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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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0answers
17 views

Additive and Bijective function on the real line

I was trying to solve the following functional equation: $f(f(x-y))=f(x)-f(y)$. And I concluded that $f$ must be additive and bijective. The question is: Let $f:\mathbb{R} \to \mathbb{R}$ be an ...
4
votes
2answers
86 views

Finding f$(x)$ from a given functional equation

Find all continuous functions $f:(0,1)\rightarrow\mathbb{R}$ such that $$f\left(\dfrac{x-y}{\ln x-\ln y}\right)=\dfrac{f(x)+f(y)}{2},$$ and $x\neq y$. I tried by solving putting $y=\frac{x}{2}$ but ...
1
vote
1answer
43 views

$f: \mathbb{R} \to \mathbb{R},\space\space\space f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2,\space\space$find $f(3)$ in terms of $f(0)$.

$f: \mathbb{R} \to \mathbb{R},\space\space\space\space f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2,\space\space\space\space$ Find $f(3)$ in terms of $f(0)$. My approach: $$f(x)-2f(\frac{x}{2})+f(\frac{...
1
vote
2answers
75 views

Find all functions $f : \mathbb{(0,\infty})\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$

Find all functions $f : \mathbb{(0,\infty)}\to\mathbb{R}$ such that $f(x+y) = xf(y)+ yf(x)$ if $f$ is continuous at $x=1$ This problem was looking quite easy at first but the domain of positive ...
3
votes
1answer
52 views

If $f(x)f(y)+f(xy)\le -\frac{1}{4},\forall x,y\in[0,1)$, show that $f(x)=-\frac{1}{2}$

Let $f:[0,1) \to \mathbb{R}$ be a function such that $$f(x)f(y)+f(xy)\le -\dfrac{1}{4} \quad \forall\, x,y\in[0,1).$$ Show that $$f(x)=-\dfrac{1}{2} \quad \forall\, x \in[0,1).$$ I have proved that ...
2
votes
2answers
56 views

Find $x$ where $f(e^{(x+1)})=x-\ln(x)$ approaches one.

Given that $$ x \in [1,\infty) \quad f(e^{(x+1)})=x-\ln(x) $$ and $$ \lim_{x \to a} f(x)=1 $$ Find $a$. I got to the point: $$ \ln(a)-\ln(\ln(a)-1)=2 $$ But from there on I could not get to $a=e^...
0
votes
0answers
18 views

How to find a monotone increasing function given its relationship with its inverse?

I am trying to find all functions $$f : [0, 1] \rightarrow [0, 1]$$ continuous monotone strictly increasing such that $$f(v) = v - \frac{v}{2} \cdot f^{-1}(\frac{v}{2}).$$ However I have no idea how ...
0
votes
2answers
73 views

Solve functional equation $f(z)=c+zf(z^2)$ with series expansion?

Let the functional equation $(1)$ be given as $$ f(z)=c+zf(z^2) \tag{1}$$ where $c \in\mathbb R$ and $c \neq 0$. How can this functional equation be solved with series expansion (power, Taylor or ...
6
votes
1answer
64 views

How to prove $f$ is $C^\infty$

Suppose $f:U \subseteq \mathbb{R}^2 \rightarrow \mathbb{R}$ is continous and $$(x^2+y^4)f(x,y)+(f(x,y))^3=1 \: \text{for all} \: (x,y) \in U. $$ Prove $f$ is $C^\infty$. This kind of exercise ...
2
votes
2answers
387 views

$2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if $f(2)=5$ find $f(5)$

This question has been asked before, however I am interested in seeing why my approach to finding a solution does not work. $2+ f(x)f(y)=f(x)+f(y) +f(xy) $, if $f(2)=5$ find $f(5)$ What I have ...
0
votes
1answer
39 views

The condition for the existence of a symmetric form for the reflection formula $f(1-x)= \chi (x) f(x)$

Suppose we have a functional equation in the form $$f(1-x)=\chi (x) f(x)$$ with given function $\chi (x)$. What is the condition on the function $\chi (x)$ so that we can write this reflection ...
3
votes
1answer
64 views

