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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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2
votes
2answers
74 views

Proof verification: system of functional equation

A problem in Putnam Competition 1992(?). The question asked: Prove that, the only solution of the system of functional equation with respect to $f:\mathbb Z\to\mathbb Z$:$$ \begin{cases} f(f(n))=n\\...
0
votes
0answers
36 views

Only zero in kernel of every element of a subspace of the dual does not imply density of said subspace for non-reflexive Banach spaces

Let X be a Banach space and $M \subset X'$ a subspace of its dual space. If X is reflexive we know the following statements are equivalent: (i) $M$ is dense in $X'$ (ii) $x \in X$ and $\phi(x)=0$ ...
5
votes
1answer
175 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 : $$ \forall x \in \mathbb{R}, \; f(x) + f(2x) + f(4x) = \lfloor 7x \rfloor \tag{1}$$ ...
0
votes
1answer
60 views

How to solve the functional equation $f(x^n+y^n)=f(x^n)+f(y^n)$ for any positive integar $n$?

I got some problems when solving the functional equation $f(x+y^n)=f(x)+[f(y)]^n, (x,y\in\mathbf{R}) $ for all positive integar $n$. I tried to solve it as following: =================================...
2
votes
1answer
79 views

Find the sum of all values of $f(2017)$ given $f^{f(a)}(b) f^{f(b)} (a) = [f(a+b)]^2$.

Let $f:\mathbb N\rightarrow \mathbb N$ be an injective function such that $$f^{f(a)}(b) f^{f(b)} (a) = [f(a+b)]^2$$ for all $a,b \in \mathbb N$. Let $S$ be the sum of all possible values of $f(2017)$...
2
votes
1answer
68 views

Find all functions such that $f(1+xf(y))=yf(x+y)$ where $x,y \in R^+$

Find all functions run over positive real numbers such that $f(1+xf(y))=yf(x+y)$ where $x,y\in R^+$ MY ANSWER: Putting $x=y=0$,we get, $f(1)=0$ Putting $x=0$ we get, $f(1)=yf(y)$ or,$yf(y)=0$ ...
5
votes
1answer
98 views

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$

Find all functions $f: \mathbb{R}\setminus\{0\} \to \mathbb{R}$ satisfying $3f(x) - f\left(\frac{1}{x}\right) = x^2$. What I did was first plug in $x = 1$ to get $2f(1) = 1 \implies f(1) = \frac{1}{...
6
votes
1answer
342 views

Integer functional equation $f(f(f(n)))=f(n+1)+1$

Can you find all functions $f:\mathbb N\rightarrow\mathbb N$ satisfying the functional equation $$ f(f(f(n)))=f(n+1)+1 $$
6
votes
1answer
127 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 ; ...
16
votes
3answers
844 views

3rd iterate of a continuous function equals identity function

If $ f: \mathbb{R} \to \mathbb{R} $ is continuous, and $\forall x \in \mathbb{R} :\;(f \circ f \circ f)(x) = x $, show that $ f(x) = x $. The condition that $f$ is continuous on $\mathbb{R}$ is ...
7
votes
4answers
5k views

If $f(f(x))=x^2-x+1$, what is $f(0)$?

Suppose that $f\colon\mathbb{R}\to\mathbb{R}$ without any further restriction. If $f(f(x))=x^2-x+1$, how can one find $f(0)$? Thanks in advance.
2
votes
1answer
175 views

About the linear functional equations: f(x+a) = bf(x) and f(ax) = bf(x).

About the linear functional equations: $f(x + a) = bf(x)$ and $f(ax) = bf(x)$, Marek Kuczma e Polyanin A.D. they got the respective solutions (http://eqworld.ipmnet.ru): $f(x) = g(x)b^{x/a}$, where $...
1
vote
2answers
115 views

Solution of the equation $f(ax) = bf(x)$

Given the equation $f(ax) = bf(x)$, with $a, b > 0$, demonstrate that the solution is: $$f(x) = g(\log x)x^{\frac{\log b}{\log a}}$$ where $g(x) = g(x + \log a)$ is an arbitrary periodic function ...
1
vote
2answers
54 views

A problem on number of solutions of a functional equations

Find all functions $f:R \rightarrow R$ such that $f(0)=1$ and for all $x\neq -1$ : $f(x)=8f(2x+1)$ (I have found only one solution: $1/(x+1)^3$. Method was by iterated substitution of $2x+1$ ...
5
votes
0answers
58 views

Does there exist a function $f_{\Box,\Box}(\Box)$ making the formula $a + (b \oplus c) = (f_{b,c}(a)+b) \oplus (f_{c,b}(a)+c)$ true?

