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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5
votes
1answer
146 views

Miklos Schweitzer 2001 5: Functional Equation conditions

Prove that if the function $f$ is defined on the set of positive real numbers, its values are real, and $f$ satisfies the equation $$f\left( \frac{x+y}{2}\right) + f\left(\frac{2xy}{x+y} \right) =f(x)+...
3
votes
1answer
200 views

Functional equation $f(f(x)+3y)=12x + f(f(y)-x)$

I found this problem on a French exchange forum : Find all the $f : \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)+3y)=12x + f(f(y)-x)$ In fact I solved the problem when $f$ is supposed to be ...
3
votes
2answers
78 views

Functional equation: $(x+y)f(x,y)=yf(x,x+y)$

Find functions $f$ on pairs of positive natural numbers satisfying: $f(x,x)=x$ $f(x,y)=f(y,x)$ $(x+y)f(x,y)=yf(x,x+y)$ It is quite easy to find that $f(1,k)=k$ for all $k$ by induction: if $f(...
0
votes
0answers
90 views

Solving the functional equation $f(f(x))=3x$ over the naturals.

I wish to find all functions $f:\mathbb{N}\to \mathbb{N}$ such that $f(f(x))=3x$ This is my progress: $f(1)\neq1$ clearly as then $f(f(1))=1$ which is false. $f(1)=3$ is not possible as then $f(f(1))...
4
votes
3answers
689 views

Find $f(5)$ where $f$ satisfies $f(x)+f(1/(1-x))=x $

Question: How do you Find $f(5)$ in which the function satisfies $$f(x)+f\left(\frac{1}{1-x}\right)=x $$ where $x\in\Bbb{R}$ and $x\neq 0,1$? My steps: Step 1) Substitute $5$ into the equation ...
1
vote
1answer
45 views

Solution for a functional equation

I'm searching for a solution to the following functional equation: $$f(u)f(u+\lambda)=\prod_{i=1}^L\rho(u-u_i)\rho(u_i-u)+\prod_{i=1}^L\rho(u+\lambda-u_i)\rho(u_i-u-\lambda)$$ where $f$ is the ...
1
vote
0answers
38 views

An interesting functional equation

$$\frac{1-f\Big(\frac{x}{x+(1-x)f(x)}\Big)}{1-f(x)} = 1-x(1-x)\frac{f'(x)}{f(x)}$$ Now, we know that $f(x)=c$ and $f(x) = \frac{a+bx}{1-x}$ are two solutions. How can I get other solutions or to ...
1
vote
2answers
71 views

Is it possible to have $f(x)f(y) = g(x)+g(y)$?

Inspired by this question I wondered whether there are any "notable" functions $f,g$ on (or on some subset $\Omega$ of) $\mathbb R$ or $\mathbb C$ that satisfy $$f(x)f(y) = g(x) + g(y) \:\forall x,y \...
1
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0answers
14 views

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

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 $h(x)$...
1
vote
1answer
74 views

Characteristic functional equation of a Theta Function

Define the following as a "simple" theta function $$ \vartheta(q) = \sum_{n=0}^{\infty} q^{n^2} = 1 + q + q^4+q^9+ \ ...$$ Defined on the open unit circle on the complex plane. I'm trying to find ...
1
vote
3answers
70 views

Prove that only quadratic functions $f$ solve the quadratic functional equation

Let $f$ be such that $f(x+y)+f(x-y)=2f(x)+2f(y)$, i.e. $f$ satisfies the quadratic functional equation. Then $f$ has to be such that $f(t)=\alpha t^2$. I am looking for an accessible proof of this. ...
2
votes
0answers
98 views

Solving $f(x/2)^2=f(x)$

Does $\left[f(\frac{x}{2})\right]^2=f(x)$ imply $f(x)=\exp(Ax)$? How can I go about finding all the solutions to this equation?
0
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2answers
58 views

Algebra problem that you have to assume certain criteria at the end.

I was trying to solve this problem: If $f(x)=\frac{ax+b}{cx+d}, abcd\neq0$ and $f(f(x))=x$ for all $x$ in the domain of $f$, what is the value of $a+d$? I start off by just plugging in and ...
0
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0answers
63 views

Prove that $\sin^2(\pi x)$ is chaotic

My approach is based on the following from the book Chaos and Fractals: New Frontiers of Science, by Peitgen, Heinz-Otto, Jürgens, Hartmut, Saupe, Dietmar. To be more specific: "If $f$ is chaotic and ...
1
vote
0answers
54 views

$f:\mathbb{R} \to \mathbb{R}$ we have $f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$ such function is polynomial of degree less than or equal to two. [duplicate]

Consider differential function $f:\mathbb{R} \to \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have $$f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$$ Then show that every such function is ...
0
votes
2answers
74 views

Find all f such that $f(f(y))+f(x-y)=f(xf(y)-x)$

Find all functions $f$ defined over real numbers to real numbers such that $f(f(y))+f(x-y)=f(xf(y)-x)$ My attempt: Set $x=y=0$ to get $f(f(0))=0$. It will be very helpful if I will able to find $f(...
6
votes
1answer
111 views

Is there a solution to this functional equation?

