The name "functional equation" is used for problems where the goal is to find all functions satisfying the given equation (and maybe some other conditions). So in this case, solving the equation means finding all functions fulfilling the equation. (This is different from the more common use of the ...

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The uniqueness of a solution of a system of differential equations

Suppose I have a system of delay differential equations $\frac{d G_1(\phi_1(s))}{ds}= F_1(\phi_1(s),\phi_2(1-s),1-s)$ $\frac{d G_2(\phi_2(1-s))}{ds}= F_2(\phi_1(s),\phi_2(1-s),s)$ where $G_1, G_2, ...
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48 views

$f(x)=xf(1)$ Doubt

I just started learning Functional Equations and I was working on a problem that asked me to find all functions satisfying a certain condition, and at some point I got $f(x)=xf(1)$, is there a way to ...
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142 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \mathbb{R} \to \mathbb{R}$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ ...
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3answers
160 views

$1992$ IMO Functional Equation problem

The problem states: Let $\Bbb R$ denote the set of all real numbers. Find all functions $f : \Bbb R \rightarrow \Bbb R$ such that $$f(x^{2}+f(y))=y+(f(x))^{2} \space \space \space \forall x, y \in ...
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3answers
147 views

How find this function $f(x)$ such $f(a+f(b))=f(a+b)+f(b)$

let function $f:R_{+}\to R_{+}$,and such $$f(a+f(b))=f(a+b)+f(b),\forall a,b\in R_{+}$$ Find $f(x)$. my try: let $a=b=1$,then $$f(1+f(1))=f(2)+f(1)$$ $a=1,b=2$,then $$f(1+f(2))=f(3)+f(2)$$ then I ...
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62 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 ...
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33 views

General solution of the recurrence equation with real shifts

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
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1answer
44 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
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22 views

Does there exists a function f:Z→Z such that f(f(n)−2n)=2f(n)+n for all n∈Z [duplicate]

Does there exists a function $f:Z→Z$ such that $f(f(n)−2n)=2f(n)+n$ for all $n∈Z$
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157 views

$f:\Bbb Z\to\Bbb Z$ such that $f(f(n)-2n)=2f(n)+n$ for all $n\in\Bbb Z$

Does there exists a function $f:\Bbb Z\to\Bbb Z$ such that $$f(f(n)-2n)=2f(n)+n$$ for all $n\in\Bbb Z$
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1answer
94 views

determine all polynomials $P(x)$ such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial

Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial. clearly we have to show $(x+1)P(x-1)-(x-1)P(x)=c$ for all values of $x$ ($c$ is a ...
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24 views

Multiplicative functional equation on Gaussian integers

Using polar form of complex numbers, it can be checked that all solutions $f:\Bbb C \to \Bbb R$ of the functional equation $$ f(zw)=f(z)f(w) \tag 1 $$ are of the form $$ ...
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1answer
27 views

Linear after change of variables

Many interesting non-linear functions of several variables like $\|x\|^2 =\sum\limits_{k=1}^n x_k^2$ become linear in "different variables". Is there a characterization of all (continuous, ...
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1answer
31 views

Similar outputs from different transfer functions

I have two rational transfer functions with the same denominator: $$ H_{0}(s) = N_{0}(s)/D(s),\,\,H_{1}(s) = N_{1}(s)/D(s)$$ I would like for the two outputs from the system, ...
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1answer
60 views

Trivial funcional equation

I'm sorry and a little ashamed to ask this very simple question. The problem is that I'm not very familiar with functional equations (I just know that they can be tricky). The question is: which are ...
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9 views

Quicker way of solving Fout = Fref * A / B / C for a given Fout and Fref

I've got a chip device that calculates frequency based on the following equation: $Fout = Fref * A / B / C$ Where: $Fout$ is the output frequency $Fref$ is a reference frequency 0 < $A$ <= ...
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1answer
61 views

Solving functional equation 2

Problem: find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2+f(y))=(f(x))^2+y^4 +2f(xy),\ \ \ \forall x,y\in\mathbb{R}$$
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85 views

How to prove subadditive function?

