Mathematics Stack Exchange
Mathematics Stack Exchange

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).

The term "functional equation" is used for problems where the goal is to find all functions satisfying the given equation(s) and possibly other conditions; e.g., the goal can be to find all continuous solutions. Solving the equation means finding all functions satisfying the given equation(s) and any additional conditions.This is different from the more common use of the word "equation", where the solutions are numbers. It is also different from the more common use of the word "functional", referring to a mapping from a space into the reals or complexes. For basic questions about functions use more suitable tags like or .

A common technique used in solving functional equations is finding some properties of satisfying functions by substituting variables for certain values in the equation. Proving properties of satisfying functions is also helpful - finding that a function is injective, surjective, involutive, and so on, is often a key step in finding all possible solutions. Other techniques such as exploiting symmetry, considering fixed points, and even using certain properties of domains (e.g. well-ordering) sometimes help.

Some well-known functional equations are:

More information can be found at Wikipedia.

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$f:\mathbb{R_{\geq 0}} \to \mathbb{R_{\geq 0}}$ 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 ask the number $a$ for $f(a)=0$ instead of the…
user628755
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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 $g(x) = g(x + a)$ is an arbitrary periodic function…
JaberMac
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$2f(x)=f(y) \Rightarrow 2f(tx)=f(ty)$

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 least one pair $(x,y)$ where the above condition…
Arnaldo
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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 $x=f(0), y=0\quad$ I got $f(0)=0$ or $1$ Suppose $f(0)=1, $…
bFur4list
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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 $f\left(x\right) = 1 - \left(1 - f\left(x+1\right)\right)^{\frac{x}{x+1}}$ where $x \geq 0$. The only progress I…
dayandnightatom
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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 functional…
Nine Thousand
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Power-like functional equation

I would like to know what are the especifications of a functional equation that give us a power function as a solution. For example, if $f:\Bbb R \to \Bbb R$ is continuous and monotonic, such that $$f(x)+f(y)=f(z)$$ iif $$f(\lambda x)+f(\lambda…
Arnaldo
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Is it possible to have $f(x)f(y) = g(x)+g(y)$?

Inspired by this question I wondered whether there are any "notable" functions $f,g$ on (or on some subset $\Omega$ of) $\mathbb R$ or $\mathbb C$ that satisfy $$f(x)f(y) = g(x) + g(y) \:\forall x,y \in \Omega$$ By "notable" I mean nontrivial…
flawr
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Find all f such that $f(f(y))+f(x-y)=f(xf(y)-x)$

Find all functions $f$ defined over real numbers to real numbers such that $f(f(y))+f(x-y)=f(xf(y)-x)$ My attempt: Set $x=y=0$ to get $f(f(0))=0$. It will be very helpful if I will able to find $f(0)$ but I failed to find it. I tried to check…
Sufaid Saleel
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Functional equation $ (n+1) f(n+1)= (a n+b) f(n) $ for $n=0,1,...$

I am looking for a solution to the following functional equation: \begin{align} (n+1) f(n+1)= (a n+b) f(n), n=0,1,... \end{align} where $a$ and $b$ are some positive constants. Moreover, $f(n)$ is positive and \begin{align} 0
Boby
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Find $f(x)$ where $f(x)(A-\frac{B}{x+B/A})+Cf(x+\frac{B}{A})=0$.

$A, B, C > 0$, $x$ is complex and $Re(x)>0$. My guess is that $f(x)=0$ but I don't know how to prove it.
Zaeem Hussain
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Functional equation $ f ( x + y ) f ( x - y ) = f ^ 2 ( x ) $

Let a function $ f $ be continuous and differentiable for all $ x $, such that it satisfies $$ f ( x + y ) f( x - y ) = f ^ 2 ( x ) \text . $$ Given that $ f ( 0 ) $ is nonzero and $ f ( 1 ) $ is $ 1 $, find $ f $. I tried replacing $ x $ by $ y $…
maveric
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Nonlinear functional equation with tangent

Please help me with this. I need to find a non-trivial function $g(x)$ which satisfy the following functional equation $$\tan(g(x))+g(x)+g(x)\tan^2(g(x))=0$$
Darek
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Prove or disprove existence of functions $f$ and $g$ (functional equations)

I came across the following problem and I have been cracking my head with it: Prove or disprove that there are no functions $f$ and $g$ such that $$f(x,g(y-x)+g(z-x))=y^2+z^2$$ for all $x,y,z \in \mathbb{R}$. The point is that I never had contact…
bertozzijr
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Funcional Equations:I'm confused

I need help with this: "Find functions $f$, $g : \mathbb{Z} \rightarrow \mathbb{Z}$, knowing that $g$ is injective and such that: $$f(g(x)+y) = g(f(x)+x), \mbox{ for all } x, y \in \mathbb{Z}.$$ Or : $$f(g(x)+y) = g(f(y)+x), \mbox{ for all } x, y…
user62189
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