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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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19 views

Solving $\sum\limits_{k=1}^n e(x-x_k) = h(x)$ for $e(x)$, where $x_k$ and $h(x)$ are given

I would like to find the function $e(x)$ which solves $\sum\limits_{k=1}^n e(x-x_k) = h(x)$, where $x_k$ and $h(x)$ are given. There are no restrictions on any of the $x_k$ or $h(x)$ except that $h(x)$...
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0answers
14 views

A third way to solve linear functional equations inspired by lacunary series

Let $(f,\omega,H)$ be complex functions $ w \subseteq \mathbb{C} \rightarrow u \subseteq \mathbb{C}$ Then it's easy to see that a "formal" solution the following functional equation $$ f(\omega(x))...
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0answers
55 views

Formal group law and Koenigs function conjecture !?

Let $f(x,y)$ be a symmetric real function and a formal group law $$G(x + y) = f(G(x),G(y)). $$ Consider the equation $$ h(2x) = f(h(x),h(x)) = A(h(x)). $$ This equation has many solutions. ...
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2answers
52 views

How many polynomial functions exist such that $f(x^2) = (f(x))^2 = f(f(x))$ [closed]

How many polynomial functions $f$ of degree $\geq1$ satisfy $f(x^2) = (f(x))^2 = f(f(x))$ for all real $x$?
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0answers
34 views

How to construct a function with these hypotheses?

I want to construct a function $f:[0,1]×[0,1]\rightarrow [0,1]$ such that $f(0,t)=t$ $f(1,t)=2t-1$ $ \forall$ $ t\geq \frac{1}{2}$ (Edited) $f(s,t)=0$ $ \forall $ $0 \leq t \leq \frac{s}{2}$ (Edited) ...
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1answer
73 views

Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for every $x \in \mathbb{R}$, $f(2x) = 2f(x)$ .

Basically I thought about a kind of modulo 2 equivalence class for real numbers, if that makes sense. With that, and noting that for each number $y \in [1,2)$, the numbers $2y$ and $y/2 $ are not in $ ...
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1answer
23 views

Smallest set of reals $\{r_{i}\}$ that ensures an additive function $f$ is linear if $f(r_{i})=f(1)r_{i}$ for all $r_{i}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an additive function. We have that $f(n)=nf(1)$ for $n\in\mathbb{Z}$ by induction, and we can extend to the rationals by setting $nf\left(\cfrac{m}{n}x\right)...
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1answer
68 views

Functional equation $ (n+1) f(n+1)= (a n+b) f(n) $ for $n=0,1,…$

I am looking for a solution to the following functional equation: \begin{align} (n+1) f(n+1)= (a n+b) f(n), n=0,1,... \end{align} where $a$ and $b$ are some positive constants. Moreover, $f(n)$ is ...
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1answer
60 views

Jensen like: $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2} $for $x,y \in \mathbb{R}$ that $x-y \in \mathbb{Z}$

Solve functional equation, where $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfy $$f(a)=\frac{f(a-\frac{k}{2})+f(a+\frac{k}{2})}{2},$$ for $a\in \mathbb{R}$ and $k\in \mathbb{Z}.$ I obtained this ...
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0answers
44 views

Find $f(x)$ where $f(x)(A-\frac{B}{x+B/A})+Cf(x+\frac{B}{A})=0$.

$A, B, C > 0$, $x$ is complex and $Re(x)>0$. My guess is that $f(x)=0$ but I don't know how to prove it.
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0answers
66 views

A Golden Ratio Functional Equation Sequence

I was looking at the equation $f^{-1}(x)=\int f(x)dx$ recently. One can note that it has an easy real-valued solution $f(x)=\phi^{\frac{\phi-1}{\phi}}x^{\phi-1}$ (by guessing for a solution of the ...
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3answers
61 views

Functional equation determination

Let a function $f$ be continuous and differentiable for all x such that it satisfies $$ f(x+y)f(x-y)= f^2(x)$$ Given that $f(0)$ is nonzero and $f(1)$ is 1. how to find f. I tried replacing $x$ by $y$ ...
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1answer
42 views

