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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30 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which is a convolution kernel. I am interested in the following equation: $0=A+B~(\varphi*v)+C~v+D~v~(\varphi*v)$, where ...
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2answers
34 views

Simplifying $\frac{1/(\frac{1}{z_1}(1-t)+\frac{1}{z_2}t) - z_1}{(z_2 - z_1)}$

This drives me mad! I am not very good in math but thought I could at least do basic things like this one, but can't figure it out and I spent a day on it. I am trying to simplify: ...
2
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0answers
42 views

What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= ...
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2answers
77 views

What is the formula and the name of the mathematical-phenomenon seen at the ending of “Around the World in Eighty Days”?

Spoiler in brief for those who don't know the ending yet: At the end, Phileas Fogg alongside with his companions realize that they have arrived back to London a day earlier than expected due fact the ...
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2answers
203 views

Find $f(x)$ satisfy $f(2x)=2f(x)+x$

I would appreciate if somebody could help me with the following problem: Find $f(x)$, given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous at $x=0$, and ...
3
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6answers
142 views

Functional equation $ f(x)+f(x+1)=x$

What functions satisfy $f(x)+f(x+1)=x$? I tried but I do not know if my answer is correct. $f(x)=y$ $y+f(x+1)=x$ $f(x+1)=x-y$ $f(x)=x-1-y$ $2y=x-1$ $f(x)=(x-1)/2$
63
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7answers
2k views

Does a non-trivial solution exist for $f'(x)=f(f(x))$?

Does $f'(x)=f(f(x))$ have any solutions other than $f(x)=0$? I have become convinced that it does (see below), but I don't know of any way to prove this. Is there a nice method for solving this kind ...
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1answer
69 views

analitycs solutions to the equation $f'(x)=f(x)f(x-1)$

As the title says I'm serching for functions ($C^n$ or analitycs $f$) that satisfies $f'(x)=f(x)f(x-1)$ some details: I've come at this equation after looking for a function $g$ satisfying for ...
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1answer
76 views

Show that f is linear

Let $f : \mathbb R \to \mathbb R$ be a solution of the additive Cauchy functional equation satisfying the condition $$f(x) = x^2 f(1/x)\quad \forall x \in \mathbb R\setminus \{0\}.$$ Then show that ...
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1answer
31 views

monotonic function. I need to show ots linear

If a real additive function f is monotonic, then it is linear. I need to show that the monotonic function f that satisfies caushy additives functional equation is linear
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0answers
33 views

Generalized Riesz theorem of operator value function

I am reading a book Gustafson-Rao's Numerical Range and come across a problem that I really don't understand. In theorem 2.1-2 of the book, it asserts that for an operator valued function ...
22
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2answers
511 views

Do there exist functions $f$ such that $f(f(x))=x^2-x+1$ for every $x$?

My question is on the existence (or not) of a function $f:\mathbb{R}\to\mathbb{R}$ which satisfy the equation: $$f(f(x))=x^2-x+1 \text{ for every }x\in\mathbb{R}$$ Supposing that such a map do exist ...
5
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1answer
77 views

Find the functions

Find all the functions $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ with the following property: $$ f(x + 3f(y)) = f(x) + f(y) + 2y, \: \forall x, y \in \mathbb{Q} $$
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2answers
88 views

functional equations.. I need hints for this problem

Find all functions $f : \mathbb{R} → \mathbb{R}$ that satisfy the functional equation $f(x + y) = f(x) + f(y) + xy$, for all $x,y ∈ \mathbb{R}$.
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2answers
81 views

Show that $f$ is a Cauchy function

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a solution of the functional equation $$|f(x + y)| = |f(x)| + |f(y)| \quad \forall x,y \in\mathbb{R}.$$ Show that $f$ is an additive function.
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2answers
157 views

Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$

Find all functions $f$ such that $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Let us plug in $n=1$ $f(f(1))+f(2)=3$ Since the function is from $\mathbb{N}$ to $\mathbb{N}$, ...
15
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4answers
296 views

Find $f''(x)$ if $f\circ f'(x) = 4x^2 + 3$

Can you tell me the solution of this question? If: $f\circ f'(x)=4 x^2 +3$ then what is $f''(x)$? This was a question in math test which I just took yesterday. One function satisfying the ...
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0answers
33 views

