# 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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### The existence of integral equations solution for a 2-dimensional unknown function

Suppose $f(\cdot,\cdot)\in C[0,1]^2$ is a kernel. $f$ is integratble $\int_0^1\int_0^1 f(x,y)dxdy<\infty$. $a,b\in C[0,1]$ are known functions, and $z(\cdot,\cdot)\in C[0,1]^2$ is a 2-dimensional ...
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### Correct math notation to show min and max of a list of numbers and sign?

I am working on a paper that uses a calculation from DIN 15018. The calculation is only described in text and not as a math equation. I would like to display as an actual equation if possible. The ...
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### Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
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### Is determining a non-constant solution to a functional inequality with polynomial arguements decidable?

So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say integer constants and $p_i$ are polynomials, say with ...
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### Multiple integrals satisfying functional equations

If $f:\mathbb{R}_+\to\mathbb{R}_+$ is a positive function which is locally $L^1$ with $\int_0^t{f(s)ds}=at^n$ for all $t\geq0$ then, by differentiating, it follows that $f(t)=a$ if $n=1$ and $f=0=a$ ...
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### Functional equation for the given function

For instance, there is functional equation for Lambert W function $z=W(z) e^{W(z)}$ And moreover, there is differential one: $z(1+W)\frac{dW}{dz}=W$. At the same time, there is no known functional ...
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### Is the gamma function a solution to this problem?

A real differentiable function on $\mathbb{R}^+$ satisfying $f(x) = x!$ for $x\in\mathbb{N}$ and having minimal derivative $\left|\frac{\partial f}{\partial x}\right|$ everywhere, say in the sense ...
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### Does mathematical modeling assist in learning how to derive equations?

I am hoping someone can provide a starting point to the question: Would mathematical modeling be a good place to start learning how to derive equations and functions from a set of data? For example, ...
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### Functional equation with only two solutions?

Recently I started studying functional equations. Now I'm trying to find all solutions to the following functional equation: $$(f(2x))^3=f(4x)((f(x))^2+xf(x)).$$ Unfortunately, I was able to show only ...
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### System of linear Volterra integral equation

Consider the Volterra integral equation $$f(t) = g(t) +\int_0^t K(t,s) f(s) ds$$ where $g(t)$ is continuous on $t\in[0,T]$ and $K(t,s)$ is a weakly singular kernel. It is well known that there ...
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### Functional equation $f(f(x)+3y)=12x + f(f(y)-x)$

I found this problem on a French exchange forum : Find all the $f : \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)+3y)=12x + f(f(y)-x)$ In fact I solved the problem when $f$ is supposed to be ...
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### Do I need to verify solutions to functional equations?

I am studying the (basics of) solving functional equations. My teacher stipulates that we check any solutions obtained by substitution. Similar guidelines are given in this IMO training material. For ...
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### Minimizing $\int_{0}^{1} (1+x^2)f(x)^2 dx$ for $f(f(x)) = x^2$

What is $$\min_{f\in D} \int_{0}^{1} (1+x^2)f(x)^2\mathrm dx,$$ where $D$ is the collection of all continuous real functions from $[0,1]$ such that in the interval $[0,1]$ we have $f(f(x)) = x^2$. ...
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### Solve the system of functional equations.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \begin{cases} f(x(1+f(x)))=f(x)^2,\\ f(x(1-f(x)))=f(x) f(-x),\\ f(x(-1+f(-x)))=f(x)f(-x). \end{cases} I have found only that for ...
An exercise in Real and Complex analysis (exercise 18, page 195,chapter 9) Show that if a function $f$ on the real line $\mathbb{R}$ satisfies \begin{equation*} f(x+y)=f(x)+f(y) \end{...
Suppose we have the following functional differential equation: $$f(a_0+a_1x+a_2f(x))(b_1+b_2f'(x))=c_0+c_1x+c_2f(x)$$ It is easy to see that a linear function: $f(x)=d_0+d_1x$, with appropriate ...