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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16 views

In which condition related to coefficients, some equation has a solution?

My purpose is to know in which condition related to the coefficients $c>0, \, n\geq 2, 0<p<1,\, a>c\, \, \, \text{and} \, \, b>0$, this equation $$ F(x)= -c x^{n+p} -bx^n + (a-c)x^p ...
3
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1answer
55 views

Solve functional equation $f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a))$

Solve functional equation $$f(a)f(b)=\frac{1}{2}f(a+b)+\frac{1}{2} f(max(|b-a|,a)),(*)$$ where the following conditions are satisfied: 1)$f:\mathbb{R}_{+}\cup \{0\}\rightarrow [0,1] ,$ 2)$f(0)=1,$ ...
2
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1answer
42 views

Soultion of a particular functional differential equation

I need to solve the following functional differential equation: $(4\mu-\lambda r - \lambda x+\lambda)f'(x) = \lambda( f(x) - f(1-x-r))$, where $x\in (0, 1-r)$, $r\in (\frac{1}{2}, 1)$ Or, a more ...
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2answers
48 views

Solve the functional equation $f(x+a+f(y))=f(f(x))+a+y$ [closed]

So let $a$ be a real number. Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $f(x+a+f(y))=f(f(x))+a+y$, for all real $x,y$.
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1answer
51 views

Cauchy's Functional Equation

Consider Cauchy's Functional Equation $$\phi(t+s)=\phi(t)+\phi(s).$$ Can we say that any right continuous with left limits (cadlag) solution is Borel measurable? Obviously continuous solutions are ...
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 $$2nf(x)=f(a^nx).$$ ...
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1answer
43 views

Find all such functions $f:R\to R$

It's my last question. Just give me advise how to start. Q: Find all such functions $$f:\mathbb R\to \mathbb R,$$ for all real x, y, the equality $$f(yf(x))=x^2y^4$$
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0answers
90 views

Solving a functional equation in $L_2(\mathbb{R})$

Let $e\left(x\right)=e^{2\pi ix}$ and let $F$ be an arbitrary complex-valued function in $L^2 (\mathbb R)$. I am trying to solve the following functional equation (or rather family of equations): ...
1
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1answer
55 views

Nonlinear Functional Equation

Find all functions $f(x)$ such for a given fixed $a\in \mathbb{R}$ such that the following functional equation holds $$f(x)^{2}=f(x/a)$$ I'm not sure how to solve this equation other then using the ...
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0answers
24 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...
3
votes
3answers
93 views

Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$(x+y)(f(x)-f(y))=(x-y)f(x+y)$$ My attempt: If $x=-y \not = 0$ then $0= 2x f(0)$ so $f(0)=0$. Suppose for the sake of ...
2
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0answers
51 views

Does the equation $\tan(x)=y$ have any non-zero rational solution?

Trivially $\tan(0)=0$ but it seems this is the "unique" solution of the equation $\tan(x)=y$ on rational numbers. In fact if we try to make $y$ rational we usually use irrational (transcendental) ...
6
votes
4answers
77 views

Find all polynomials with real coefficients that satisfy $(x^2-6x+8)P(x)=(x^2+2x)P(x-2)$

Find all polynomials with real coefficients that satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)\forall x\in\Bbb R$$ My work; $$\frac{P(x)}{P(x-2)}=-\frac{4}{x-2}+\frac{12}{x-4}+1\tag{1}$$ ...
4
votes
1answer
223 views

About an integral equation

I would like to obtain $g$ by solving the following integral equation $$ \int_s^T R(u) dg(u) + f(s,T)\int_s^T g(u)du =0$$ where $f,R:\mathbb R _+ ^*\rightarrow \mathbb R _+ $and $g: \mathbb R _+ ...
17
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2answers
863 views

A functional equation with no solution

Let $f:\mathbb{R}\to (0,\infty)$ be a differentiable function satisfying $$f(f(x))=f^\prime(x)$$for each $x$. Show no such function exists. I got this problem in an exam. I haven't done anything ...
0
votes
3answers
54 views

functional equation for $x^2$ $f(f(x))=x^4$

If $f(f(x))=x^4$ for all real $x$ and $f(1)=1$ find $f(0)$. It seems that $f(x)=x^2$ but can we solve without this explicit form of $f$?
3
votes
4answers
102 views

Functional equation $f(x)-f(y)=\frac{1}{(x-y)^{2}}$

Please help me to solve the functional equation $$ f(x)-f(y)=\frac{1}{(x-y)^{2}} $$ for all real $x\neq y$. I have reduced it to $$ f(x+h)-f(x)=\frac{1}{h^{2}} $$ for all real $h\neq 0$. But what to ...
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0answers
144 views

Solving an integral (or series) equations system

Peace be upon you, In the question A late-diverging "approximating solution" for a system of functional equations, I have asked for an approximating solution for a system of functional ...
1
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3answers
79 views

If $f$ is continuous, $f(1) >1$ and $f(x+y)=f(x)f(y)$, then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
3
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3answers
108 views

How to solve the functional equation $f(2x) = (e^x+1)f(x)$?

