Let $b=0$, $f(a,c)=f(a,0)-f(c,0)$. Define $g(a)=f(a,0)$. We get the answer. ...
dodo's user avatar
  • 828
5 votes
4 answers
212 views

Functional equation with $f(x+y)=f(x)f(1-y)+f(1-x)f(y)$

Let $f:\Bbb R\to\Bbb R$ such that $f(x+y)=f(x)f(1-y)+f(1-x)f(y)$, $f(x)$ is strictly increasing on $[0,1]$. Find the solution set to $f(x)\ge\frac12$. This is a problem form a mock test with no ...
youthdoo's user avatar
  • 1,475
1 vote
1 answer
48 views

Solutions to functional equation $\phi(x)f(x) = af(bx)$ for a given function $\phi(x)$.

For a given function $\phi(x)$, can we always find a function $f(x)$ satisfying \begin{align} \phi(x)f(x) = af(bx), \end{align} for some $a$ and $b$? Some examples of functions $\phi(x)$ where I ...
Lyle's user avatar
  • 138
2 votes
2 answers
95 views

Let $f \colon \Bbb{N} \to \Bbb{N}$ such that $f^{f(m)}(n) = n + f(m)$. Prove that $f(0) = 1$.

I am trying to prove that the unique solution to the functional equation $$f^{f(m)}(n) = n + f(m)$$ (where the exponent indicates composition) is $f(n) = n + 1$. To do so, the last step left in my ...
user avatar
0 votes
2 answers
115 views

What did I miss while solving $f(x+f(x)) = x+f(x)$?

What is the number of linear functions $f$ satisfying $$f(x+f(x)) = x+f(x)$$ $\forall x \in R$ ? I began by setting $g(x) = x+f(x)$. Then I applied $f$ on both sides, which gave me $$f \circ g(x) = f(...
AryanSonwatikar's user avatar
1 vote
0 answers
28 views

Functional derivative of a function with nested integrals

I'm trying to solve a calculus of variations problem to find the cross-sectional area of a bar as a function of its length, which minimises its volume but has some fixed displacement at the free end. ...
Ben's user avatar
  • 53
1 vote
0 answers
35 views

A functional equation for a prime divisor finding algorithm's complexity [closed]

I wanted to estimate the computational complexity of a basic algorithm to find all prime divisors of a given number $N$. We'll look for potential divisors for up to $\sqrt{N}$ which would be about $O(\...
Loading - 146 Complete's user avatar
1 vote
0 answers
89 views

functional equation on the set of real numbers

I have a functional equation as follows find $f:R\to R$ satisfy: $f(x+f(xy))=f(x)+xf(y)$ for all real numbers x,y I don't see the feasibility to prove this is an injective function or this is a ...
bestty's user avatar
  • 147
1 vote
0 answers
81 views

$n(fx)=f(nx)$ for natural number $n$ and $f(x)$ is an increasing real function. So $f$ is linear?

It is known that, for real increasing function $f:[0,1]\to\mathbb R$, if $f(ax)=af(x)$, then $f$ is linear. Now consider $a=n$ where $n$ is natural. Is it possible that $f(x)$ is not linear on a ...
dodo's user avatar
  • 828
6 votes
3 answers
177 views

Proving $f(x-y)+f(\frac{2xy+y^2}{x+y})+f(-\frac{2xy+x^2}{x+y})=2f(\frac{x^2+xy+y^2}{x+y})∀(x,y)\in\mathbb{C}^2 \iff f(x)\in${$ax^2+bx^4:(a,b)\inℂ^2$}

For some convenient help analyzing the problem from the partially helpful point of view of real numbers, I've made a model in desmos, which also serves to bolster confidence that what we seek to prove ...
Simon M's user avatar
  • 887
1 vote
1 answer
53 views

Decomposition of a function

Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function with compact support on $\mathbb{R}$. Under which hypotheses on $f$ the functional equation (in the $w$ variable) $$ w(2t)-w(t)=f(t) $$ admits at ...
Kosh's user avatar
  • 1,510
7 votes
3 answers
250 views

Which holomorphic maps $f:\mathbb{H}\to\mathbb{H}$ satisfy $f(z+1)=f(z)-1$?

