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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6
votes
3answers
156 views

functions satisfying $f(x)=2f(2x)$

How can I prove every smooth real function f satisfying $f(x)=2f(2x)$ is of the from $f(x)=k/x$ where $k$ is a constant? I have tried this for integers that can be divided by $2^n$, but then I cannot ...
0
votes
0answers
27 views

A functional equation for a Dirichlet serie

I'm looking for a functional equation for the following Dirichlet serie $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ? Thank you !
1
vote
0answers
22 views

What is the function $f$ verifying : $f(\frac{x}{2}+\frac{x}{2}\cos(\frac{v\pi}{x}))=\frac{x}{2}\sin(\frac{v\pi}{x})$

What are the solutions to the functional equality (for a constant $v$): $$ \forall\, x > 0, \ \ \ \ ...
0
votes
1answer
53 views

Functions such that $f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$

What are the continuous functions $f:\mathbb{R} \to \mathbb{R}$ such that for every $x,y\in \mathbb{R}$ $$f(\frac{x+y}{2})=\frac{1}{2}f(x)+\frac{1}{2}f(y).$$
4
votes
4answers
115 views

Find all functions $f(x)$ such that $f(x^2-y^2)=(x-y)(f(x)+f(y))$.

find all functions $f(x):R\to{}R$ such that $f(x^2-y^2)=(x-y)(f(x)+f(y))$. I have derived this clues:- $f(0)=0$, $f(x^2)=xf(x)$, $f(x)=-f(-x)$ but now I am confused. I know solution will be ...
0
votes
1answer
46 views

Functional equations and cubes

Problem $10728$ from Amer. Math. Monthly "Preserving the sum of three cubes" says: Determine all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying $f(x^3+y^3+z^3)=(f(x))^3+(f(y))^3+(f(z))^3$ ...
-2
votes
1answer
74 views

Solve this functional equation [closed]

A function $f: \mathbb R \rightarrow \mathbb C$ is such that $f(0) = 1$, $f(-t) = \overline{f(t)}$ and $\mathrm{Re} f(t) = f(t) \overline{f(t)}$. Solve for $f$. If you don't mind.
1
vote
0answers
18 views

I want to solve these inequalities with respect to $f$

Let $f\colon\mathbb C\to\mathbb C$ be an entire non constant function. We consider its values on the real line. The function $f$ has infinitely many real zeros and there is infinitely many real ...
0
votes
2answers
57 views

Find all $f:\mathbb R\to\mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $xf(y)+yf(x)=(x+y)f(x)f(y)$.

Find all $f:\mathbb R\to \mathbb R$ such that $\forall x,y\in\mathbb R$ the given equality holds: $$xf(y)+yf(x)=(x+y)f(x)f(y)$$ My try: whenever $y=0$, we have $$x\cdot f(0)\cdot(1-f(x))=0$$ ...
7
votes
1answer
110 views

All functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$

I just thought about the following question: Find all functions $g:\mathbb{N}\to\mathbb{Z}$ such that $m+n~|~g(m)+g(n)$ for all $m,n\in\mathbb{N}$. Clearly every polynomial $g(X)\in\mathbb{Z}[X]$ in ...
0
votes
1answer
41 views

Does this functional equation have a unique solution?

I wish to prove/disprove that there exists a unique solution to the functional equation $$xyF(xy^2, y) = F(x, y), \quad x \ne 0, \quad |y| < 1, \quad y \ne 0,$$ where $F(x, y)$ is continuous. I ...
3
votes
1answer
60 views

Are Exponential and Trigonometric Functions the Only Non-Trivial Solutions to $F'(x)=F(x+a)$?

