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).

1
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0answers
52 views

Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ so that $f(x)f(y)- \frac{4}{9} xy= f(\!x+ y\!)\,(\!\forall x,\,y\in \mathbb{R}\!)$ .

Problem. Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)f(y)- \dfrac{4}{9}\,xy= f(x+ y)\,\,(\!\forall x,\,y\in \mathbb{R}\!)$ (1). My above problem given a solution, and I'm ...
0
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2answers
85 views

Find all functions for $f:\Bbb{N}\to\Bbb{N}$ such that $f\left(m^2+f(n)\right)=f\left(m^2\right) +n$

I would have given my approach but i didnt get anywhere. I just substituted zeroes and got $f(f(n)) =n$ and I'm just lost. Any help would be appreciated
1
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1answer
72 views

$f:\mathbb{R_{\geq 0}} \to \mathbb{R_{\geq 0}}$ such that for all $x$ we have $xf(1+xf(y))=f(f(x)+f(y))$

Find all nonnegative real number $a$, such that $f(a)=0$ for any function $f$ satisfying: $xf(1+xf(y))=f(f(x)+f(y))$ with all $x,y$ are nonnegative real number. I don't know why this problem only ...
0
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0answers
48 views

Functions $f$ that $f(f(x))=x$, but $f:S^1\to S^1$

Background Denote $e_A$ the identity map from $A$ to itself. Questions such like solving $f$ in the functional equation $f\circ f=e_\mathbb{R}$ or $f\circ f=e_{\mathbb{R}\setminus\{a_1,\ldots,a_n\}}$ ...
2
votes
3answers
57 views

Functional equation for tan

If $f$ is a differentiable function on $\mathbb{R}$ and $f'(0)=2$ satisfying $$f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)},$$ then to prove that $f(x)=\tan 2x$. I know that we must prove using the first ...
0
votes
1answer
73 views

Number of solutions of the equation $e^{f(x)}=f(x)+2$ [on hold]

Let $f$ be an everywhere differentiable function, and suppose that $f(x)=0$ has a unique solution, and suppose that $f$ has no local extreme points. What is the number of solutions of the equation $...
0
votes
1answer
88 views

$f(x+y)=f(x)f(y)-f(xy)+1$ , $f\left(\frac {2017}{2018}\right)=\frac mn$ .Then $(m-n)=?$ [duplicate]

For any rational numbers $x$ and $y$, $f(x)$ is a real number and $$f(x+y)=f(x)f(y)-f(xy) +1$$ We also have that $f(2018)$ is not equal to $f(2017)$. Given that $f\left(\frac {2017}{2018}\right)=\...
0
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2answers
51 views

Function of a Function Differential Equation [duplicate]

Is there any function, $f(x)\neq x$, for which $f(f'(x))=f'(f(x))$?
0
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0answers
63 views

$f(e^x) = e^{f(x)}$, what is f? [duplicate]

Find all functions $f$ and their domains, such that $f(e^x) = e^{f(x)}$ I have verified that the functions below satisfy the equation for certain domains. Would these be the only solutions? But how ...
6
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2answers
148 views

Finding all positive real functions satisfying $xf(y)+f(f(y))\leq f(x+y)$

Find function $f: \mathbb{R}_{> 0}\rightarrow \mathbb{R}_{> 0}$ such that: $xf(y)+f(f(y))\leq f(x+y)$ for all positive $x$ and $y$? That problem made me think a lot. This is the first time I ...
3
votes
1answer
81 views

$f:\mathbb{R} \rightarrow \mathbb{R}$, $f(xf(y)+f(x))=2f(x)+xy$

So far I've only got that $f(x) = x + 1 \qquad\forall x \in\mathbb{R}$ is probably the only solution, and that Substitute (1,y): $f(f(y)+f(1))=y+2f(1) \implies f\text{ surjective}$ $f(x)=f(y) \...
-1
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1answer
64 views

Solve the functional equation $f(x+1)-f(x)=x*\sin(x) $ [closed]

Solve $f(x+1)-f(x)=x*\sin(x) $
2
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2answers
74 views

find functions $f$ such that $f(x)+f(y) = f(g(x,y))$, $g$ is given and symmetric

I want to find solutions $f$ of the following functional equation given a function $g(x,y)$, which is symmetric ($g(x,y)= g(y,x)$) and strictly monotonic $\forall x,y \in $ Reals: $f(x)+f(y) = f(g(x,...
0
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0answers
48 views

I can compute a function $F(x)$ such that $F(x(1/2-x))= F(x)/2$, It is analytic on a filled Julia set.

I have been studying a function $F(x)$ obeying $F(p(x))=F(x)/2$. I did numerical work for $p(x)=x(1/2-x)$, although a similar functional equation could be solved for any polynomial with an attractive ...
2
votes
1answer
63 views

How many functions can be defined from natural number to natural number such that LCM(f(n),n) -HCF(f(n),n) is less than 5.

