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).

Filter by
Sorted by
Tagged with
11 votes
3 answers
201 views

$f(xy) = f(x+y)$ : A variation of an old problem

I have seen the following problem in various places, such as Toppr and Quora. Suppose that the function $f:\mathbb{N}\to\mathbb{N}$ satisfies $f(x+y)=f(xy)$ for all $x,y\in\mathbb{N}$. Show that $f$ ...
Timothy Chow's user avatar
2 votes
0 answers
92 views

Solutions to $f(x^2) = 2f(x)^2-1$

I'm looking for solutions over $\mathbb{R}$ to the functional equation: $$f(x^2) = 2f(x)^2 -1$$ The left hand side is very similar to the expansion for $\cos(2\theta)$. But the right hand side ...
mtheorylord's user avatar
  • 4,182
0 votes
0 answers
49 views

Doubt in an alternate solution to an IMO problem from 1987

Consider the following problem from 1987 IMO Show that there is no function $f:\mathbb N→\mathbb N$ Such that $f(f(n))=n+1987$ In the book that I was referring the solution is given as follows: " ...
Aarush Saharan's user avatar
1 vote
0 answers
40 views

Alternate solution to an IMO 1993 problem

Let $\mathbb N=\{1,2,3,…\}$. Determine if there exists a strictly increasing function $f:\mathbb N→\mathbb N$ such that $f(1)=2$ $f(f(n))=f(n)+n$ for all $n$. Recently I posted the following solution ...
Aarush Saharan's user avatar
1 vote
0 answers
61 views

IMO problem from 1993 on functional equations

Let N={1,2,3,…}. Determine if there exists a strictly increasing function $f:N→N$ such that $f(1)=2$ $f(f(n))=f(n)+n$ for all n." Here is the solution I came up with We can see $f(1)=2$ $f(2)=...
Aarush Saharan's user avatar
1 vote
0 answers
61 views

Analysis of the functional inequality $f(x-s)f(y+s) \geq f(x)f(y)$

I would like to find references or analysis of the two functional inequalities: $f(x-s)f(y+s)\geq f(x)f(y)$ $f(x-s)+f(y+s) \geq f(x) + f(y)$ where $x,y\in \mathbb{R}$, $y>=x$ and $s>0$. ...
Hushus46's user avatar
  • 952
3 votes
0 answers
63 views

Is there a name for equations like $f(n) = f^{(n)}(0) $?

So far, I've encountered three questions here on MSE that involve a function evaluated at a natural argument $n$ on the one left side of the equation, and the same function that has been ...
Max Muller's user avatar
  • 6,346
1 vote
3 answers
61 views

functional equation polynomial with real roots

if $f(f(x))=0$ has atleast one real root,and there be a real root $a$ of $f(x)$,is it necessary for the equation $f(x)=a$ to have atleast one real solution/root.if so why? my contention is if $f(a)=0$,...
mike dennes's user avatar
0 votes
0 answers
17 views

Looking for a Functional Form Satisfying Certain NonLinearites

I'm working with a functional form $\alpha = F(x,y,t)$ where t represents time and $x,y $ other factors affecting $\alpha$. I want to find a functional form for F (that also has some parameters) such ...
esos's user avatar
  • 1
1 vote
0 answers
89 views

Must this function necessarily be decreasing?

Let $f: \mathbb{R} \to \mathbb{R}$ be a strictly monotone function satisfying $$ f(f(x)) - f(x) = e^x + x - 1 $$ for all $x \in \mathbb{R}$. Can we conclude that $f$ must be decreasing? (I don't know ...
user23571113's user avatar
  • 1,254
0 votes
1 answer
80 views

Increasing functions for natural numbers

Is there an increasing function which satisfies $f(1) = 3$ and $f(f(n)) = f(2n)+ 1$ for all natural numbers? I first tried doing this problem using the subbing values, by subbing values of $n=-1,0,1$,...
Alan Gardiner's user avatar
0 votes
0 answers
16 views

Solving "time evolution" partial differential eq using Lagrange shift operator

I was reading about the Weierstrass transform $(W[\cdot])$, and how it's related to the difussion equation in one dimension. It's relation is given by that $W[f]$ is the convolution with the Heat ...
Daniel Muñoz's user avatar
3 votes
1 answer
143 views

Calculating the functional derivative of an implicit functional

We want to calculate the functional derivative of the following wrt $\rho$: $$F=\int X dx$$ where $X$ is implicitly defined as: $$X=\frac{1}{1+\overline \rho(x) \int \rho(x) X dx}$$ and $\overline \...
Ali's user avatar
  • 347
1 vote
1 answer
75 views

Let ℝ+ be the set of positive numbers. Determine all functions $f:\mathbb R^+\to\mathbb R^+$ such that $f(x^2+xf(y)) = f(f(x))(x+y)$ [closed]