Let $f: [0, \infty) \to \Bbb R$ be a function satisfying the following conditions: (1) For any $x,y \geq 0, f(x+y) \geq f(x) + f(y)$. (2) For any $x \in [0,2], f(x) \geq x^2 - x$. Prove that, for ...
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1answer
33 views

continuous functions on rationals

Let $\Bbb Q$ be the set of rationals and $f:\Bbb Q\to \Bbb R$ be a continuous function. Then $f$ is bounded on some interval? If not, what happen if in addition $f$ satisfies $f(xy)=f(x)f(y)$ for all ...
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1answer
57 views

Solving functional equation 1

find all functiions $f:\mathbb{R}\to\mathbb{R}$ such that $f'$ exists and $$f(x)=f\left(\frac{x}{2}\right)+\frac{x}{2}.f'(x),\forall x\in\mathbb{R}$$
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1answer
41 views

Solve a functional equation

Find all functions $f:[0,+\infty)\to [0,+\infty)$ such that $f(x)\geq \frac{3x}{4}$ and $$f(4f(x)-3x)=x,\forall x\in[0,+\infty)$$
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1answer
124 views

If $f(2x)=2xf'(x)$, then find $f(x)$

If $f(x)$ is Analytic functions on $R$,and such $$2xf'(x)=f(2x)$$ Find all $f(x)$ My idea: let $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n$$ so I can't Thank you
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1answer
61 views

Solution to a Functional Equation $g'(x)=\dfrac{g(bx)-g(x)}{(b-1)x}$

What approaches can we take to solve the functional equation, $g:\mathbb{R}\to \mathbb{R}$ is a differentiable function, such that, $$g'(x)=\dfrac{g(bx)-g(x)}{(b-1)x}$$ Where, $b \in \mathbb{R}$ ...
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28 views

I tried this problem in the following way; is it right?

Let, f(x) be a twice differentiable function defined on (-1, 1) and f(0) = 1. Let, f(x) ≥ 0, f'(x) ≤ 0 and f''(x) ≤ f(x) for all x ≥ 0. Show that, f'(0) ≥ -√2. I am telling you what I did. First, ...
6
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2answers
107 views

Show that $f(x) = x$ if $f(f(f(x))) = x$.

If $f: \mathbb{R} \to \mathbb{R}$ and $f$ is strictly increasing, show that $f(x) = x$ if $f(f(f(x))) = x$. So this compulsorily ESTABLISHES that $f(x) = x$ only, and no other solution. So, merely ...
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56 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
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2answers
63 views

How do you find two functions $f$ and $g$ such that $f(x) \cdot g(x)=f(x)-g(x)$?

This was inspired by this question ( Logarithms with trigonometric inequality ) I already know the answer ( $f(x)=\tan^2 x$ and $g(x)=\sin^2 x$). However I am interested in how to find this answer. ...
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1answer
126 views

An IMO inspired problem

This problem from IMO 1988 is said to be one of the most elegant ones in functional equations. Problem : The function $f$ is defined on the set of all positive integers as follows: \begin{align} ...
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1answer
32 views

About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions. I notice that in the energy ...
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17 views

Logarithmic functional equation on positive rational numbers

Let $\Bbb Q^+$ be the set of positive rationals and $f:\Bbb Q^+ \to \Bbb R$ be a continuous function satisfying $$ f(rs)=f(r)+f(s) $$ for all $r, s\in \Bbb Q^+$. Then $f$ has only the form $$ f(r)=c ...
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1answer
37 views

Multiplicative function on rationals [duplicate]

Let $\Bbb Q^+$ be the set of positive rational numbers. Find all solutions $f:\Bbb Q^+ \to \Bbb R$ of the functional equation $$ f(xy)=f(x)f(y), \quad x, y\in \Bbb Q. $$ Is $f(x)=x^a$ the only ...
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0answers
24 views

Solving differential functional equations with a restricted solution

I have a variable vector $X=\{x_1,x_2,...,x_n\}$, and a constant vector $V=\{v_1,v_2,...,v_n\}$. $f(x_i,X)$ is a function that takes X and xi as the parameter, for example: $f(x_i,X) = ...
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2answers
55 views

Solve functional equation [closed]

find all functions $f:\mathbb{R^{*}}\to \mathbb{R}$ such that $$f(y^2f(x)+x^2+y)=x(f(y))^2+f(y)+x^2,\;\forall x,y\in \mathbb{R^{*}}$$ ($\mathbb{R^{*}}=\{x\in\mathbb{R},x\ne 0\})$
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0answers
51 views

Solving functional equation

Problem:find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$$
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1answer
41 views

Functional equation with $f(1)$ integer

Here is a nice problem: Let $f:R\rightarrow R$ be a function, R is the set of real numbers, satisfying the following properties: $ f(1)$ is an integer and $xf(y)+yf(x)=(f(x+y))^2-f(x^2)-f(y^2)$, for ...
4
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2answers
97 views

Functional Equation : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x).

