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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65 views

Determine the smallest number P

I have here a hard problem, which I couldn't solve. Denote $M$ the set of all functions $f:[0,1]\to\mathbb{R}$ with the following properties: $f(x)\ge0, \forall x$ in $[0,1]$, $f(1)=1$, $f(x+y)\ge ...
2
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0answers
30 views

Is determining a non-constant solution to a functional inequality with polynomial arguements decidable?

So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say integer constants and $p_i$ are polynomials, say with ...
6
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1answer
79 views

submodular-like functions on $\mathbb{R}$

The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for ...
2
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2answers
272 views

Proof of continuity $f(x+y) = f(x) + f(y)$

I am trying to prove that if I have $f:\mathbb{R} \to \mathbb{R}$ satisfying $\forall x,y\in\mathbb{R},f(x+y) = f(x) + f(y)$. Which is assumed continuous at $0$, that $f$ is continuous on $\mathbb{R}$ ...
93
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1answer
3k views

Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square

Question: For any $a,b\in \mathbb{N}^{+}$, if $a+b$ is a square number, then $f(a)+f(b)$ is also a square number. Find all such functions. My try: It is clear that the function $$f(x)=x$$ ...
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1answer
52 views

Largest value in the functional $\int_0^\infty e^{-rt}( x^2+2x+\dot x^2)dt$?

I am trying to understand the second order linear differential equation and the answer here (Finnish) that I have translated below. Translation Problem What is the value of $x(t)$ where the ...
2
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1answer
60 views

The “trick” in the Herglotz trick

In How does the Herglotz trick work?, is explained as in "Proofs from THE BOOK" by Aigner and Ziegler, but the "trick" itself I found to be not so clear. The trick says: It follows from (4) ...
1
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0answers
35 views

Functional equation on a ring

Let $R$ be a ring with identity. Find all functions $f: R \to \Bbb R$ satisfying the functional equation \begin{equation} f(xp+yq, yp+xq)=f(x, y)f(p, q) \end{equation} for all $x, y, p, q\in R$, and ...
4
votes
1answer
387 views

Inverse Function Differential Equation [duplicate]

For the differential equation $$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$ where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x). I gave up on finding the solution analytically pretty quickly and decided ...
3
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1answer
74 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 $$f(n+f(n))=n$$
2
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1answer
137 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 (...
3
votes
3answers
368 views

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 ...
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3answers
54 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
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2answers
334 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
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1answer
39 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
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2answers
95 views

Problem solve functional equation

Solve functional equation:find all strictly monotone functions $f:(0,+\infty)\to(0,+\infty)$ such that $$(x+1)f(\dfrac{y}{f(x)})=f(x+y),\forall x,y>0$$
2
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0answers
74 views

Creating a monotonic function

I have $n$ functions $f_i(x) \{i = 1 ,...,n\} $that does not preserve the monotonic mapping order. i.e. if $x_1 < x_2$, then in general, $f_i(x_1)$ is not less than $f_i(x_2)$ (for all $i = 1 ... ...
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1answer
111 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 ...
4
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0answers
47 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
350 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 $$m=2,n=2\Longrightarrow f^2(2)=f(4)+f(3)$$ $$m=1.n=4,f(1)f(4)=f(4)+...
4
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1answer
104 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 $4f(x)=f(2x+1)$
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3answers
662 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 $f(2013)$?
3
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0answers
84 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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0answers
44 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
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1answer
58 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 ...
6
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2answers
350 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 $f(x) = ax^b$ ...
2
votes
3answers
269 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 ...
5
votes
3answers
276 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 ...
10
votes
0answers
141 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 ...
1
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1answer
93 views

General solution of a finite-difference equation with real non-commensurable differences.

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 ...
0
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1answer
72 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
7
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4answers
173 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 $n\in\Bbb Z$
5
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1answer
122 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 ...
1
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1answer
30 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
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1answer
45 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, $Y_0(t)=\mathcal{L}^{-1}(...
4
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1answer
70 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 ...
2
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1answer
86 views

Solving functional equation 2

Problem: find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(x^2+f(y))=(f(x))^2+y^4 +2f(xy),\ \ \ \forall x,y\in\mathbb{R}$$
9
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1answer
198 views

How to prove that subadditive function has this property?

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
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1answer
44 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 $...
4
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1answer
71 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
53 views

Solve a functional equation

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

Solve the functional equation $\,\,f(2x)=2x f^\prime(x)$

If $f(x)$ is a real analytic functions on $\mathbb R$, and $$2xf'(x)=f(2x),$$ then find $f(x)$. My idea: express $f$ as: $$f(x)=\sum_{n=0}^{\infty}a_{n}x^n.$$ Thank you
2
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1answer
75 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, $b \in \mathbb{R}$ ...
6
votes
2answers
173 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 ...
2
votes
1answer
111 views

Another functional equation: $f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor$

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
71 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
252 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} f(1)...
7
votes
3answers
470 views

Why computing Fatou coordinate is so hard?

I'm trying to make images of Fatou coordinate for some polynomial maps. If I'm not wrong there is no explicit general formula/method for computing Fatou coordinate near parabolic fixed point. Is ...
2
votes
1answer
49 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
1answer
103 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 ...