How many functions $f:\mathbb{N}\to\mathbb{N}$ can be defined from natural number to natural number such that $$\text{LCM}(f(n),n) -\text{HCF}(f(n),n) <5$$ LCM is least common multiple HCF is ...
3
votes
1answer
44 views

Restricted Cauchy equation on a non-dense domain

It is really well known that if $f: \mathbf{R}\to \mathbf{R}$ is continuous and $$ \forall x,y \in \mathbf{R},\,\,\,\,f(x+y)=f(x)+f(y) $$ then $f$ is linear, i.e., there exists $a \in \mathbf{R}$ such ...
1
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1answer
27 views

Uniqueness of the solution of non-linear ODE of second order

Let $n$ be an integer with $n>3$ and $f \colon [0,\infty ) \to \mathbb{R}$ be a solution of $t^{1-n}(t^{n-1}f'(t))'=f(t)|f(t)|^{\frac{4}{n-2}}$ with initial values $f(0)=a$ and $f'(0)=0$. Then, is ...
0
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0answers
59 views

If $f: N\rightarrow N $ be such that $f(f(n))+f(n+1)=n+2$ for all $n\in N$ find $f(1)$ and $f(2)$. [duplicate]

Putting $n=1$, we get $f(f(1))+f(2)=3$. Thus there are two possibilities. Either $f(f(1))=1, f(2)=2$ or $f(f(1))=2, f(2)=1$. Also, we observe that $f(f(n))=n+2-f(n+1)\leq n+1$ and $f(n+1)=n+2-f(f(n))\...
1
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2answers
78 views

How do I find a function $f$ such that $f(x^2)=2f(x)$? [duplicate]

Does there exist a continuous function $f$ such that $f(x^2)=2f(x)$ and $f(0)=0$? How do you solve this? I understand that this is nothing like a normal equation, because you can't solve for $x$, or $...
1
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0answers
97 views

What is a function $f$ such that $f(f(x))=2f(x)$ and $f(0)=1$?

Is there a function $f$ such that $f(f(x))=2f(x)$ and $f(0)=1$? I don't know how to attack this problem. How do I solve an equation for a function?
2
votes
2answers
34 views

$f(x,y) - f(x,z) = g(y,z)$ implies $f(x,y) = a(x) + b(y)$

A result I need is: If $f(x,y) - f(x,z) = g(y,z)$ for all $(x,y,z)$, then $f(x,y) = a(x) + b(y)$ for some functions $(a,b)$. This seems almost obvious, and I've constructed a proof, but that proof ...
2
votes
1answer
56 views

Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ subject to the conditions: [duplicate]

Find all functions $f:(0,\infty)\rightarrow(0,\infty)$ subject to the conditions: $f(f(f(x)))+2x=f(3x)$ for all $x>0$ and $\displaystyle\lim_{x\to\infty}(f(x)-x)=0$ I tried as follows: Suppose $...
28
votes
3answers
790 views

$f(ax)=f(x)^2-1$, what is $f$?

Suppose $f(ax)=(f(x))^2-1$ and suppose that $f$ is analytic in some neighborhood of $x=0$. Expanding in power series, we get $a=1+\sqrt{5}$ or $1-\sqrt{5}$. We take positive $a$. If $f\neq{\rm const}$ ...
4
votes
3answers
232 views

Complicated rational number functional equation

Let $\mathbb{Q}^+$ denote the set of positive rational numbers. Let $f : \mathbb{Q}^+ \to \mathbb{Q}^+$ be a function such that $f \left( x + \frac{y}{x} \right) = f(x) + \frac{f(y)}{f(x)} + 2y$ for ...
10
votes
1answer
336 views

Which positive continuous functions satisfy $F(x) = F(e^x)-F(e^{-x})$ for $x\geq 0$?

There is at least one such function. It is the cdf of the equilibrium probability distribution for the chaotic sequence $x(n+1) = |\log x(n)|$ with $x(1) = 2$. Its graph (approximation) is pictured ...
2
votes
1answer
66 views

Find all function $f$ : $f(x+y)=e^{2xy}f(x)f(y)$

Problem: Suppose $f\colon\mathbb{R}\to\mathbb{R^+}$ is differentiable function which satisfies $f'(0)=1$ and $$\forall x, y \in \mathbb{R}, \quad f(x+y)=e^{2xy}f(x)f(y)$$ Where $\mathbb{R^+}$ is set ...
0
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0answers
32 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 $\phi(x)=0$ ...
6
votes
1answer
100 views

Differential Equation with inverse function $\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$

$$\frac{1-f^{-1}\left(\frac{f(x)}{x}\right)}{1-x} = 1- \frac{f(x)}{xf'(x)}$$ I know $f(x) = ax+b$ is a solution. How can I find other solutions?
1
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2answers
53 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
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0answers
46 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
2answers
109 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
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0answers
16 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
69 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 ...
2
votes
1answer
127 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 $...
17
votes
2answers
628 views

Solve the functional equation $f(xf(y)+yf(x))=yf(x)+xf(y)$

Let $f:\mathbb{R}\to \mathbb{R}$ and such for any real numbers $x,y$ we have $$f(xf(y)+yf(x))=yf(x)+xf(y)$$ Find $f(x)$. I have let $x=y=0$ have $$f(0)=2f(0)\Longrightarrow f(0)=0$$ and I guess the ...
2
votes
4answers
307 views

$2f(x)=f(y) \Rightarrow 2f(tx)=f(ty)$

Find all continuous and strictly monotonic function $f:[0,\infty)\to \Bbb R$ such that: If there is a pair $(x,y)\neq (0,0)$ such that $2f(x)=f(y)$ then $2f(tx)=f(ty)$ for all $t>0$; There is at ...
1
vote
1answer
15 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^{\...
0
votes
2answers
63 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
73 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 ...
1
vote
1answer
40 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)...
0
votes
4answers
81 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 ...
0
votes
0answers
37 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-\...
1
vote
0answers
36 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
112 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
3answers
46 views

Functional equation with three variables

I have a functional equation with three variables. $f(x,y,z)$ is a real function with three variables where y is different from z i.e., $f(x,y,z)$ defined only for $y \neq z$. This function satisfies ...
0
votes
1answer
32 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
48 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 $$(...
1
vote
3answers
82 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
104 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
40 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 $f(x)=f(x-1) +2x$