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).

Filter by
Sorted by
Tagged with
1
vote
1answer
69 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.
1
vote
1answer
23 views

Pexider's (/ Cauchy's) functional equation over a bounded domain

I am looking at Pexider's equation $f(x+y)=g(x)+h(y)$, where $f,g,h$ are continuous functions but are defined over bounded domains. Specifically, $f,g,h$ each is defined on a real interval (of length ...
0
votes
0answers
31 views

Cauchy's functional equation from complex to reals

I am looking for the general continuous solutions $f:D \rightarrow \mathbb R$ of the multiplicative Cauchy functional equation $f(x)f(y) = f(xy)$ for the domain $D=\{x \in\mathbb C: |x|<1\}$. (...
2
votes
1answer
44 views

Solving tricky functional equation resembling quadratic equation

I have the following functional equation in hand, I can easily solve it for the case $(a, b ,c)=(1, 1,0)$ which gives $f(x)$ to be $x^2+x$. $\begin{aligned}{g(x)=a\left[f(x)\right]^2+bf(x)+c, \text{...
13
votes
1answer
260 views

Find all pairs of functions $(f,g)$, $\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$

Find all pairs of functions $(f,g)$ : $\mathbb{R} \to \mathbb{R}, g : \mathbb{R} \to \mathbb{R}$ satisfying : $$\forall x, y \in \mathbb{R}, f(x+g(y))=x f(y) - y f(x) + g(x)$$ I am really stuck ...
8
votes
1answer
65 views

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(f(x)+yz)=x+f(y)f(z)$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(f(x)+yz)=x+f(y)f(z)$ I was told to do this by proving $f$ is injective and surjective. I have proved it this ...
1
vote
1answer
55 views

Do I need to verify solutions to functional equations?

I am studying the (basics of) solving functional equations. My teacher stipulates that we check any solutions obtained by substitution. Similar guidelines are given in this IMO training material. For ...
5
votes
1answer
160 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
79 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
96 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
695 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
39 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
vote
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
76 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
73 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
100 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
votes
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
votes
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
75 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
117 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
43 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
59 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
201 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
103 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
94 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
109 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
62 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
655 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
61 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
335 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
473 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
539 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
84 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
28 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
69 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
97 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 ...