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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5
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1answer
113 views

find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $ [duplicate]

This is a very hard functional equation. the problem is this : find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that : $ f(f(x))=x^2-2 $ to solve it i have no idea! can we solve it ...
3
votes
1answer
106 views

General Solution of Functional Equation

What is the general solution to: $$ \frac{f(x+h)-f(x)}{h} = \frac{1}{x} \tag{1} $$ Obviously the solution to this for the limiting case of $h\to 0$ is $f(x) = \ln(x) + c$ Attempting to solve the ...
3
votes
2answers
68 views

calculation of all function which satisfy $f(x – y) = f(x) f(y) – f(a – x) f(a + y)\;,$ where $f(0)=1$

A real valued function $f(x)$ satisfies the functional equation $f(x – y) = f(x) f(y) – f(a – x) f(a + y)$ Where $a$ is a given constant and $f(0) = 1\;,$ Then prove that $f(2a – x) = -f(x)$. ...
0
votes
0answers
9 views

Measuring availability of a service

I have a task to measure availability of some IT-components. The problem is that I need to create a equation for measuring this. For now the availability is going to be measured through life-cycle ...
3
votes
0answers
61 views

Continuous functions such that $f(x)+f(x^2)=x$ [duplicate]

Find all continuous $f :[0,1]\to \mathbb R$ such that $\forall x\in [0,1], f(x)+f(x^2)=x$ I suspect there are none. I made little progress so far, but it's worth noticing that $f(0)=0$ and ...
5
votes
2answers
175 views

Find $f$ such that $\frac{d^2}{dx^2}f(x)=f(\sqrt{x})$

Which non-constant functions $f$ (if any) satisfy $\dfrac{d^2}{dx^2}f(x)=f(\sqrt{x})$ for $x>0$? I suspect there is no $f$ which satisfies the differential equation, but I cannot prove this.
7
votes
1answer
149 views

Functional equation and fixed points

Let $f$ be strictly increasing and such that $f(x)+f^{-1}(x)+1=e^x$. Is it true that $f$ has at most one fixed point? I am told the answer is yes, but I am having trouble proving it. It's obvious ...
11
votes
1answer
305 views

Additive functional inequality

The function $f:R_+\to R_+$ is continuously differentiable and increasing. Also, $f(0)=0$ and $f(\infty)=\infty$. Continuity and differentiability of higher orders can be assumed if necessary. ...
8
votes
4answers
211 views

Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
1
vote
0answers
21 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
7
votes
4answers
169 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
0
votes
1answer
32 views

A logarithm-like functional equation

Suppose we are given that a monotonically decreasing smooth function $f$ on $(0,\infty)$ obeys the functional equation $f(x) = -f(\frac{1}{x})$, and satisfies $f(\frac{1}{3}) = \frac{1}{2}$ and ...
0
votes
1answer
37 views

Understanding an Equation and how to implement it

A common method for linking language with psychological variables involves counting words belonging to manually-created categories of language. One counts how often words in a given category are used ...
1
vote
0answers
61 views

Are even Dirichlet series constants?

I consider a Dirichlet series with an absolute abscissa of convergence $\sigma$ that can be meromorphically extend to $\mathbb{C}$: $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ and the analytic ...
1
vote
0answers
15 views

Implementing Equation on current data

I am trying to implement Personality, Gender, and Age in the Language of Social Media equation. I have 5 patterns and one list of 100 text = 900 words. The result of find a Match in the 900 to the ...
9
votes
2answers
242 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
6
votes
2answers
111 views

Problem about functional equation (Bulgarian selection team test)

This problem was taken from the bulgarian selection team test for the 47th IMO and appeared in a chinese magazine, I came across it in my own training. http://www.math.ust.hk/excalibur/v10_n4.pdf ...
0
votes
0answers
25 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
6
votes
3answers
157 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
28 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
54 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
121 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
47 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
58 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
113 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
42 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
64 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
169 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
304 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
120 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
48 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
165 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
34 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
162 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
124 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
114 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
40 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
69 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
94 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 $f(x)=k+x$ ...