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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35 views

Prove That $f(n+f(n))=n$

if $f:\mathbb{N}\rightarrow\mathbb{N}$ is a function such that $f(1)=1$ and $$f(n)=n-f(f(n-1))\,\,\: \forall n \ge 2$$ Prove That $$f(n+f(n))=n$$
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1answer
32 views

Find a relation between $a$ and $b$?

I would appreciate if somebody could help me with the following problem: Let $f(x)=x^2-2ax+b$, $a,b\in \mathbb{R}$ Q: Find a relation between $a$ and $b$ ? If $|x|\leq 1$ then $|f(x)|\leq1 $
2
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1answer
34 views

Suppose a function $f : \mathbb{R}\rightarrow \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$.

Suppose a function $f : \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$. Show that (a) $f$ is one-to-one. (b) $f$ cannot be strictly decreasing, and ...
2
votes
2answers
53 views
+50

Help with functional equation $F(x,x)+F(x,c-x)-F(c-x,x)-F(c-x,c-x)=0$

How can we find $F$ satisfying: exists a $c$ such that $$F(x,x)+F(x,c-x)-F(c-x,x)-F(c-x,c-x)=0 \text{ for all } x,y $$ Several quadratic polynomials in $x,y$ satisfy the above property. I'm trying to ...
0
votes
3answers
36 views

Find all polynomials p with real coefficients

Find all polynomials $p$ with real coefficients such that $p(x+1)=p(x)+2x+1$. I feel like in this question you let $x+1=x'$.
3
votes
2answers
50 views

Find all functions that satisfy the following conditions

Find all functions $f:\mathbb Z\to \mathbb Z$ that satisfy the following conditions: (i) $f (0) = 1 $ (ii) $f(f (x)) = x$ for all integers x (iii) $f(f(x + 1)+1) = x$ for all integers x How ...
0
votes
1answer
19 views

Functional Equation and totally multiplicative functions

Find all $f:\mathbb{C}\rightarrow\mathbb{C}$ s.t. $\forall a, b\in \mathbb{C}, f(ab) = f(a)f(b)$. I could deduce a lot of things about what happens at the roots of unity, and 0, but I can't find out ...
5
votes
2answers
54 views

Problem solve functional equation

Solve functional equation:find all strictly monotone functions $f:(0,+\infty)\to(0,+\infty)$ such that $$(x+1)f(\dfrac{y}{f(x)})=f(x+y),\forall x,y>0$$
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0answers
42 views

How to solve integral equations like this?

sorry for such a non-specific question and lack of research effort, but I'm new to integral equations and don't know where to start. How does one go about solving equations of the form ...
1
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1answer
33 views

Contour approach to Riemann zeta functional equation

I have a question regarding Riemann's first proof of the functional equation that was given in his paper on the Riemann zeta function. I am an undergraduate working on a fairly short undergraduate ...
3
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0answers
21 views

Wronskian different from zero and solutions of ODE.

Let $a_0, \ldots , a_{n-1}$ continuous functions in an interval $I$.Consider the equation $$x^{(n)} = a_{n-1}(t)x^{(n-1)}+\cdots+a_0(t)x. \tag 1$$ Let $\phi_1, \phi_2, \ldots,\phi_n$ $n$ are ...
3
votes
3answers
85 views

if $f(mn)+f(m+n-1)=f(m)f(n)$How find $f(n)$

let $f:N^{+}\to Z$,and $f$ is monotonic nondecreasing,and such $$f(m)f(n)=f(mn)+f(m+n-1),f(4)=5$$ Find all $f(n)$ My try: let $$m=2,n=2\Longrightarrow f^2(2)=f(4)+f(3)$$ ...
4
votes
1answer
50 views

Solve the functional equation, $f(x+y+1)= (\sqrt{f(x)} + \sqrt{f(y)})^2$

Solve the functional equation, find all real valued functions $f(x)$ s.t. $$f(x+y+1)= (\sqrt{f(x)} + \sqrt{f(y)})^2$$ given $f(0)=1$. I reached to a point that $4f(x)=f(2x+1)$
7
votes
3answers
235 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
2
votes
0answers
65 views

Is this a field of study?

Is there a name for an equation that takes the following form? $$F(f(x),f^{-1}(x),x)=0$$ A nice example being $$f(x)-f^{-1}(x)=0$$ because the solutions of this equation are their own inverses. ...
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vote
0answers
24 views

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, ...
0
votes
1answer
41 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 ...
5
votes
0answers
87 views

Function whose inverse is also its derivative?

What are some good examples of a function $f : \Re \to \Re$ where its derivative is equal to its inverse? I attempted to find a monomial that satisfied it by starting with $f(x) = ax^b$ and showing ...
3
votes
3answers
122 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 ...
4
votes
3answers
137 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 ...
6
votes
0answers
49 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 ...
0
votes
0answers
24 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 ...
1
vote
1answer
25 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
0
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0answers
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 $n∈Z$
7
votes
4answers
142 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 $n\in\Bbb Z$
4
votes
1answer
83 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 ...
0
votes
0answers
20 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 $$ ...
1
vote
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, ...
0
votes
1answer
28 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, ...
4
votes
1answer
59 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 ...
0
votes
0answers
8 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 < $A$ <= ...
3
votes
1answer
48 views

Solving functional equation 2

Problem: find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2+f(y))=(f(x))^2+y^4 +2f(xy),\ \ \ \forall x,y\in\mathbb{R}$$
6
votes
0answers
72 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 ...
2
votes
1answer
28 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 ...
3
votes
1answer
52 views

Solving functional equation 1

find all functiions $f:\mathbb{R}\to\mathbb{R}$ such that $f'$ exists and $$f(x)=f\left(\frac{x}{2}\right)+\frac{x}{2}.f'(x),\forall x\in\mathbb{R}$$
0
votes
1answer
39 views

Solve a functional equation

Find all functions $f:[0,+\infty)\to [0,+\infty)$ such that $f(x)\geq \frac{3x}{4}$ and $$f(4f(x)-3x)=x,\forall x\in[0,+\infty)$$
0
votes
1answer
106 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
2
votes
1answer
52 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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vote
0answers
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
votes
2answers
97 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 ...
1
vote
0answers
48 views

Another functional equation

I would like to find all continuous functions $f \, : \, \mathbb{R} \, \longrightarrow \, \mathbb{R}$ such that : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
2
votes
2answers
60 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. ...
10
votes
1answer
109 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} ...
2
votes
1answer
31 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 ...
0
votes
0answers
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 ...
0
votes
1answer
34 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 ...
0
votes
0answers
22 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) = ...
4
votes
2answers
50 views

Solve functional equation [closed]

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

Solving functional equation

Problem:find all continuous functions $f:[0,+\infty)\to [0,+\infty)$ such that $$f(y^2f(x)+x^2f(y))=xy(f(x)+f(y)),\;\forall x,y\in [0,+\infty)$$
1
vote
1answer
29 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 ...