I need to solve $f(2x)=(e^x+1)f(x)$. I am thinking about Frobenius type method: $$\sum_{k=0}^{\infty}2^ka_kx^k=\left(1+\sum_{m=0}^{\infty}\frac{x^m}{m!}\right)\sum_{n=0}^{\infty}a_nx^n\\ ...
3
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2answers
106 views

How to derive an proof for this infinite square root equation?

Here is continuous square root, namely: $\sqrt {1 + a \sqrt {1+b \sqrt {1+c\sqrt {1 +...}}}}$= any integer Find $a,b,c,d,e,f,...$ in general Uh, very interesting algebra pre-calculus problem, yet ...
2
votes
1answer
49 views

Functional-equation for strictly increasing functions

Let $n,c$-given natural numbers .Let $f(x)$ - strictly increasing function , domain of definition and set of values of ​​which are non-negative integers ,$f(0)=0 , f(1)=c$ and \begin{align} ...
2
votes
7answers
131 views

$f\left(x + \frac1x\right)= x^3+x^{-3},$ find $f(x)$

$$f\left(x + \frac1x\right)= x^3+x^{-3},$$ find $f(x)$. What i do know at this state is that.. express x as a function of y : $y= x + 1/x$ $x^2−xy+1=0$ Quad formula: $x= (y ± \sqrt {y^2-4}) / 2$ ...
3
votes
1answer
136 views

How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
1
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3answers
71 views

Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$

Suppose that $f: \mathbb R \to\mathbb R$ satisfies $f(x + y) = f(x) + f(y)$ for each real $x,y$. Prove $f$ is continuous at $0$ if and only if $f$ is continuous on $\mathbb R$. Proof: ...
1
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0answers
52 views

How to demonstrate a particular functional equation solution

In order to find a prior probability distribution I have to solve the following functional equation: $$af\left(\frac{a\theta}{1-\theta-a\theta}\right)=(1-\theta+a\theta)^2f(\theta)$$ the solution of ...
3
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1answer
70 views

Failing to reproduce specific Functional derivative

I'm failing to reproduce an (indirect) result in a paper, namely $${δF[g]\overδg(x,y,z)}={r^4\over\ell^5} $$ where $F[g]=\iiint \frac{2dxdydz}{\ell g(x,y,z)} $ and $g(x,y,z)=-{\ell^2 \over r^2} $. ...
3
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1answer
47 views

Are there algorithms for solving simple functional equations?

So somebody posted yesterday asking a question for continuous solutions $f$ satisfying $f(x+y) = f(x)f(y)f(xy)$. Continuity could be used for a simpler proof but then somebody posted a solution ...
2
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0answers
24 views

Integers and funtional equation [duplicate]

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
6
votes
1answer
94 views

Functional Equation $f(mn)=f(m)f(n)$.

If $f: \mathbb N \mapsto \mathbb N$ is one-to-one and $f(mn) = f(m)f(n)$, what is the smallest possible value of $f(999)$? Easily $f(1)=1$, and I think $f(n)=n$ must be the only map, but not able to ...
1
vote
1answer
30 views

Differentiability Problem

Supposing we are given relation that $$f(xy + 1)= f(x).f(y) - f(y) - x +2$$ and also given that $$f(0)=1$$ for a differentiable function then is function one-one onto? I partially differentiated ...
1
vote
2answers
151 views

Functional equation involving gamma function

Recently, I found the following functional equation: $$ n^{nx-1}\cdot\prod_{k=0}^{n-1}{\Gamma{\left(x+\frac{k}{n}\right)}}=\Gamma{(nx)}\cdot\prod_{k=1}^{n-1}{\Gamma{\left(\frac{k}{n}\right)}} $$ Now ...
1
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1answer
29 views

Proof involving functional equation

I'm trying to prove that if $$f(x+n)=f(x)f(n)$$ for all $x\in \Bbb R$ and $n \in \Bbb N$, then it also holds for $x,n \in \Bbb R$. One "argument" I came up with was regarding the symmetry. There's no ...
10
votes
2answers
227 views