Let $a$ and $b$ denote the resistances of two resistors. If they're put in series, the total resistance is $a+b$. If they're put in parallel, the total resistance is $$a \oplus b := \frac{1}{\frac{1}{...
1
vote
0answers
18 views

How to extract associative binary operations from a class of binary operations on $\mathbb{R}$

I would like to find (all) associative binary operations of the form $$u_{1}*u_{2}=\ln{\left[e^{u_{1}}+e^{u_{2}}\right]}+Q\left(u_{2}-u_{1}\right),$$ where $Q$ is an arbitrary function. My effort: ...
1
vote
1answer
73 views

Prove that $\forall x \in \mathbb{R}, f(x)=0$ [duplicate]

Suppose $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable function and satisfies $$\forall x \in \mathbb{R}, \vert f'(x)\vert \leq \vert f(x)\vert, \quad f(0)=0.$$ Prove or disprove $f(x)=0$ How ...
1
vote
1answer
18 views

Suppose $L$ has a regular parametrix . Assume $U$ is a distribution given in an open set $\Omega \subset R^d$ and $L(U)=f$ , then $U$ is $C^{\infty}$

Suppose $L$ has a regular parametrix . Assume $U$ is a distribution given in an open set $\Omega \subset R^d$ and $L(U)=f$ , with $f$ a $C^{\infty}$ function in $\Omega$ , then $U$ agrees with a $C^{\...
20
votes
5answers
4k views

Prove that function is constant

Prove that a function $f:\mathbb{R}\to\mathbb{R}$ which satisfies $$f\left({\frac{x+y}3}\right)=\frac{f(x)+f(y)}2$$ is a constant function. This is my solution: constant function have derivative $0$ ...
0
votes
2answers
64 views

Existence of $f(x)$

Suppose $g(x)$ is cubic which has two local extrema. Is there differentiable function $f(x)$ which satisfies $\forall x \in \mathbb{R}, g(f(x))=x$ exist? I know if I make $f$ piecewise inverse of $g$...
1
vote
1answer
77 views

I would like to go for search for this given functional equation using either java or python $f(x+1) = f(x)^2-1$

I would like to go for search for this given functional equation using either java or python $$ f(x+1) = f(x)^2-1 $$ $$ f(0) = 1 $$ I don't know where to start. I know how to graph in pycharm. I know ...
2
votes
1answer
54 views

What is the solution of this recursion, that's defined in terms of a sum, but with this $1$ odd twist?

$$ F(n) = \sum_{i=0}^{n} F\left(\left\lfloor\frac{i}{5}\right\rfloor\right) $$ I encountered this odd looking functional equation, while perusing the site yesterday. I'd be interested in seeing a ...
0
votes
4answers
83 views

Existence of the function $f(x)$

Let $f\colon\mathbb{R}\to\mathbb{R}$ be differentiable function that satisfies $$ f(0)=1 \\ \forall x \in \mathbb{R}, \quad f(x+1)=\exp(3x^2+1)f(x)$$ I think a function $f$ exist which satisfies ...
1
vote
1answer
43 views

Functional equation $f((xf(x))^2 + f(y))=-x^4 + y$

Problem Functional equation Suppose $f\colon\mathbb{R}\to\mathbb{R}\quad$ $\forall x, y \in \mathbb{R}, f((xf(x))^2 + f(y))=-x^4 + y$ What I found : Put $x=y=0,$ then $f(f(0))=0$ And put $x=f(0)...
2
votes
0answers
186 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. ...
0
votes
0answers
39 views

Functional Equation with two conditions

Let function $$ f\colon \mathbb{R}\to\mathbb{R}$$ is continuous function and satisfies both conditions : $$ \forall x\in\mathbb{R}, f(f(x))=x \tag{1}$$ $$ \forall x\in\mathbb{R^+}, \int_{-x}^0f(t)dt-\...
6
votes
2answers
286 views

$f(\alpha x) = f(x)^{\beta}$ under different constraints

With $\alpha > 0,\, \beta \in \Bbb R^*,\, \alpha, \beta \neq 1$ and $f : \Bbb R \to \Bbb R_+^*$, let's consider the functional equation $$ f(\alpha x) = f(x)^{\beta} \tag{$\Xi$}$$ or equivalently ...
1
vote
0answers
37 views

Solve the functional equation $f\left(x\right) = 1 - \left(1 - f\left(x+1\right)\right)^{\frac{x}{x+1}}$

Trying to find a concave function defined on the positive reals, satisfying some inequalities, I came up with the following relation $f\left(x\right) = 1 - \left(1 - f\left(x+1\right)\right)^{\frac{x}...
5
votes
2answers
116 views

How to solve this D.E $y''(\frac{x}{2})+y'(\frac{x}{2})+y(x)=x$

I know how to slove $y''(x)+y'(x)+y(x)=x$ But I couldn't solve this $$y''(\frac{x}{2})+y'(\frac{x}{2})+y(x)=x$$ any hint to help me? Thanls
0
votes
1answer
41 views