Let $\mathbb R^+$ be the set of positive numbers. Determine all functions $f:\mathbb R^+\to\mathbb R^+$ such that $$f(x^2+xf(y)) = f(f(x))(x+y)$$ for all positive real numbers $x$ and $y$. I think ...
sellll's user avatar
  • 21
4 votes
0 answers
65 views

Functional equation of complex numbers

If $f:\Bbb C\to\Bbb C$ satisfies $f(f(x))=\left(x\overline x-x-\overline x\right)^2$ and $f(1)=0$, find $f(\mathrm i)$. If $f(a+b\mathrm i)=a^\prime+b^\prime\mathrm i$, define $g:\Bbb R^2\to\Bbb R^2$ ...
youthdoo's user avatar
  • 705
1 vote
1 answer
69 views

Find all injective functions from reals to reals that satisfy $f(x + f(y)) = f(x) - f(y)$

Find all injective functions from reals to reals that satisfy $f(x + f(y)) = f(x) - f(y)$, from reals to reals. I tried subbing in values to cancel some variables out, like $y = x$ and $y=-f(x)$, but ...
lightningjay's user avatar
1 vote
1 answer
64 views

What field should I search to solve delay like equations?

I want to solve equations like: $$ t f(t) + u(t-2) f(t-2) = \sin(t) $$ It's like delay differential equations but without the derivative. What field should I search in to solve this kind of equation?
Mohamed Mostafa's user avatar
0 votes
0 answers
38 views

Proving a Functional Equation for the JacobiTheta Function

Let $\Theta(t) = \sum_{k = - \infty}^{\infty} e^{-\pi k^{2} t} $. How can it be proved that $\Theta(\frac{1}{t}) = \sqrt{t}\Theta(t)$? I have read a proof here https://scholarship.claremont.edu/cgi/...
Robert's user avatar
  • 59
0 votes
0 answers
13 views

Analytic continuation of sums of Hypergeometric $_3F_2$ function pairs

While investigating the relationship $$\Gamma\left[f,s\right]\zeta(s)=\Gamma\left[\hat{f},1-s\right] \zeta(1-s)\tag{1}$$ where $\Gamma\left[f,s\right]$ denotes the Mellin transform $$\Gamma\left[f,s\...
Steven Clark's user avatar
  • 6,703
0 votes
0 answers
66 views

Obtaining probability generating function by solving a functional equation

For investigating a queueing model in my research, I need to obtain the probability generating function (PGF), $G(z)$, by solving the following functional equation $$\lambda(z(1-p)+p)G((1-p)z)-(\mu+\...
גבי חנוכוב's user avatar
2 votes
1 answer
78 views

Find all $f:\mathbb{Q^+}\rightarrow \mathbb{Q^+}$ so that $\forall x,y\in\mathbb{Q^+}$ : $f(f(x)^2y)=x^3f(xy)$ [duplicate]

Find all $f:\mathbb{Q^+}\rightarrow \mathbb{Q^+}$ so that $\forall x,y\in\mathbb{Q^+}$ : $f(f(x)^2y)=x^3f(xy)$ so $f(0)=0$ when $x=y=0$ $f(f(1)^2)=f(1)$ when $x=y=1$ $f(f(x)^2)=x^3f(x)$ when $y=1$ and ...
user avatar
0 votes
0 answers
61 views

How to solve $f(ix)=i f^{-1}(x)$

Is there some method to solve this equation? $$f(ix)=i\cdot f^{-1}(x)$$ I found these solutions: $x$ $c_1\arctan\left(\tanh\left(\frac{x}{c_1}\right)\right)$ $c_2\arcsin\left(\sinh\left(\frac{x}{c_2}\...
Math Attack's user avatar
  • 2,373
-3 votes
1 answer
65 views

Functions satisfying $Af(x) + B = f(Cx)$ [closed]

Let $f: [0,\infty) \to [0,\infty)$ be a continuous function such that $Af(x) + B = f(Cx)$ for all $x \in [0,\infty)$. Here the constants satisfies $A \in (0,1), B>0, C>1$. Can we find $f$, or ...
June Kim's user avatar
0 votes
1 answer
60 views

Is there a Newton-Raphson method for finding the zeros of scalar functions of several variables?