A matrix version of Fredholm integral equation of the second kind

Fredholm integral equation are often encountered in physics. I have to deal with the following Fredholm matrix equation \begin{equation} \big[ L (x,y) \big]_{ij} = \big[ A (x,y) \big]_{ij} + \sum_{k} ...
3
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3answers
101 views

Find $f(2)$ if $f$ satisfies $2f(x)-3f(\frac1x)=x^2$

The following expression is given, and we are asked to find $f(2)$. \begin{equation} 2f(x)-3f\left(\frac{1}{x}\right) =x^2 \end{equation} Does a unique and well defined answer exist? Why? and what ...
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4answers
86 views

Determine all functions (functional equation) [closed]

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the functional equation $$f(x + y) + f(z) = f(x) + f(y + z)$$ for all $x, y, z \in \mathbb{R}$.
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31 views

show the following: (functional equations)

If f : R to R is a solution of the additive Cauchy functional equation, then show that f is either everywhere or nowhere zero.
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2answers
1k views

Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?

If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
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2answers
149 views

Find value of a functional equation

Find $f(x)$ such that $$2 f(n) + \frac{1}{3}f\left(\frac{1}{n}\right) = 12.$$ Can anybody suggest me a way to solve this kind of functional equations?
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0answers
22 views

Periodic Function With A Given Functional Equation

Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant '$a$' the equation $f(x+a)=1/2+\sqrt{f(x)-(f(x))^2}$ holds for all $x$. Then prove that $f$ is ...
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0answers
50 views

Differential equation involving composition

I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ...
3
votes
2answers
101 views

Find all real functions $f:\mathbb {R} \to \mathbb {R}$ satisfying the equation $f(x^2+y.f(x))=x.f(x+y)$

Find all real functions $f:\mathbb {R} \to \mathbb {R}$ satisfying the equation $f(x^2+y.f(x))=x.f(x+y)$ My attempt - Clearly $f(0)=0$ Putting $x^2=x,y.f(x)=1$, we have $f(x+1)=x.f(x+y)$. Now ...
2
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2answers
176 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 $\mathbb{R}$ ...
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1answer
34 views

How to solve a set of equations where the unknowns are a function and some parameters?

I'd like to know how to solve something like this: $$\begin{eqnarray} f(f(x_2)-f(x_1)) & = & 27.5\\ f(f(x_3)-f(x_1)) & = & 21.6\\ f(f(x_4)-f(x_1)) & = & 15.1\\ f(f(x_5)-f(x_1)) ...
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1answer
25 views

Checking that a defined map is satisfied by some given condition

Suppose I have a group homomorphism $f: (\mathbb{R}^{2}, +) \rightarrow (\mathbb{R}, +)$ defined by $$f(x,y)= f(\frac{x+y}{2}, \frac{x+y}{2})+f(\frac{x-y}{2}, \frac{y-x}{2}) \text{,}$$ where $x, y ...
4
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1answer
71 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 ...
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1answer
28 views

Functional equation over $f(x) = \int_0^{ax}f(t)dt + g(x)$

Let $a\in(-1,1)$ and $g\in C^{\infty}(\mathbb{R}, \mathbb{R})$. Let $S(a, g)$ the set of all f such that : $$f(x) = \int_0^{ax}f(t)dt + g(x)$$ The first part was to show that : $$S(a, 0) = ...
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3answers
507 views

Improvement IMO 1988 $f(f(n))=n+1987$

The following problem was given at IMO 1987. Prove that there is no function f from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$. So I tried to ...
0
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1answer
47 views

Technique to compositive functional equation

What is function $f,g:\Bbb R^+\rightarrow\Bbb R$ sought that satisfies $$\forall x\in\Bbb N,\,f_{(r)}(x)=\underbrace{f(f(\dots(f(f(x)))\dots))}_{r \,\mathsf{times}}=2^{(\log x)^c}$$ $$\forall ...
3
votes
1answer
97 views

Functional equation defined over non-negative real numbers

I'm new to this forum and I don't know how to write mathematical symbols. I have the following functional equation: $f$ defined on $[0, +\infty)$ with values in $[0, +\infty)$ $f$ is bijective and ...
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1answer
145 views

Functional Equation $f(f'(x))=-f(x)$

Assume $f:\mathbb{R}_{>0}\to\mathbb{R}$ is differentiable and satisfies $\forall x>0:f(f'(x))=-f(x)$. What is $f(x)$? I know that $f(x)=\ln x$ is a solution, but I don't know if there is ...
1
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1answer
90 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
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2answers
185 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get ...
15
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4answers
542 views

Find all bijections $\,\,f:[0,1]\rightarrow[0,1],\,$ which satisfy $\,\,f\big(2x-f(x)\big)=x$.