I was going through my old notebooks and I found a sheet of paper with this problem on it. I thought it would be a shame to let such an unreasonably difficult question go to waste, so I decided I ...
0
votes
2answers
42 views

How to solve a functional equation involving log?

It's given that $$f(xy)=\frac {f (x)}{y}+\frac {f (y)}{x}$$ Also $x,y>0$ and $f(x)$ is differentiable for $x>0$ such that $f(e)=\frac{1}{e}$. By the look of the functional equation I am sure ...
4
votes
0answers
101 views

Functions $f:\mathbb R \to \mathbb R$ which satisfy $f(x^2+f(y))=y+(f(x))^2$ [duplicate]

Find all the functions $f:\mathbb R \to \mathbb R$ which satisfy $$f(x^2+f(y))=y+(f(x))^2$$ for all $x, y$ in $\mathbb R$. I have the following proof from my math book and want to see if I can ...
0
votes
2answers
57 views

Two functional equations

Is there a systematic approach that can be used to solve these two functional equations? $$af(x) = f(bx), \quad\text{where }\ f\colon \mathbb{R}\to\mathbb{R} \tag{1}$$ $$ag(y) + ay = g(ay),\quad\...
3
votes
3answers
200 views

A functional equation defined on the real axis.

$$ f: \mathbb{R} \to \mathbb{R}\qquad \frac{f(x+y)}{x+y} = \frac{f(x)-f(y)}{x-y}, \qquad \forall x,y\in \mathbb{R}, \left|x\right| \neq \left|y\right| $$ Can I prove anything interesting about this ...
2
votes
3answers
102 views

$ \int_0^x f(t)dt=\int_0^{ax}f(t)dt+ \int_0^{bx}f(t)dt$ implies $f$ constant

Let $a,b \in (0,1)$ be such that $a+b=1$ and $f:[0,1] \to \mathbb R$ be a continuous function such that $ \int_0^x f(t)dt=\int_0^{ax}f(t)dt+ \int_0^{bx}f(t)dt$. We have to prove that $f$ is constant. ...
2
votes
3answers
93 views

Find all polynomials $P(x)$ with $P(x)P(1/x)=P(x)+P(1/x)$

Find all polynomials $P(x)$ with $$P(x)P({1\over x})=P(x)+P({1\over x})$$ First I choose $x=1$, so $P(1)=0$ or $P(1)=2$. So I choose $x=-1$ too, but it's the same. I'm very stuck on this because ...
4
votes
1answer
108 views

If $f(x) + f(2x)$ is continuous, is $f$ continuous or not?

True or false: If $g(x)=f(x)+f(2x)$ with $g:\mathbb{R}\rightarrow \mathbb{R}$ is continuous, then $f$ is continuous. My idea was to find a counterexemple since, first, I claim that this is false. ...
1
vote
0answers
49 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}$ $f(s,t)=0$ $ \forall $ $0 \leq t \leq \frac{s}{2}$ $f(s,\frac{s}{2}...
3
votes
1answer
59 views

find all fucntions such that $f(x+y) \geq f(x) + yf(f(x)) $

Find all functions $f:\mathbb{R}_+ \to \mathbb{R}_+$ (not necessarily continues function) where $\mathbb{R}_+ = ${$r \in \mathbb{R} : r \geq 0$}, such that $$f(x+y) \geq f(x) + y f(f(x)) \quad\...
3
votes
0answers
64 views

What problems are related with the following type of FDE?