Let $\mathbb{H}$ denote the upper half-plane. Which holomorphic maps $f:\mathbb{H}\to\mathbb{H}$ satisfy $f(z+1)=f(z)-1$ for all $z\in\mathbb{H}$? My guess is that none exist. I think there might be ...
4plus4man's user avatar
  • 377
17 votes
3 answers
374 views

$f(a)+b$ is a perfect square iff $f(b)+a$ is a perfect square too

Let $f:\mathbb{N}\to\mathbb{N}$ be a function such that $f(a)+b$ is a square iff $f(b)+a$ is also a square. Show that $f$ is injective. Note that, in this context, $0\notin\mathbb{N}.$ Clearly, $f$ ...
aqualubix's user avatar
  • 2,145
1 vote
0 answers
38 views

find $g(x,n)$ , where $g(x,n)$is a compostion of a function $f(x)$ $n$ times.

let $g(x,n)$ be when you have a compostion of a function $f(x)$ $n$ times. The method I used to solve simple ones is by using sequences, for example. $a_{n+1}=ra_{n}+b$ $a_{n+2}=r^2a_{n}+br+b$ $a_{n+3}...
abdul muhaimin's user avatar
8 votes
4 answers
367 views

Functional equation arising in computing an integral

Let $f(x)$ be a twice differentiable function (with a continuous second derivative) satisfying the identity: $$f \left(\frac{x}{2} \right)+f \left(\pi-\frac{x}{2} \right)=\frac{f(x)}{2}$$ Determine $f(...
Cognoscenti's user avatar
1 vote
2 answers
86 views

Don't understand the solution

Question from chapter 1 of Putnam and Beyond Show that there does not exist a strictly increasing function $f :\mathbb N \to\mathbb N$ satisfying $f (2) = 3$ and $f (mn) = f (m) f (n)$ for all $m, n \...
Jason's user avatar
  • 13
2 votes
5 answers
100 views

Can a function $f$ exist such that $f(x)f(y) = xy + 1$ for positive real $x$ and $y$

Problem. Can a function $f$ exist such that $f(x)f(y) = xy + 1$ for positive real $x$ and $y$ If not, is there something like a next best thing, perhaps only for integers, finite field, or maybe only ...
Zero's user avatar
  • 33
-1 votes
1 answer
39 views

calculate the terms of a function using base 2, number theory

I have the following problem from the book Teoria dos Numeros:um passeio com primos e outros numeros familiares pelo mundo inteiro. Let $f : \mathbb{N} >0 → \mathbb{N}$ be a defined function of the ...
amkpm90's user avatar
  • 352
2 votes
1 answer
94 views

Solutions of the functional equation $f(x) = f(x/2) + 1$

Is there a nice characterization of all functions $f : (0, +\infty) \to \mathbb{R}$ such that $f(x) = f(x/2) + 1$? Obviously, there are many solutions involving the base-2 logarithm. For example, any $...
Electro's user avatar
  • 365
1 vote
0 answers
44 views

Problem 1.131 Robert Megginson [duplicate]

Let $E$ be a normed vector space of infinite dimension. If $U \subset E^*$ is a nonempty set which is open with respect to the weak-* topology, how can we show that it is not strongly bounded? I have ...
Jason Jacob's user avatar
2 votes
1 answer
94 views

$\Pi_n \circ T$ is continuous and T linear, them T is continuous.

For each $n \in \mathbb{N}$, let $\Pi_n: \ell^1 \rightarrow \mathbb{R}$ be the functional defined by: $$\Pi_n(\{x_i\})=x_n \quad \forall\{x_i\} \in \ell^1$$ and let $E$ be a Banach space. Prove that ...
Mark Carpio's user avatar
0 votes
1 answer
112 views

Isomorphism Reflexive Spaces [closed]

This afternoon I have been trying to try this exercise from a Book of Functional Analysis, but I have not obtained any results. I quote the exercise. Let $E$ and $F$ be Banach spaces. a) Prove that if ...
Mark Carpio's user avatar
0 votes
1 answer
47 views

When does a sum of an odd and even function gives us either an odd or an even function?

Greetings, I'm currently trying to find a way to prove that this functional equation has no solution: $$f(x+yf(x))+f(xf(y)−y)=f(x)−f(y)+2xy^2$$ I know that an eventual solution has to be odd: to do so,...
Nerincet Vonwthaud's user avatar
2 votes
3 answers
159 views

The functional equation $f(x^2+f(y))+f(y^2)=2x^2y$

I have a functional equation problem: $f(x^2+f(y))+f(y^2)=2x^2y$ I have attempted many things but here was the furthest I have gotten: Substitute $x=0$: $f(f(y))+f(y^2)=0$ Substitute $y=0$: $f(x^2+f(0)...
Slare's user avatar
  • 56
2 votes
1 answer
99 views

Extremal of a functional with one variable endpoint

Consider the functional $$J[y] = \int \limits_{0}^{b} {\frac{\sqrt{y'^2 + 1}}{y} \text{d}x}$$ with $y(0) = 0$ and the other endpoint somewhere along the circle $(x-9)^2 + y^2 - 9 = 0$ (call this ...
JOlv's user avatar
  • 99
0 votes
0 answers
139 views

