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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Solving functional equation $1=e^{−\frac{2}{k}f(ax+bx^2)}+2f(x)$

I want to solve the functional equation $$1=e^{−\frac{2}{k}f(ax+bx^2)} + 2f(x),$$ where $k>0$ and $a,b\in\mathbb R$. Do you have any ideas on how to tackle this problem?
Marcel
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Help with a Functional Equation / Eigenfunction Problem

I am trying to find eigenfunctions (if there are any to be found!) of the operator: $$T\left\{ f\right\} \left(x\right)=\frac{2}{9}f\left(-a-\frac{9}{2}x\right)+\frac{2}{9}f\left(b-\frac{9}{2}x\right)$$ where $a$ and $b$ are positive rational…
MCS
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Find $f(x)$ such that $f^\prime(x) = f(f(x))$

Is there such a differentiable function $f: R\rightarrow R$ that for each real $x$ we have $f (x)> 0$ and $f' (x) = f (f (x))$;
Ben
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Find $h(85)$ if $h(x^2+x+3)+2h(x^2-3x+5)=6x^2-10x+17$

let : $h: \mathbb{R}\to \mathbb{R}$ ane for any real number $$h(x^2+x+3)+2h(x^2-3x+5)=6x^2-10x+17$$ then : $$h(85)=?$$ My Try: $$x=0:h(3)+2h(5)=17\\x=1:h(5)+2h(3)=13\\+\\3h(3)+3h(5)=30\\h(3)+h(5)=10$$ now ?
user402750
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Doubt in problem on functions

I'm stuck with a problem here, because i could not justify my way around a situation. THE PROBLEM Let $f:R \to R$ be a function satisfying: $|f(x+y) -f(x-y) -y| \le y^2$ for all $x,y \in R$. Then show that $f(x) = \frac{x}{2} + c$ for some constant…
Lelouch
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Find all $f:\mathbb{N} \to \mathbb{N}$ such that $f(m)^2 + f(n) \mid (m^2+n)^2;\; \forall n,m \in \mathbb{N}$

Find all $f:\mathbb{N} \to \mathbb{N}$ such that $f(m)^2 + f(n) \mid (m^2+n)^2;\; \forall n,m \in \mathbb{N}$ I tried to write $k(f(m)^2 + f(n)) = (m^2 + n) ^2$ and tried to find if there is any fixed $k$ but failed. I can't find a way to start.
Rezwan Arefin
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Find all $f:\mathbb{N} \to \mathbb{N}$ such that $f(f(n)) = 2n$

Find all $f:\mathbb{N} \to \mathbb{N}$ such that $f(f(n)) = 2n\; \forall n \in \mathbb{N}$ If it was $f:\mathbb{R} \to \mathbb{R}$ then the solution was simply $f(n) = \sqrt{2}n$. But how to solve when $f : \mathbb{N} \to \mathbb{N}$ ?
0123456789101112
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Find the general solution to $f(z)=f(z/2)f(z-1)$

Find the general solution to $f(z)=f(z/2)f(z-1)$ where $z$ is a complex number.
mick
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Functional equation (N to N)

Find all $f : \mathbb{N} \to \mathbb{N}$ which satisfy the equation: $f(d_1)f(d_2)...f(d_n)=N$ Where $N$ is a natural number and $d_i, 1 \leq i \leq n$ are all of the divisors of $N$.
J.Does
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Cauchy-like functional equation $f(x+g(x)y)=f(x)+f(g(0)y)-f(0)$

I have the functional equation $$ f(x+g(x)y)=f(x)+f(g(0)y)-f(0) $$ where $f$ is monotone increasing and continuous $g$ is continuous and positive The domain of both functions is a closed interval that includes 0. The obvious solutions are: $g$…
mike
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Solve functional equation $f(x^4+y)=x^3f(x)+f(y)$

I need help solving this equation, please: $$f(x^4+y)=x^3f(x)+f(y),$$ such that $ f:\Bbb{R}\rightarrow \Bbb{R},$ and differentiable I've found that $f(0)=0$ and $f(y+1)=f(1)+f(y)$, but I couldn't continue, I think the solution is $f(x)=ax$. Thanks…
Aymane Gr
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Proving a function to be a difference

I am concerned with a function $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(f(x_1,x_2),f(x_3,x_2))=f(x_1,x_3)$$ for any $x_1,x_2,x_3\in \mathbb{R}$, and $$f(x,0)=x$$ $$f(x,x)=0$$ for any $x\in\mathbb{R}$. For example, such a…
Enrico
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If $f(1-x) + 2f(x) = 3x$, what is $f(0)?$

I did the following: $$f(1-0) + 2f(0) = 3\cdot 0$$ $$f(1) + 2f(0) = 0$$ This reminds me of the equation of the straight line in the plane, then: $$\left< \begin{pmatrix} {1}\\ {2} \end{pmatrix} , \begin{pmatrix} {f(1)}\\ {f(0)} \end{pmatrix}…
Red Banana
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Solving the functional equation $x[f(x+1)-f(x-1)]=1$

Possible Duplicate: Solving the functional equation $f(x+1) - f(x-1) = g(x)$ How do I approach this problem $x[f(x+1)-f(x-1)]=1$.
Benji
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How to solve functional equations using the given tips

I have a midterm tomorrow and have been able to cover all other topics except this. I don't even have an idea how to start these questions. If someone could give me some tips I would very much appreciate it. The questions I'm having trouble with…
The_Questioner
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