Are exponential & trigonometric functions the only non-trivial solutions to $F'(x)=F(x+a)$? $F(x)=0$ would be the trivial solution. Then, for $a=0$ (or $a=2\pi i$), we have $F(x)=e^x$, and ...
1
vote
1answer
163 views

A Dangerous Function [closed]

Recently, I have started solving many ques on functional equations. But, this ques for me was tough, $ f(y)^{x^2}+f(x)^{y^2}=f(y^2)^x+f(x^2)^y $ I've started substituting some values, trying to ...
10
votes
1answer
300 views

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
3
votes
1answer
117 views

$f,g$ such that $\int fg = \int f \int g$

Suppose $f,g$ are real valued on $\mathbb{R}$ (and no further restrictions apart from the obvious requirement that the integrals exist), then when does $\displaystyle\int f(x)g(x)\,dx = \int f(x) \, ...
-3
votes
1answer
67 views

Find all the $f:\mathbb{R}\rightarrow\mathbb{R}$,satisfying

$f(f^3(x)+y^3)=x^2+f^3(y)$ where $f^3(x)$ stands for $[f(x)]^3$. I really don't know where to start off$\cdots$
1
vote
0answers
43 views

Proof that $\forall x \in \mathbb{R}, \forall n \in \mathbb{N}: (f(x))^n = f(nx)$ only has solutions of the form $f(x) = c^x$ for some constant $c$

I am looking for a proof (or a counterexample I suppose) that the recurrence equation (is that the right term here?) $(f(x))^n = f(nx)$ only has a solution of the form $ f(x)=c^x$ for some ...
-1
votes
1answer
46 views

An functional equation [closed]

Find all the functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$, such that $\forall w,x,y,z\in\mathbb{R}^+,~wx=yz$, and $$\dfrac{f^2(w)+f^2(x)}{f^2(y)+f^2(z)}=\dfrac{w^2+x^2}{y^2+z^2}$$ $f^2(x)$ ...
10
votes
4answers
164 views

If $f(3x)=f(x)$ and $f$ is continuous, show that $f(x)$ is a constant function.

If $f(x)$ is a continuous function such that $f(3x)=f(x)$ and the domain of $f$ is all non-negative real numbers. Prove that $f$ is a constant function. What I did: ...
0
votes
1answer
34 views

If $f$ is not identically zero and $|f(x)| \le 1$ for $x$ $\in \mathbb{R}$, then also $|g(x)| \le 1$ for $x \in \mathbb{R}$

Suppose that $f$ and $g$ satisfy the equation $f(x+y)+f(x-y)=2f(x)g(y)$, $x$,$y$ $\in \mathbb{R}$. Show that If $f$ is not identically zero and $|f(x)| \le 1$ for $x$ $\in \mathbb{R}$, then also ...
2
votes
2answers
89 views

Functional inequation on $\mathbb{R}$: $f(x+y^2)-f(x)\geq y$

I have the following equation: $$f:\mathbb{R}\rightarrow\mathbb{R}$$ $$\forall (x,y)\in\mathbb{R}^2,\ f(x+y^2)-f(x)\geq y$$ f is not necessarily differentiable/continuous/... (In fact, we can prove ...
1
vote
0answers
33 views

Can a linear solution make a non-linear (functional) differential equation linear?

I was inspired by this question Does a non-trivial solution exist for $f'(x)=f(f(x))$? And tried coming up with similar problems, one interesting case I found was $f'(x) +f(x)=f(f(x))$ which has ...
2
votes
3answers
161 views

Does a solution to this functional equation exist and if so can we construct it?

For $x\geq 0 $ we have $f(x) +xf(1/x) = x/(1+x)$ as well as the conditions $\lim_{x\rightarrow 0} f(x) = 0$ and $\lim_{x\rightarrow \infty} f(x) = 0$. Clearly $f(1) = \frac{1}{4}$. What is the ...
7
votes
2answers
118 views

Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$ f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0. $$ Can someone give explicit ...
4
votes
3answers
112 views

Functional equation $f(x)=f(\sqrt{x})$

If we take an equation $f(x)=f(\sqrt{x})$ defined for positive $x$ then it is quite easy to see that it is constant; $f(x)=f(0)$ if continuous at zero. My question is: What would happen if we take ...
0
votes
1answer
34 views