Solutions of $f^2(x)-f^2(y) = f(x-y)f(x+y)$ [duplicate]

I am searching for all functional solutions of equation $$(f(x))^2-(f(y))^2 = (f(x)-f(y))(f(x)+f(y)) =f(x-y)f(x+y),\quad x,y \in \mathbb{R}.$$ A few properties can be derived from the equation: If $...
Sam's user avatar
  • 3,360
2 votes
1 answer
105 views

Functional Equation $f(xy+f^2(y))=f(x)f(y)+yf(y)$

How do you solve this problem? I've only managed to prove that $f$ is injective, but that's not enough: Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ for which this equality holds: $f(xy+f^2(...
TheDoubleO7's user avatar
2 votes
2 answers
82 views

A neat functional equation $f(x)=f(x+c)-f'(x+c)$

While investigating I thought of this functional equation $$ f(x)=f(x+c)-f'(x+c) $$ and I wondered if there are any real analytic non-constant solutions. Here $c$ is some constant. Are there any real ...
zeta space's user avatar
1 vote
1 answer
65 views

Finding the extremal of a non-linear, second order functional.

The question reads as follows: Consider the functional $$J[y] = \int \limits_{0}^{1} {(y'' ^2 - 2k y)} \text{d}x$$ where $k$ is a constant. Find the extremal of $J$ satisfying the conditions $y(0) = y'...
JOlv's user avatar
  • 99
2 votes
0 answers
67 views

How to solve this functional equation in a nice way? [duplicate]

I have an equation: $$\frac{h(e^x)}{h(x)} = e^x$$ I want to find the nicest way of solving this for $h$. Preferably an analytic solution. The best thing I was able to find yet is this (sorry for image,...
Усердный бобёр's user avatar
1 vote
2 answers
56 views

Determine $f:\mathbb R \to \mathbb R$ s.t. $f(x+y) = 2^xf(y)+2^yf(x)$

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that f(1) = 1 and satisfies $$f(x+y) = 2^xf(y)+2^yf(x)\,\,\,\forall x \in R,\,\,\forall y \in R.$$ Determine $$1.)\lim_{x \to 0} \frac{f(x)...
Ark's user avatar
  • 139
0 votes
1 answer
37 views

Skeptical Proof of Functional Equations Question

Question: The function $f$ assigns to each non-negative integer $n$, the non-negative integer $f(n)$, such that: $f(mn) = f(n) + f(m)$ for $m,n \geq 0$ $f(n) = 0$ if the unit digit of $n$ is 3 $f(10) =...
Ayush Maurya's user avatar
4 votes
0 answers
119 views

Functional equation $f(x+2024)=\frac{1+f(x)}{1-f(x)}$

Find all continuous functions on $\mathbb{R}$ satisfying the equality$$f(x+2024)=\frac{1+f(x)}{1-f(x)} \forall x\in \mathbb{R}.$$ I have an idea for a solution, but I'm not sure if it's correct $$f(x)...
didenko jack's user avatar
0 votes
0 answers
35 views

Help with a functional equation: $f\left(f(x)^2 + f(y)\right)= xf(x) + y$ [duplicate]

This is a question posed to me by a friend two days ago. It goes as follows: Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that $f\left(f(x)^2 + f(y)\right)= xf(x) + y$ for all $x, y \in \...
xoxo's user avatar
  • 119
1 vote
1 answer
101 views

Are there other solutions to the functional equation $f(x^t) = t f(x)$ besides logarithms?

Are there other solutions to the functional equation $f(x^t) = t f(x)$ besides logarithms? Here $x$ and $t$ are real variables with $x>0$. I know that given the property of logarithms $\log(x^t) = ...
Joako's user avatar
  • 1,634
2 votes
0 answers
124 views

Is there a sequence of non-trivial continuous functions $f_{n+1}(f_{n+1}(x))=f_{n}(x)$, $f_2(x)= \frac{1-x}{1+x}$?

Just for curiosity I was trying to find a sequence of non-trivial continuous functions at $\mathbb{R}$ except finite many points such that $f_{n+1}(f_{n+1}(x))=f_{n}(x)$, $f_1(x)=x$ and by non trivial ...
pie's user avatar
  • 6,581
0 votes
0 answers
90 views

What are the roles of $x$ and $y$ in the functional equation: $f(x+y)=f(x)+f(y)$ for all $x,y$ in R?

I am slightly confused by the roles of input variables for certain functional equations. Kinda new to this study of functions. Knowing the role of variables in standard equations such as $ax+by-c=0$, ...
webtolight's user avatar

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