Tagged Questions

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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Integral functional equation.

$f(x)= \Big(\int _1^x g_1(t)g_2(t)dt\Big)\Big(\int _1^x g_3(t)g_4(t)dt\Big)-\Big(\int _1^x g_1(t)g_3(t)dt\Big)\Big(\int _1^x g_2(t)g_4(t)dt\Big)$ $\forall$ $x$ $\in R$ Where $g_1(x),g_2(x),g_3(x)$ ...
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Functional inequality $f(y)+xf(x)≤yf(x)+f(f(x))$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function sucht that: $$f(y)+xf(x)≤yf(x)+f(f(x))$$ for all $x,y\in\mathbb{R}$. Show that $$f(x)+yf(x+y)≤0$$ for all $x,y\in\mathbb{R}$. I tried some ...
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Cauchy's functional equation for involutions

It is well known that Cauchy's functional equation for $f:\mathbb{R}\to\mathbb{R}$ $$f(x+y)=f(x)+f(y)\space\forall x,y\in\mathbb{R}$$ admits highly pathological solutions if no further conditions ...
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Functional equation $f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$

$$f(x^3) - f(x^3 - 2) = (f(x))^2 + 12$$ Given the functional equation above, I am trying to find the value of $f(3)$. I do not remember the exact statement of the problem precisely, so I am not sure ...
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How to prove a function is periodic from a given functional equation?

Given that $f: \mathbb R \to \mathbb R$, and that for some $a \in \mathbb R$, $f(x+a)={1\over 2}+\sqrt{f(x)-f^3(x)}$; prove $f(x)$ is a periodic fucntion. I know that to prove a function is ...
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Re-arrangement of equation

This type of question might be voted down or frowned upon, but if anyone could point me in the right direction I would be very grateful. I have worked through a question getting the following result: ...
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Formulation of a rule

Im new to this forum(hence, formatting issues). I come from CS background. I am trying to figure out a formula to code set of rules. The requirement is as below. Imagine two parameters X and Y ...
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If $g(x) = \frac{x}{x+1}$, and $f\circ g(x) = x^2$, find $f(x)$? [closed]

$$g(x) = \frac{x}{x+1}$$ $$f\circ g(x) = x^2$$ $$f(x) =\text{ ?}$$ Hello, I don't have any clue how to solve that. Do you have any ideas how to solve that? Thanks in advance.
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Solve the functional equation $f(x) + f(x^2) = 2$ [closed]

What are the solutions of the functional equation $f(x) + f(x^2) = 2$? Will they be one to one or many to one? Will they be periodic or not?
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Polynimials $P(x)$ and $Q(x)$ satisfying $P(x^2+1)=(Q(x))^2+2x, Q(x^2+1)=(P(x))^2$

I am looking for all polynomials $P(x)$ and $Q(x)$ satisfying the simultaneous functional equations given in the title, i.e. $P(x^2+1)=(Q(x))^2+2x$ $Q(x^2+1)=(P(x))^2$ I've already found one ...
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Can one do anything useful with a functional equation like $g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$?

I got $$g(x^2) = \frac{4x^2-1}{2x^2+1}g(x)$$ as a functional equation for a generating function. Is there a way to get a closed form or some asymptotic information about the Taylor coefficients ...
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Extremal of a functional $I=\int\limits_0^{x_1} y^2(y')^2dx$

The extremal of the function $$I=\int\limits_0^{x_1} y^2(y')^2dx$$ that passes through $(0,0)$ and $(x_1,y_1)$ is a constant function a linear function of x part of a parabola part of ...
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Can any functions be expressed in one formula? (without conditions)

This is the extension of my previous inquiry: Is it possible to describe the Collatz function in one formula? Can each of all functions be expressed in one formula? That is, can any function ...
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Solving $f(x)=f(\frac{1}{x}),\,\,\,x>0$

What are the non-trivial solutions of the following functional equations: $$f(x)=f(\frac{1}{x}),\,\,\,x>0$$ given $f$ if differentiable on $(0,\infty)$. $\textbf{Edit:}$ Do all the solutions ...
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Functional equation $f(h(y)x+y)=g(y)f(x)$
(Note: this is a simplified version of my previous question, which was not answered). I am seeking the solution for the functional equation $f(h(y)x+y)=g(y)f(x)$ where $f,g,h$ are continuous. ...