Upper estimate for an integral in norms

Is there any reference to obtain an upper bound for $$ \int_{\Omega}v_{t}^2(v^2+1/v^2)dx $$ where $v\in H^{2}(\Omega)\cap H_{0}^1(\Omega) \setminus \{0\}$, $v_{t}\in H_{0}^1(\Omega) \setminus \{0\}$ ...
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1answer
102 views

Find all polynomials $f (x)$ such that $f (x^2+x+1)$ divides $f (x^3-1)$

I had come across a question in which involved finding polynomials (with real coefficients) satisfying the division criteria stated above. By inspection, it was easy to see that polynomials like $x$, $...
2
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1answer
42 views

A Functional equation in one variable

Let $ f: \mathbb R\setminus\{2\} \rightarrow \mathbb R$ be a function satisfying the following functional equation: $$ 2f(x) + 3f\left(\frac{2x+29}{x-2}\right) = 100x+80$$ Find $f(x)$. I tried ...
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3answers
75 views

Finding all of the solutions for a functional equation

Let $P(x+2)+P(x-2) = 8x+6$ . Find all of the solutions for $P(x)$ . If we put $P(x) = ax + b $ the answer is obvious but how to determine all of the solutions ?
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2answers
95 views

Determine all functions $f(x)$ such that $f(f(x+y))=f(x)+f(y)$

The question is from here: Find all continuous functions $f:\mathbb R\to \mathbb R$ such that for any real $x$ and $y$, $$f(f(x+y))=f(x)+f(y).$$ I'm totally new to functional equations so please ...
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0answers
40 views

Solving a complex valued functional equation

I have an equation in the following form: $\frac{1}{r}F(s-j) = F(s) - F^{2}(s)$ Where $r$ is a real valued constant, $j$ is the unit imaginary number, $s$ is a complex variable and $F$ is a complex ...
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1answer
29 views

Nonlinear functional equation with tangent

Please help me with this. I need to find a non-trivial function $g(x)$ which satisfy the following functional equation $$\tan(g(x))+g(x)+g(x)\tan^2(g(x))=0$$
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3answers
54 views

Existence of a real valued function satisfying $f'(x)=f(x+1)$ where $f(x)\ne0$

I am wondering about the existence of a real valued function that satisfies $f'(x)=f(x+1)$ other than the trivial solution $f(x)=0$. I thought a likely solution would be a repeating (perhaps ...
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0answers
26 views

What is the asymptotic upper bound of a variable in this functional equation?

We are given a recursive function $f(x) = \lceil \frac {f(x + 1)}{\lceil \log_2(f(x + 1)) \rceil} \rceil$. We know that $f(1) = 2$ and $f(a) = n$. What is the asymptotic upper bound of $a$ expressed ...
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1answer
47 views

A particular Functional equation

Bonjour, Find all continuous functions, $f$, such that $f(x)-1999f\big(\frac{2x}{1-x^2}\big)=18$ for $|x|\neq 1$. My method: Putting $x=\tan{h}, h\in]\frac{-\pi}{2},\frac{\pi}{2}[ ,h\neq \pm \pi/4$...
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0answers
13 views

Existence of solution of some functional equation involving integral

I want to prove that there exists always solutions to this equation $$g(x) = \int\limits_a^b {f(s,x)ds} $$ where $g \in {L^2}(0,1)$ is given and $f \in {L^2}((a,b) \times (0,1))$ is the uknown, $0<...
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1answer
30 views

Interpolating data.

Consider the following formula $$_{n}q_x=1-exp[-n\times _{n}m_x-.008 \times n^3 \times _{n}m_x^2]\ldots(1).$$ Page 867 of this book shows values of $_{5}q_x$ associated with $_{5}m_x$ by the above ...
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1answer
79 views

About $ f(x^2) = f(x) + f(x/2) $

I was thinking about the equation $$ f(x^2) = f(x) + f(x/2) $$ This should be consistant for $x>1$. And probably for all reals. But I focus on $x>1$. This equation implies that $f$ grows ...
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1answer
50 views

Find $a$ if $h(2018)=a^3$

Suppose $$\begin{cases}f(x)=g(x+1)\\f(y)=2018y+2016y+\cdots+2y\\g(x)=h(2x)-x\end{cases}$$ If $h(2018)=a^3$ and $a\in\mathbb Z$, what is the value of $a$? The answer is $1009$. I found it by ...
9
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4answers
175 views