A friend of mine gave me the following problem: Find all functions $f:[0,1]\to[0,1]$, which are one-to-one and onto and satisfy the following functional relation: $$ f\big(2x-f(x)\big)=x, \tag{1} $$ ...
3
votes
2answers
153 views

To find an extremal of a given functional

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
4
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1answer
445 views

non-continuous function satisfies $f(x+y)=f(x)+f(y)$

As mentioned in this link, it shows that For any $f$ on the real line $\mathbb{R}^1$, $f(x+y)=f(x)+f(y)$ implies $f$ continuous $\Leftrightarrow$ $f$ measurable. But how to show there exists such ...
3
votes
2answers
682 views

Finding all continuous functions such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$

I've been working on the following homework problem: Find all continuous functions $f : \mathbb{R} → [0,∞)$ such that $f^2(x + y) − f^2(x − y) = 4f(x)f(y)$ for all real numbers $x, y$. The first ...
0
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0answers
39 views

Prove $\lim\limits_{x \to +\infty} \frac{f_1(x)}{f_2(x)} = \text{Constant}$

Let $g(x)$ be a real-analytic strictly rising function for $x>0$. Define for $x>1$ two real analytic functions $f_1,f_2$ such that : $$f_1(x) - f_1(x-1) = f_1(g(x))$$ $$f_2'(x) = f_2(g(x))$$ ...
3
votes
1answer
228 views

What is known about functions of type $f(n+1) = f(n)^ {f(n)}$?

Update : having looked at Knut's Double Arrow Notation ( Thank you DJC), it seems that this question is nothing more than a frivourless wondering that should be undertaken by whom ever wonders it, it ...
7
votes
2answers
131 views

For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

Let $G$ be a group and $f : G \to G$ a function such that for all $x,y\in G$: $$f(x f(y)) = f(x) y.$$ Prove that $f$ is an isomorphism. There are two problems here: we don't know that $f$ is a ...
2
votes
4answers
372 views

Problem in solving functional equation.

To find all functions $f$ which is a real function from $\Bbb R \to \Bbb R$ satisfying the relation $$f(x^2 + yf(x)) = xf(x+y)$$ It can be easily seen that the identity function $i.e.$ $f(x)=x$ and ...
9
votes
4answers
168 views

Solve $f\left(\frac{x-3}{x+1}\right)+f\left(\frac{x+3}{x+1}\right)=x$ $\forall x\neq -1$

Given function $y=f(x)$ such that $$f\left(\frac{x-3}{x+1}\right)+f\left(\frac{x+3}{x+1}\right)=x \quad\forall x\neq -1$$ find $f(x)$ and $f(2007)$.
4
votes
2answers
70 views

Solve the functional equation $2f(x)=f(ax)$ for some $a$.

I am trying to solve the following functional equation, and could use some help.$$ 2f(x)=f(ax)$$ For some $a\in\mathbb{R}$. By repeated adding $2f(x)$ together we notice that $$2nf(x)=f(a^nx).$$ ...
0
votes
2answers
93 views

$f(xy)=\frac{f(x)+f(y)}{x+y}$ Prove that $f$ is identically equal to $0$

For all $x,y\in\mathbb{R}$. also $f : \mathbb{R} → \mathbb{R}$ and $x+y\not=0$. My attempt: I restated it as $a[x^2 y^2 (\frac{x}{y}+\frac{y}{x}-\frac{1}{y^2}-\frac{1}{x^2})] + ...
0
votes
1answer
126 views

Solutions to functional equation $ \gamma(s,t)=f(t \cdot g(s))+h(t) $

Let $$ \gamma(s,t)=f(t \cdot g(s))+h(t) $$ where $\gamma$ is a known function of $s \in \mathbb{R}$ and $t \in \mathbb{R}$ while $f$, $g$, and $h$ are unknown functions. Assume ...
4
votes
0answers
49 views

Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...