Find all real functions such that $(x + 1)f(xf(y)) = xf(y(x + 1))$

Find all real functions of real variable such that $$(x + 1)f(xf(y)) = xf(y(x + 1))$$ Let $a=f(0)$. For $y=0$ we get $(x+1)f(ax) = ax$, so if $a\ne 0$ we get $$f(x) = {ax\over x+a}$$ which is actual ...
0
votes
0answers
18 views

Difficulties with Inner products and polarization identities

I am discussing the general inner product space. Here is what Polarization Identities mean. I denote the inner product by $(x,y)$. I am having a difficult time with the polarization identities. Of ...
6
votes
2answers
176 views

Multivariate polynomial functional equation

I’m having some difficulties solving the following functional equation: Find all polynomials $P(x,y)\in\mathbb{R}[X,Y]$ for which: $P(x,y)$ is homogeneous (so $\exists n\in\mathbb{N}, \...
2
votes
1answer
50 views

Functional equation in single-variable calculus

Suppose we know that $f(x) \in C^2$ and $f(x)$ defined for all real numbers. Furthermore, $f(x)$ has following property: $$ \forall x,y \in \Bbb R \quad f(x+y) - f(x) = yf'(x + \frac{y}{2})$$ How to ...
2
votes
2answers
146 views

An equation of rational functions

I'm trying to get the set of solutions of the following equation, whose unknowns are the rational functions $f$ and $g$ : $\forall x\in\mathbb{R}$ such that the LHS and RHS are both defined, $$f(x)f(...
0
votes
0answers
24 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 $D=\{x \in\mathbb C: |x|<1\}$. (...
4
votes
3answers
110 views

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y,z \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$

Find all functions $f:\Bbb{R} \to \Bbb{R}$ such that for all $x,y, \in \Bbb{R} $ , $f(xf(x)+f(y))=x^2+y$ We can easily get a strong condition $f(f(y))=y $ by setting $x=0$ . By this equation we ...
2
votes
1answer
39 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{...
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votes
2answers
102 views

Functional Equation $f(x)f(f(x)+\frac{1}{x})=1$

I'd like to ask how to find all solutions to the functional equation $f(x)\cdot f(f(x)+\frac{1}{x})=1$, where $f: (0, +\infty)\to\mathbb{R}$ is strictly increasing?
8
votes
1answer
63 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 ...
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(...
6
votes
2answers
116 views

Find $f(x)$ if $f(x)+f\left(\frac{1-x}{x}\right)=1-x$

If $f: \mathbb{R} \to\mathbb{R} $, $x \ne 0,1$ Find all functions $f(x)$ such that $f(x)+f\left(\frac{1-x}{x}\right)=1-x$ My try: Letting $$g(x)=x+f(x)$$ we get $$g(x)+g\left(\frac{1-x}{x}\right)=\...
1
vote
0answers
33 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
63 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
1answer
41 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
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
69 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
63 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
89 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
53 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
60 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
52 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
68 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
97 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 ...
19
votes
4answers
383 views

Solving $f(yf(x)+x/y)=xyf(x^2+y^2)$ over the reals

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that $f(1)=1$ and for all real numbers $x$ and $y$ with $y \neq 0$, $$f\Bigg (yf(x)+\frac{x}{y}\Bigg)=xyf(x^2+y^2)$$ This seems quite hard. $f(x)=...
0
votes
2answers
36 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 ...
0
votes
2answers
55 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\...
2
votes
3answers
97 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
86 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
104 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. ...
3
votes
1answer
58 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\...
6
votes
0answers
91 views

Find all functions that satisfy $f(x+f(y))=f(x)-y$

here is the problem Here is my solution : $x=y=0$ gives $f(f(0))=f(0)$ $x=0; y=f(0)$ gives $f(f(f(0)=0=f(0)$ (because $f(f(0))=f(0) \iff f(f(f(0)))=f(f(0))=f(0)$) $x=0$ gives $f(f(y))=-y$ $x=0 ; ...
3
votes
3answers
197 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 ...
1
vote
2answers
49 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
261 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
51 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-...
0
votes
1answer
36 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 ...
7
votes
1answer
131 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(...
2
votes
0answers
122 views

Romanian Master of Mathematics 2019 Day 2 Prob. 5. (Still lack of solution 2019 Feb. 28th)

I found this problem is interesting, but I do not know how to do it. I want to know in general, how can one deal with such a functional problem. Are there any recommend books, lecture notes and etc. ...