Consider the following type of functional differential equations: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{x,t}), & (x,t) &\in [a,b] \times [0,T] \end{align}$$ where $u(x,t)$ is ...
5
votes
3answers
641 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the Euler ...
1
vote
2answers
58 views

Euler-Lagrange equations for dependent multiple functions

Find the extremals for the functional: $$ J(x) = \int_{0}^{1}\left[x\left(t\right)\dot{x}\left(t\right) + \ddot{x}^{2}\left(t\right)\right]\mathrm{d}t $$ where $x(0)=0$, $\dot{x}(0)=1$, $x(1)=2$, $...
4
votes
2answers
322 views

Find $P(7/8)$ given ${P(5)}^2=P(6)$ and $(x-1)P(x+1)=(x+2)P(x)$

There's a polynomial $P(x)$, we know that ${P(5)}^2=P(6)$ and $$(x-1)P(x+1)=(x+2)P(x)$$ Find the value of $P(\frac{7}{8})$. Any hints? I know that $P(1)=0,P(0)=0,P(-1)=0$ and $P(5)=0$ or $P(5)=\...
-1
votes
1answer
52 views

Let $g(z) = 1/(1+e^{-z})$ be the logistic function. Show that $1-g(z)=g(-z)$

I am having trouble with this problem. I am able to work it out to the point where I have either an extra $1$ or with $e^z$ and $e^{-z}$ and also the extra $1$ Let $g(z)= 1/ (1+e^{-z})$ Show that $1-...
7
votes
1answer
132 views

Find $f$ such that $f(a-b)+f(c-d)=f(a)+f(b+c)+f(d)$

Denote the set of non-negative real numbers by $\mathbb R^+_0$. Find all functions $f:\mathbb R \rightarrow \mathbb R_0^+$ s.t. $\forall a,b,c,d\in\mathbb R$ satisfying $ab+bc+cd=0$ we have $$f(a-b)+f(...
0
votes
1answer
39 views

Limit involving iterated function $f_a(x)=x^2+a^2$

I have long ago give up trying to find a nice formula for the $n$th iteration of functions in the form $$f_a(x)=x^2+a^2$$ However, it would be interesting to consider the asymptotic growth of the ...
0
votes
2answers
471 views

Approximating the Digamma fucntion near 1

Peace be upon you, I had the following system of equations to be solved \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ \psi(\beta)-\psi(\alpha+\beta)=c_2 \end{cases} \end{align*} ...
2
votes
3answers
355 views

How can I find y?

The following equation is given: $$ye^y=e^{x+1}$$ when $x=0$. I tried to solve it as logarithmic equation but I can't go further. I know $y=1$ but I don't know how to prove it. Any idea? Thank you😊
2
votes
1answer
43 views

Existence and uniqueness of a solution of a sort of ODE

Suppose $F:\left[0,1\right] \times C^{1}\left(\left[0,1\right]\right)\rightarrow \mathbb{R}$ be a Lipschitz-Function with Lipschitz-constant $L>0$ so that \begin{align} \left\vert F\left(t,u\...
1
vote
2answers
533 views

Find all multiplicative continuous functions on $(0,\infty)$ [duplicate]

If $x>0,y>0$ and if $f(xy)=f(x)f(y)$, then $f=\, ?$ I tried the problem. And got it as $f(x)^n=f(x^n)$. But answer is $f(x)=x^n$. How? $f$ is a continuous function.
0
votes
3answers
68 views

A function describes $g(x + y) = g(x)g(y)$ for all $x, y$. If $g(4) = + 3,$ find the value of $g(–8)$? [closed]

I tried solving the question, but I always ended up getting my answer wrong. I'm also not sure if the given options are correct. Here are the options that were given: A. 1/3 B. 1/9 C. 9 D. 6
1
vote
0answers
83 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. ...
0
votes
0answers
27 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 $h(x)$...
0
votes
0answers
26 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))...
0
votes
2answers
67 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 $x$?
1
vote
0answers
96 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 ...
7
votes
2answers
128 views

Find all $f$ that satisfies $f:\mathbb{R}\rightarrow\mathbb{R};f(x+y)+f(x)f(y)=(1+x)f(y)+(1+y)f(x)+f(xy)$

Find all $f$ that satisfies: $1, ~f:\mathbb{R}\rightarrow\mathbb{R};\\ 2,\forall x,y\in\mathbb{R},f(x+y)+f(x)f(y)=(1+x)f(y)+(1+y)f(x)+f(xy); $ Maybe we can prove it's derivable or it's a linear ...
11
votes
1answer
188 views

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying a functional equation [duplicate]

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying: $f\left ( x \right )f\left ( y \right )+ f\left ( xy \right )+ f\left ( x \right )+f\left ( y \right )= f\left ( x+y \right )+ 2\,...
0
votes
1answer
57 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 ...
5
votes
1answer
85 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 $ ...
0
votes
1answer
82 views

Request for interesting Functional Equations, of a specific type

I am looking for interesting functional equations of a specific type, and I thought that perhaps the math SE community would be able to deliver a good amount of them. When I look up "functional ...
0
votes
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)...
1
vote
1answer
74 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 ...