Functional equation involving $f(x^4)+f(x^2)+f(x)$

Find all increasing functions $f$ from positive reals to positive reals satisfying $f(x^4) + f(x^2) + f(x) = x^4 + x^2 + x$. It's easy to show that $f(1)=1$, and I was also able to show that $$f(x)-x ...
5
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0answers
148 views

Solution of advanced functional differential equation

Statement Consider an advanced functional differential equation $$ Lf(x) = f(2x+\pi)+f(2x-\pi),\quad L\equiv\frac{d^2}{dx^2}+1. \tag{1} $$ Let's construct a solution of Eq. $(1)$ with finite ...
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4answers
148 views

Find all continuous functions in $0$ that $2f(2x) = f(x) + x $

I need to find all functions that they are continuous in zero and $$ 2f(2x) = f(x) + x $$ About I know that there are many examples and that forum but I don't understand one thing in it and I ...
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1answer
83 views

Functional equation with $f(2x)$

Any other solutions(advice) are welcome. For any $x>0, \;\;\; 2f(\frac{1}{x}+1)+f(2x)=1$ Find all possible f(x). I wish you luck on a good thing in $2019$.
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0answers
49 views

$ t(n) = t( x_1 x_2 x_3 …) = t(x_1) + t(x_2) + t(x_3) + … + t( x_1 + x_2 + x_3 + … ) $

Let $ n > 1 $ be an integer. Consider The prime factorization $$ n = x_1 x_2 x_3 ... $$ Now define $$ t(n) = t( x_1 x_2 x_3 ...) = t(x_1) + t(x_2) + t(x_3) + ... + t( x_1 + x_2 + x_3 + ... ) $$...
5
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2answers
75 views

If $f(\frac{x+y}{x-y})=\frac{f(x)+f(y)}{f(x)-f(y)}$, which of the following statement is correct

If $f(\frac{x+y}{x-y})=\frac{f(x)+f(y)}{f(x)-f(y)}$ for $x \ne y$, $x$ and $y$ are integer. Which of the following statement is correct : (1) $f(0)=0$ (2) $f(1)=1$ (3) $f(-x)=-f(x)$ (4) $f(-x)=f(x)...
2
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2answers
69 views

What is wrong with my approach in this problem? Find all functions $f$ such that $f(x + y)f(x − y) = (f(x) + f(y))^2 − 4x^2 f(y)$ for real $x$, $y$

Problem: Find all functions $f : R → R$ such that $$f(x + y)f(x − y) = (f(x) + f(y))^2 − 4x^2 f(y)$$ for all $x, y ∈ R$, where R denotes the set of all real numbers. My approach: I ...
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1answer
36 views

Proof periodic equation

I don't understand how he get $b- \sqrt2, -a, -b$.... and what is the periodic function?
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0answers
27 views

Prove or disprove existence of functions $f$ and $g$ (functional equations) [duplicate]

I came across the following problem and I have been cracking my head with it: Prove or disprove that there are no functions $f$ and $g$ such that $$f(x,g(y-x)+g(z-x))=y^2+z^2$$ for all $x,y,z \in \...
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0answers
37 views

Fixed point of unusual integral equation

I am a little rusty in this area so please forgive the slowness. I am trying to prove or disprove the existence of fixed points for the following integral equation. Throughout I am interested in the ...
4
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2answers
80 views

Find all continuous function $f: \mathbb R \rightarrow \mathbb R$

Find all continuous function $f: \mathbb R \rightarrow \mathbb R$ for which $f(3)=5$ and for every $x,y \in \mathbb R$ it is truth that $f(x+y)=2+f(x)+f(y)$. I tried to find some dependence before $x$...
3
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0answers
47 views

$f(x)f(\frac{1}{x})=1\hspace{1 em}\forall x\in\mathbb{R}$ [duplicate]

Find all functions $f:\mathbb{R^*}\to\mathbb{R^*}$ that satisfy $$f(x)f(\frac{1}{x})=1\hspace{1 em}\forall x\in\mathbb{R^*}$$ I only found that every function $f(x)=x^n, n\in\mathbb{N^*}$ is a ...
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1answer
51 views