Problem : If $(x-y)f(x+y) -(x+y)f(x-y) =4xy(x^2-y^2)$ for all x,y find f(x). My approach : The given equation can be written as $$(x-y)f(x+y) -(x+y)f(x-y) =4xy(x-y)(x+y)$$ $$\Rightarrow ...
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1answer
32 views

Solution to a general scaling problem $G(\lambda z)=\frac{G(z)}{\gamma z^n}$

When playing with the scaling problem $$G(4z)=\frac{G(z)}{2z}$$ (see also this question) I discovered, that the general problem $$G(\lambda z)=\frac{G(z)}{\gamma z}$$ with two constants ...
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2answers
50 views

A scaling functional equation

I want to find a closed form of a function satisfying $$G(4z)=\frac{G(z)}{2z},$$ unfortunately I am not experienced with scaling problems, so I have no idea and would be thankful for any hints. ...
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1answer
69 views

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$? [duplicate]

Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?
5
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5answers
139 views

$\forall x\in\mathbb R$, $|x|\neq 1$ it is known that $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$. Find $f(x)$.

$\forall x\in\mathbb R$, $|x|\neq 1$ $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{3+x}{1-x}\right)=x$$Find $f(x)$. Now what I'm actually looking for is an explanation of a solution to this ...
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0answers
42 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
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49 views

Does there exist a nontrivial cumulative distribution function $F$ on $\mathbb{R}_+$ for which $(F^2)'= (F')^{\ast k}$ for some $k > 0$?

This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ...
3
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1answer
63 views

Functional inequality and one identity

I'm a high school student from Bonn, Germany and I have to solve the following problem: If $g:R \rightarrow R $ is a function with the property $g(ab)-ag(b)\leq bg(a)$, for all real numbers a and b, ...
2
votes
1answer
79 views

Find $g(x)$ if $f(g(x))=f(x)g(x)$ and $g(2)$=37, $f(x)$ and $g(x)$ are polynomials

Suppose $f(x)$ and $g(x)$ are non-zero polynomials with real coefficients, such that $f(g(x))=f(x)\times g(x)$. If $g(2)=37$, find $g(x)$. I tried plugging $f(x)$ and $g(x)$ as $n$ and $m$ ...
4
votes
1answer
60 views

Find all function satisfying a condition with $\min$ and $\max$

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$ \forall (x,y)\in\mathbb{R}^2,x\ne y,\quad \min (f(x),f(y)) \leq \frac{f(x)-f(y)}{x-y} \leq \max(f(x),f(y)) $$ I have started with ...
0
votes
4answers
97 views

Functional equations

Let $f:\mathbb{R}\to \mathbb{R}$ is a function such that for all real $x$ and $y$, $f(x+y)= f(x) + f(y)$ and $f(xy)= f(x)f(y)$, then prove that $f$ must be one of the two following functions: ...
2
votes
1answer
51 views

Functional equation of non-negative function

Find all $ f:[0,\infty)\rightarrow [0,\infty) $ such that $ f (2)=0 $, $ f (x)\not= 0 $ for $ x\in [0, 2) $ and $$ f (xf (y)) f (y)=f (x+y) $$ for all $ x, y\ge 0 $. I tried plugging in values ...
7
votes
2answers
79 views

Non trivial solutions of $g\circ f-f\circ g=g\circ f\circ g$

While thinking of perfect numbers, I came across the functional equation $g\circ f-f\circ g=g\circ f\circ g$ where the unknowns $f$ and $g$ are functions from $\mathbb{R}$ to itself. I only know one ...
1
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1answer
81 views

A Funtional Equation

Find all functions ${\rm f}:{\mathbb N}\times{\mathbb N} \rightarrow {\mathbb N}$ satisfying $$ \begin{array}{rrcl} a) & {\rm f}\left(n,n\right) & = & n \\[2mm] b) & {\rm ...
1
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2answers
109 views

On a function with a (complicated) functional equation.

Let $g(x,y)$ be a function such that: I. $-1\lt g(x,y)\lt1.$ II. $$\ln(\frac{1+g(x,y)}{1-g(x,y)})+2y\tan^{-1}(yg(x,y))=2(y^2+1)x,$$ for $x\in\mathbb R, y\gt1.$ Then i. Show that $g(x,y)$ ...