I have seen countless examples of using the Newton-Raphson method to find the roots of a system of equations. But I cannot find anything on the possibility of finding the zeros of a scalar function of ...
Sophie's user avatar
  • 343
1 vote
0 answers
10 views

Min, max, location and scale invariant family of continuous distributions with finite parameters

I want to find a family of continuous distributions $H$, where if $X$ and $Y$ are random variables from $H$, $aX+b$, $max(X,Y)$, $min(X,Y)$ also have distributions from $H$. The distributions of $H$ ...
László T.'s user avatar
2 votes
0 answers
39 views

Polynomials with a given property

Find all polynomials with real coefficients such that $$xp(x) + yp(y)\geq2p(xy)$$ for all $x, y\in\mathbb{R}$. My attempt: I noticed the similarity between the given expression and the square of the ...
Tassandro Cavalcante Leitão's user avatar
5 votes
1 answer
166 views

Finding all functionals over $\mathbb{N}$ such that $f(f(n)) = an$ for which $a\in\mathbb{Z}^+$

I want to find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ for which there is a $a\in\mathbb{Z}^+$ satisfying $$f(f(n)) = an$$ for all $n\in\mathbb{N}$ My attempt: First, taking the case $a=1$,...
Dee's user avatar
  • 457
2 votes
0 answers
27 views

Solution of the functional equation $f\left( x,\, y \right)=x^{f(x,\,y-1)}\wedge\{x,\,y,\,f( x,\, y)\}\in\mathbb{C}$ and $x\ne0$

$$\text{Question}$$ What is the general solution of the functional equation: $$ \begin{align*} f\left( x,\, y \right) &= x^{f\left( x,\, y - 1 \right)} \tag{1} \label{eq: 1}\\ \end{align*} $$ ...
Kevin Dietrich's user avatar
-1 votes
2 answers
88 views

Solving functional equation using theoretical techniques [closed]

I know the answer to the functional equation $$\frac{f(x)}{C}=f(Cx)$$ is $f(x)=\frac{C_{2}}{x}$ , where $C$ and $C_{2}$ are constants. How can we solve this algebraically using differential equation ...
Pele Pure's user avatar
6 votes
2 answers
179 views

Condition about functional equation has a solution

Problem : Let $P(x, y)$ be a polynomial respect to $x, y$. Find a condition about $P$ where functional equation of $f$ $$f(x+y)=f(x)+f(y)+P(x,y)$$ has a solution. (Where $f\colon \mathbb{R}\to\mathbb{...
bFur4list's user avatar
  • 2,417
-1 votes
1 answer
34 views

Existence of Positive Integer Solution for a Polynomial Equation [closed]

Let $f(x)$ be a polynomial of degree n, and it is known that $f(x)$ has no positive integers as its roots. In other words, there are no positive integer values of x for which $f(x)=0$. Consider the ...
Krishna Mishra's user avatar
1 vote
0 answers
68 views

Functional equation $f(2x)=2(f(x))^2 -1$

Given that $f(x)$ is differentiable, $$f(2x)=2(f(x))^2 -1$$ I can assume that $f(x)=cos(ax)$, but I can’t think of a way to prove that it is the only solution. In fact, I am not sure that $f(x)=cos(ax)...
John. P's user avatar
  • 571
5 votes
1 answer
121 views

Functional equation $f(f(x))-2f(x)+x=0$. [duplicate]

We consider the equation $$f(f(x))-2f(x)+x=0,$$ where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function. The question is to prove that $f(x)=x+c$. I proved that $f$ is strictly increasing and ...
Hamza's user avatar
  • 3,741
5 votes
1 answer
141 views

Nontrivial solutions to simple functional equation

I am interested in the functional equation $$ f(x^2) = 2f(x)^2 - 1, $$ where the domain of $f$ is nonzero real numbers. Are there any nontrivial continuous solutions? I know $f(x) = 1$ and $f(x) = \...
jackson's user avatar
  • 967
3 votes
1 answer
58 views

Linear functional equations

Linear functional equations like $(x + 1)P (x) = (x − 10)P (x + 1)$ are fairly common in competition math, and there are some general techniques for proving things about the solutions, such as degree ...
Yly's user avatar
  • 14.9k
4 votes
2 answers
130 views

IMC 2023 problem 1

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that have continuous second derivative and for which the equality $f(7x+1)=49f(x)$ holds for all $x \in \mathbb{R}$. I found a solution online to ...
Victor's user avatar
  • 203
1 vote
0 answers
52 views

A periodic functional equation

I am not experienced with functional equations, and I have been wondering about equations of the form $$ f(x) = f(mx+c) $$ For positve $m,c$, where we further restrict the solutions to be continuous. ...
Carlyle's user avatar
  • 1,226
4 votes
1 answer
173 views

Solve continuous function: $f(bx)-f(by)=f(x)-f(y)$

Solve for continuous function $f$ where $f(bx)-f(by)=f(x)-f(y)$ for all positive number $b$. $f$ is defined on positive numbers. Example: $f(x)=\log x$ Check: $f(bx)-f(by)=\log x+\log b-\log b-\log y$ ...
High GPA's user avatar
  • 3,580
0 votes
0 answers
69 views