Weird integral equation with non convolution kernel

Let $f$ and $g$ two rugular functions. My question is the following: Under what condition can we say that for given $g$, there exists $f$ such that we have: $$\int\limits_0^1 {f(x - s,s)ds = g(x)} $$ ...
2
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3answers
65 views

let $f(x)$ be rational non constant polynomial and $f\circ f(x)=3f(x)^4-1$ then find the $f(x)$

let $f(x)$ be rational non constant polynomial and $f\circ f(x)=3f(x)^4-1$ then find the $f(x)$ . My Try : $$f(f(x))=3f(x)^4-1$$ Let $f(x)=ax^n+g(x)$ so : $$a(ax^n+g(x))^n+g(ax^n+g(x))=3(ax^n+g(x))...
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0answers
32 views

Relation between two functions defined by and integral

Let us consider the two functions $$F(x) = \int\limits_x^1 {h(s - x,s)ds - } \int\limits_{1 - x}^1 {h(2 - s - x,s)ds} $$$$G(x) = \int\limits_0^x {h(x - s,s)ds - } \int\limits_0^{1 - x} {h(x + s,s)ds} $...
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1answer
38 views

A book for questions on functions

I recently learnt functional analysis and I'm pretty comfortable at solving its questions but when it comes to more complex statements(though it may not be complex but for me they seem they are)I get ...
3
votes
1answer
56 views

A function satisfying a given condition

Is there a continuous real valued function $f$ satisfying $f(x+1)(f(x)+1)=1$ for all $x$ in the domain of $f$(possibly $\mathbb{R})$? Clearly the image of $f$ does not include $0$ and $-1$. By the ...
1
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1answer
103 views

Functional Equation simple problem

How do I show that if there are functions $f,g$ such that$$ f(g(x)+g(y))=bx+cy $$holds for all $b,c\in\mathbb{R}$, then we necessarily have $b=c$? Is this even true? It seems so, but I'm just not sure ...
1
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1answer
113 views

$f(x+y) = f(x) + f(y) + f^k(x) f(y) + f^k(y)f(x) $

Let $k>1$ be an integer. Consider equations of type $$f_k(x+y) = f_k(x) + f_k(y) + f_k^k(x) f_k(y) + f_k^k(y) f_k(x) , f(-z) = f(z), $$ where $f_k$ is the $k$ th function and $f_k^k $ is the $k$...
3
votes
1answer
42 views

prove $f$ is bounded given $f(t^2+u)=tf(t)+f(u)$

I have been trying to solve the functional equation $f:\Bbb R \to \Bbb R$ $f(t^2+u)=tf(t)+f(u)$. So far i have managed to show that $f$ is additive i.e. $f(a+b)=f(a)+f(b)$ which means that the ...
0
votes
3answers
38 views

Functional Equation Solved Using Differentiation

Let $f$ be a function with domain $R$ that satisfies the conditions: $$f(x+y)=f(x)f(y), \forall x,y $$ and $$f(0) \neq 0$$ (a) Show that $f(0)=1$ (b) Prove that $f(x) \neq 0$, for all $x\in R$ (c) ...
0
votes
0answers
27 views

Find the form of functional

I know the form of f(x) and that of g(x) and I would like to find an expression for the function H such that H[g(x)] = f(x). f is a polynomial and g is more complex and involves some exp and cos. Is ...
1
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1answer
82 views

Find all functions satisfying $f(x+1)=\frac{f(x)-5}{f(x)-3}$

Find all functions satisfying $$f(x+1)=\frac{f(x)-5}{f(x)-3}$$ My try: We have $$f(x+1)=1-\frac{2}{f(x)-3}$$ Letting $g(x) =f(x+1)-3$ We get $$g(x+1)=-2-\frac{2}{g(x)}$$ Any clue here?
7
votes
3answers
443 views

A functional equation of two variables

Solve the following functional equation : $f:\Bbb Z \rightarrow \Bbb Z$, $f(f(x)+y)=x+f(y+2017)$ I have no prior experience with solving functional equation but still tried a bit. I set $x=y=0$ to ...