Solving the functional equation $f(xy)=f(f(x)+f(y))$

Find all functions from $f: \mathbb{R} \to \mathbb{R}$ such that for all $x$ and $y$ $$f (xy)=f (f (x)+f (y))$$ I've put $x$ and $y$ as $0$ and $1$. How to proceed after substituting if we don't ...
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2answers
93 views

Approximating the Digamma fucntion near 1

Peace be upon you, I had the following system of equations to be solved \begin{align*} \begin{cases} \psi(\alpha)-\psi(\alpha+\beta)=c_1\\ \psi(\beta)-\psi(\alpha+\beta)=c_2 \end{cases} \end{align*} ...
2
votes
2answers
96 views

Additive functional equation

Find all function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$ f(x+y) = f(x) + f(y)$$ and $$ f(f(x)) = x$$ for all $x, y \in \mathbb{R}$ This is one problem involving additive functional ...
2
votes
2answers
88 views

Functional equation - Understading an easy step in my solution.

I am trying to solve the equation and find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that: $f(m+f(n))=f(f(m))+f(n)$ for all $n, m \in \mathbb{N_{0}} $. A reasonable approach to begin with ...
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2answers
78 views

Find all functions $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, f^n(x)=-x$

I got this problem: Prove that the only function $ f:\Bbb{R}\to\Bbb{R}$ with the intermediate value property such that $\exists 1\leq n\in\Bbb{Z}, \forall x\in\Bbb{R}, f^n(x)=-x$ where $f^n =f\circ ...
1
vote
1answer
51 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
2
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1answer
43 views

How equal are two given numbers

I have two numbers x & y for N different readings and wish to find how close they are from each other and would like to rank the reading in order of they equalness. If I were to have the ...
3
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1answer
54 views

Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$

I got this problem: Find examples of continuous functions $f:[0,1]\to[0,1]$ that satisfy $\forall x\in[0,1], f(f(x))=f(x)$ other than $f(x)=x$. I proved that $f([0,1])=\{x\in[0,1]|f(x)=x\}$, But I ...
0
votes
0answers
18 views

Solving functional equation $b(x)=\int b(xy)f(y)dy$

I want to prove that given a real-valued smooth function $f$, the set of functions $b$ solving $b(x)=\int_0^{\infty} b(xy)f(y)dy$ is given by linear combinations of $x^{\sigma}$ where $\sigma$ is a ...
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4answers
127 views

Example of a non-trivial function such that $f(2x)=f(x)$

Could you give an example of a non-constant function $f$ such that $$ f(x) = f(2x). $$ The one that I can think of is the trivial one, namely $\chi_{\mathbb{Q}}$, the characteristic function on the ...
4
votes
2answers
242 views

Polynomial satisfying $ P \big(P (x)\big)=P (x)+ P\big(x^2\big)$

If $P(x)$ is a polynomial with integer coefficients such that for all integer $x$, $$P (P (x)) = P (x)+P (x^2).$$ I've tried solving it putting it as a function instead. Not much though. How do you ...
7
votes
3answers
173 views

Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$

I got this problem: Prove that there is no function $f:\Bbb{R}\to\Bbb{R}$ with $f(0)>0$ such that $\forall x,y\in\Bbb{R}, f(x+y)\geq f(x)+y f(f(x))$. (Hint: the solution involves limits at ...
9
votes
3answers
134 views

How find this all function $f(x^n+2f(y))=(f(x))^n+y+f(y)$

Question: Given a positive integer $n\ge 2$ . Find all functions $f:R\to R$, such that $$f(x^n+2f(y))=(f(x))^n+y+f(y)$$ let $x=0,y=0,a=f(0)$ then $$f(2f(0))=(f(0))^n+0+f(0)\Longrightarrow ...
9
votes
4answers
554 views

What functions satisfy this functional equation?

$$f(x)-g(x)=f(g(x))$$ How could I find an f(x) and g(x) that satisfy this?
5
votes
1answer
109 views

At most n functions

Some background: I was trying to solve the functional equation f(f(x))=sin(x). I realized that $f(\pi n)$ is a root of f for all integers n, because $f(f(\pi n))=\sin(\pi n)=0$. Thus, we can write f ...
5
votes
1answer
84 views

Finding all real valued functions that satisfy $f(f(y) + xf(x)) = y + (f(x))^2$

I would like some help with finding all real valued functions that satisfy this equation: $f(f(y) + xf(x)) = y + (f(x))^2$ I tried the usual substitutions like $x = y = 0$, but my experience with ...
0
votes
0answers
30 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...