$f(x) + f(y) = \frac{1}{f(xy)}$

Suppose $f(x) + f(y) = \frac{1}{f(xy)}$ holds for all $x,y$ and $f(x) > 0$ for all $x$. which are real. I understand that you can choose for example, $x=1$ and $y=4$ to get $f(1) + f(4) = \frac{1}{...
Nav Bhatthal's user avatar
  • 1,081
1 vote
0 answers
42 views

Proving that powers are the only solutions to the functional relation $g(r)g(1/r)=1$

As stated above, I am trying to prove that the only solutions to the functional relation $$ g(r)g(1/r)=1, \quad \text{for}\; r>0, $$ are of the form $g(r)=r^a$ for $a \in \mathbb{R}$. Other ...
thecommutator's user avatar
-2 votes
1 answer
42 views

Cauchy functional equation under certain conditions [closed]

Does the equation $f(x+y)=f(x)+f(y)$ (with $x \in\mathbb{R^+}$ and $y \in\mathbb{R^-}$) ($0$ in included in $\mathbb{R^+}$ and $\mathbb{R^-}$) have the same solution of the normal equation knowing ...
Math player's user avatar
0 votes
0 answers
31 views

Difference of two sub additive function

A real valued function $ f : \mathbb{R} \to \mathbb{R} $ is said to be sub additive if $ f( x+y ) \leq f( x ) + f( y ), \quad x,y \in \mathbb{R}. $ There are some nice properties related to sub ...
swapan Jana's user avatar
-2 votes
1 answer
103 views

$f(xy) = f(x) +f(y)$ [duplicate]

If $f(xy) = f(x) + f(y)$ for all positive reals $x,y$, what can $f$ be? If given the starting conditions $f(10) = 14$ and $f(40) = 20$. We know $f(40) = f(10) + f(4)$ giving $f(4) = 6$, then $f(4) = ...
Nav Bhatthal's user avatar
  • 1,081
2 votes
0 answers
35 views

About a counterexample for an integral-functional equation in number theory.

I was reading http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf How did the counterexample for the equation on page 8 look like ?? Specificly : (quote) “Tur´an’s lecture (probably a quite ...
mick's user avatar
  • 14.6k
4 votes
1 answer
161 views

Sequences $(a_n)$ such that $i^2+j^2=k^2+\ell^2\Longrightarrow a_i+a_j=a_k+a_{\ell}$

In trying to solve the problem posed in this old message, I asked myself the following question : Which sequences $(a_n)_{n\in\mathbb{N}}$ (with $\forall n\in\mathbb{N}\,,\,a_n\in\mathbb{Z}$) satisfy ...
uvdose's user avatar
  • 107
0 votes
1 answer
35 views

Existence of solution of functional equation on $S^1$.

I want to know if there exists a non-trivial ($h \neq 0$) continuous function $h:S^1 \rightarrow \mathbb{C}$ such that for all $z \in S^1$, $$ h(z) = h(\sqrt z) + h(- \sqrt z). $$ My suspicion is that ...
Tomás Pacheco's user avatar
1 vote
1 answer
152 views

How to solve this functional equation: $f(x^2) = f(x) f^{-1}(x)$?

We know that the equation admits the particular solutions: $f(x) = x$, $f(x) = x - 1$ and $f(x) = \frac{1}{x}$ But how to get these results? The method of guessing a function and substituting it into ...
JaberMac's user avatar
  • 557
6 votes
1 answer
126 views

Function on the real plane that can be expressed in terms of a function on the real line

Let $f:\mathbb{R}^2\to\mathbb{R}$ such that $f(x,y)+f(y,z)+f(z,x)=0 \text{ } \forall \text{ } x,y,z \in \mathbb{R}.$ Show that there exists a $g:\mathbb{R}\to \mathbb{R}$ such that $f(x,y)=g(x)-g(y).$ ...
aqualubix's user avatar
  • 1,295
0 votes
1 answer
84 views

Find all functions over $\mathbb{Z}^+$ satisfying this divisibility criterion

The problem is to find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $[f(m)]^2 + f(n)$ divides $(m^2+n)^2$ for all $m$,$n\in\mathbb{Z}^+$. I already have my claim and attempted ...
Dee's user avatar
  • 457
2 votes
1 answer
161 views

Shortcut Collatz function satisfies a particular functional equation. Has this approach been studied yet, and if so where are the reference articles?

Let $X = 2\Bbb{Z} + 1$ or $2 \Bbb{N} + 1$ where $0 \in \Bbb{N}$, this approach will probably play well with both forms. See Extending the Collatz function to larger domains. Define the shortcut ...
MathCrackExchange's user avatar

1
2 3 4 5
77