# 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).

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### Solving infinite coupled equations

I need to solve symbolically an infinite system of coupled algebraic equations that I tried to do analytically but I could not. Solutions of these equations, $V_l^m$, define the coefficients of a some ...
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### Does $f(g(s))=g(s)$ and $g(f(s))=f(s)$ imply $f(f(s))=f(s)$ and $g(g(s))=g(s)$?

I have a set $S$ and two one-to-one functions $f:S \to S$ and $g: S \to S$ such that for all $s \in S$ we have $f(g(s))=g(s)$ and $g(f(s))=f(s)$. If $S$ is infinite, does it follow that $f(f(s))=f(s)$...
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### How to solve a first order differential equation with parameters inside functions: $ag'(cy)+bg'(ey)=\alpha$

I am trying to solve a first order differential equation with the condition that $g(y)=0$ iff $y=0$: \begin{align*} &ag'(cy)+bg'(ey)=\alpha\\ &g(0)=0, \tag{1} \end{align*} where ...
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### How to find the solutions of this functional equation

$$f(\tfrac{1}{2}+x)+f(\tfrac{1}{2}-x)=8xf\big(4(\tfrac{1}{2}+x)(\tfrac{1}{2}-x)\big)\qquad\text{for}\qquad x\in(0,\tfrac{1}{2})$$ I have no idea about how to tackle this equation. The original ...
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### Functional equation problem; chain of functions

I have tried more than an hour but couldn't solve it, can somebody please give me a clue? $$f:\mathbb R\rightarrow\mathbb R$$ $$f(f(f(X)))+f(f(X))+X=3f(X).$$ Find $f(X)$ I know that $f(X)=X$ is a ...
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### Show that the functional equation $f(f(x))=-x^{3}+g(x)$ has no continuous solution.Here $g$ is a continuous periodic function with positive period.

I want to show $f(f(x))=-x^{3}+g(x)$ has no continuous solution $f:\mathbb{R}\to\mathbb{R}$.Here $g$ is a continuous periodic function with its period $T>0.$ Any help will be thanked.
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### Functional Equations from information theory

I looking for functional equations which are out coming from real life An example from real life to explain and relate to uncertainty is as follows: Example: 3 candidates A, B and C are sitting ...
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### Solve for all possible functions f: $|f(x)-f(y)|=2|x-y|$. [closed]

I'm getting $f(x)=2x+f(0)$ and $f(x)=f(0)-2x$ by setting $y=0$, but I'd like to verify. Am I right?
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### Given $a$ and $c$ positive numbers, is there a function $h$ such that $h(-ax)= -ch(x)$ for all $x$?

Given $a>1$ and $c$ a positive number, is there a (no trivial) function $h$ such that $$h(-ax)= -ch(x)$$ for all $x$?. I know that $h(x)=c^{\log_a|x|}$ satisfies $h(ax)=ch(x)$, however I am not ...
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### Problem with existence of discontinuous additive function.

I think I have made a mistake in my justification but I can't see where. Let's assume that $f$ is a discontinuous additive function. From the basic properties of additive functions we know that $f$ ...
Is there a subgroup of the real numbers under addition of index 2? (and if so, can we classify them somehow?) [I am trying to solve the functional equation $f(x)f(y)=f(x-y)$ for all real $x,y$. If $f$...