Differentiating Exponential Functional Equation [duplicate]

The Functional Equation satisfied by the exponential $f(x)=e^{kx}$ is of the form: $$ f(x+y)=f(x)f(y), \quad f(0)=1, f'(0)=k $$ Show that $f'(x) = kf(x)$. Attempt I tried applying Chain Rule to ...
0
votes
2answers
49 views

Question on Function of Function.

$$f(x)=\frac{x+2}{1-2x}$$ $$g(x)=\frac{2x+1}{2-x}$$ Find $$(fofofo...ofOgogo...og)=\frac{1}{x}$$ {fofo... are 101 times and gogo.. are 100 times} Then Find $x$? I calculated as follows Since $$(...
2
votes
1answer
70 views

Find all function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$

If $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and $a,b,c>0$, then find all function such that : $$f(ax)f(by)=f(ax+by)+cxy,\quad \text{where } a,b,c>0 \text{ for all } x,y\in \Bbb{R}.$$ My attempt ...
1
vote
2answers
64 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
3answers
93 views

Is there any functional equation $f(ab+cd)= f(a)+f(b)+f(c)+f(d)$?

I am looking for a real, continuous function that satisfies the functional equation $$ f(ab+cd)= f(a)+f(b)+f(c)+f(d) $$ where $a,b,c,d$ are real. This is equivalent to a function satisfying these two ...
4
votes
2answers
107 views

find all possible functions : $f(a)f(b)-6ab=\frac{3}{2}f(a+b)$

I'm trying to find all possible functions that satisfy this functional equation: $f(a)f(b)-6ab=\frac{3}{2}f(a+b),$ $f\in \mathbb{R}.$ My attempt : $a=b=0$ then $f(0)=0$ or $1.$ But I don't ...
0
votes
1answer
44 views

Functional equations in one variable.. [closed]

How do you solve the functional equation involving only one variable...what if and if not given that $f(x)$ is a polynomial... Say for example $f(x)=f(x-1) +2x$
1
vote
0answers
32 views

$‎\lim_{n\to\infty} (xf(n) + \sum_{k=1}^n f(k) - f(k+x))‎$‎ is‎ ‎convergent for $x\geq 1$. [closed]

‎‎Let $f:[1‎, +‎\infty)\rightarrow\mathbb{R}$ be a function such that ‎$‎\lim_{n\to\infty} (f(n) - f(n+1)) = 0‎$‎‎. ‎Suppose that $\lambda:[1‎, +‎\infty)\rightarrow\mathbb{R}$‎ ‎is an exp-convex ...
2
votes
1answer
98 views

IMO 1993 b2 proof

"Let $\mathbb{N}=\{ 1,2,3,\ldots \}$. Determine if there exists a strictly increasing function $f:\mathbb{N}\to\mathbb{N}$ such that 1) $f(1)=2$ 2) $f(f(n))=f(n)+n$ for all $n$." Of the solutions ...
3
votes
2answers
103 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 ...
0
votes
2answers
143 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 ...
2
votes
1answer
134 views

Find functions $f,g$ such that $f(x+y)=g(x·y)$ for all $x,y$? [closed]

My approach towards this question was that first I putted x=0 and then y=0 which yields f(x)=f(y)and f(x)=g(0), and again for x=1 & y=1 it gives g (x)=f(1+x), and so on. My query is that what ...
4
votes
0answers
78 views

Solving the equation $f(x)=f^{-1}(x)$. [duplicate]

Exactly under what conditions would the equality $f(x)=f^{-1}(x)$ hold? The proof attached below considers a special case when $f$ is strictly increasing. The theorem then says that the set of ...
4
votes
2answers
98 views

Functional problems: Find all functions such that $f(x)f(y) = f(xy + 1) + f(x - y) – 2$

Find all functions such that $f(x)f(y) = f(xy + 1) + f(x - y) – 2$ for all $x, y $ are real numbers. I put y=0 into the equation and get $(f(0)-1)f(x)=f(1)-2$. If $f(0)≠1$, then $f(x) = (f(1)-2)/...
0
votes
2answers
64 views

Power-like functional equation

I would like to know what are the especifications of a functional equation that give us a power function as a solution. For example, if $f:\Bbb R \to \Bbb R$ is continuous and monotonic, such that $$...
2
votes
0answers
53 views

Functions satisfying $f(2x)+f(3x)\leq k f(x)$

Given $a<0$ such that $k:=2^{a}+3^{a}\in (0,1)$ and define $f(x):=x^{a}$ for all $x>0$. Then, $f(2x)+f(3x)=x^{a}(2^{a}+3^{a})=kf(x)$. Somebody know another example of a function $f:(0,+\infty)\...
0
votes
0answers
33 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 ...
2
votes
2answers
466 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 ...
6
votes
1answer
82 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
61 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
27 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 ...