Finding equations when given new center of a circle

$y = −x + \sqrt{2}$, $y = −x − \sqrt{2}$, $y = x + \sqrt{2}$, and $y = x − \sqrt{2}$. These equations determine lines, which in turn bound a diamond shaped region in the plane. Construct a diamond ...
4
votes
1answer
59 views

Solving the functional equation $f(x+y)-f(x)f(y)+g(x)g(y)=0$

As in the title I want to solve the functional equation $$f(x+y)-f(x)f(y)+g(x)g(y)=0 \tag{1} $$ provided that $f,g$ are differentiable for all real values, and that $f$ is an even function. My ...
2
votes
1answer
33 views

A problem about functional equations

We want to find all continuous functions $f:R→R$ that satisfy the equation $f(x^2+1/4)=f(x)$ for all real x. Of course -If I am right- constant functions satisfy the equation mentioned, and as well ...
1
vote
2answers
67 views

Functional equation (is solution unique)

Let $f:[0,1]\rightarrow \mathbb{R}$ - cont. diff. function. Is it true that equation $\cos{t}f(\sin{t})+\sin{t}f(\cos{t})=1, t \in [0,\pi/2]$ has only solution $f(x)=\sqrt{1-x^2}$? How can we prove ...
0
votes
0answers
24 views

ODE in case of discontinuity

Consider a function $\varphi(x) : \mathbb{R} \rightarrow \mathbb{R}$ bounded and continuous and $c \in \mathbb{R}$. Question 1 Is there a unique $f : \mathbb{R} \rightarrow \mathbb{R}$, among all ...
4
votes
1answer
92 views

Tricky probability problem

I am having trouble with proving the following assertion: $X,Y$ are i.i.d. with mean $0$ and variance $1$. If $X+Y$ and $X-Y$ are independent then $X,Y$ are normally distributed. Should I be ...
3
votes
1answer
89 views

$f_{n+1}(x)=f_n(x+1)-f_n(x)$ functional equation and “classification of functions”

Doing a quiz I found a question of this kind "given $a_0, a_1, a_2, ...,a_n$ find $a_{n+1}$" In order to find the $f$ such that $f(a_n)=a_{n+1}$ I tryed for a function like $f(x)=k+x$ ...
1
vote
1answer
42 views

$f(x + y)f(x − y) = ( f(x) + f(y) )^ 2− 4x^ 2 f(y)$ so why $f(2) \neq 2$?

Let $f : \mathbb R → \mathbb R$ be a function that satisfies for all $x,y ∈ \mathbb R $ defined as $f(x + y)f(x − y) = ( f(x) + f(y) )^ 2− 4x^ 2 f(y)$. So, why $f(2) = 2$ is impossible? Why $f(0) = ...
3
votes
2answers
60 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
votes
0answers
22 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 ...
4
votes
1answer
63 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 ...
0
votes
1answer
49 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 ...
0
votes
0answers
27 views

Solving a functional equation with a boundary condition (involving probabilites)

Ok, I am getting a functional equation in $z$ domain given by $F(z)= F(G(z))$ where $G(z)= e^{-a(1-z)}$. I want to get $f(n)$ ($F(z)$ is the $z$ transform of $f(n)$) where $f$ is some pmf, hence we ...
2
votes
1answer
40 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
vote
0answers
23 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 ...
3
votes
1answer
58 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 ...
0
votes
0answers
13 views

Validation of proof,for a functional equation

Sorry for bad English in advance,I had to translate most of things from another language that's why it's pretty messy,I feel like I made a mistake about this Given a natural number $k$.Let ...
2
votes
1answer
53 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
votes
1answer
60 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
3answers
137 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 ...
1
vote
3answers
42 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
74 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
24 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
68 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
votes
0answers
64 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 ... ...