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 ...

learn more… | top users | synonyms

1
vote
0answers
42 views

if I get the asymptotic solution of a certain equation involving $ f(x)$ does it mean that the solution exists

Let's take a complicated functional equation $ f(g(x))=f(1-x)g(x) $. Let us suppose that by using a) Analytic method b) Numerical method I can prove that for example $ f(x) \sim x $ as $ ...
1
vote
0answers
111 views

where can I find a good Proof for Cauchy's functional equation

Do you know where can I find good proofs for Cauchy's functional equation?
1
vote
0answers
687 views

Solving a functional discrete equation.

I was to solve the following functional discrete equation (I arguing that $a_k$ is a discrete function): \begin{equation}f\left[a_{k+1}\right]-f\left[a_{k}\right]=0\end{equation} where ...
1
vote
0answers
115 views

solution of d’Alembert’s equation.

i know that equation for d’Alembert’s equation. is looking so $g(x+y)+g(x-y)=2*g(x)*g(y)$ so am trying to find actual solution for this equation,first i took $x=y=0$ and i got $2*g(0)$=$2*g(0)^2$ ...
1
vote
0answers
123 views

Could anyone derive a formula for this?

Edited: I want to get a sentiment score of various sentences and I've tried coming up with an equation that could satisfy the conditions that are inherent to each sentence (It's estimated mood as ...
1
vote
0answers
407 views

functional derivative of an integral of the function itself

I have the following $$ \frac{d}{dn(x)} \int_{x \in \cal{R}^3} {n(x) dx} $$ I know that this additional relationship holds $$ \int_{x \in \cal{R}^3}{n(x) dx} = N $$ where N is a constant. My ...
0
votes
0answers
13 views

Stability of Cauchy exponential functional equation

If f : R → R is a function satisfying |f(x + y)- a^xy f(x) f(y)| ≤ δ for all x,y ∈ R and for some positive δ, where a is a positive real constant, then show that either the function $f(x) ...
0
votes
0answers
24 views

D'Alembert's functional equation. I need to solve this problem

Let $λ$ be a nonzero real constant. Find all functions $f,g \colon \mathbb{R} \to \mathbb{R}$ that satisfy the functional equation $$ f(x + y) + g(x-y) = \lambda f(x) g(y) $$ for all $x,y \in ...
0
votes
0answers
9 views

Functional equation in distribution

Neagu(1984) reformulated the Pompeiu functional equation (PE) in distribution and determined the distributional solution of (PE) My question is what is meaning of "functional equation (PE) in ...
0
votes
0answers
34 views

Unique solution of a simple functional equation

Let $x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R}$ be two smooth functions ($x,y\in C^{\infty}([a,b])$). How can I prove that there is a unique function $\theta:[a,b]\to\mathbb{R},\ \theta\in ...
0
votes
0answers
30 views

The most general solution to the functional equation

Suppose we have the following functional equation $$X(x_1,x_2)Y(x_1,x_3)=Z(x_2,x_3).$$ Is the most general solution given by $$X(x_1,x_2)=A(x_1)B(x_2), Y(x_1,x_3)=\frac{C(x_3)}{A(x_1)}, ...
0
votes
0answers
25 views

Approach on solving limit equation systems and finding some f given assymptotes?

This is a "reverse" question of finding the asymptote of a function Recently, I am interested in doing some sort of modelling which involve equations of the form $$@(t)=1-f(t)$$ where $f(t)$ is ...
0
votes
0answers
16 views

Iterative Functional Equation? (discrete, increment, logarithm)

Say $f(n)$ is defined as discrete iterative process, $n \in [0,1,2,\ldots,N]$, and $g(x)$, $x \in [-\infty,+\infty]$ is "small": $|g(x)|<1, \forall x$: $$f(n+1)=f(n) + \log(1 + g(f(n)))$$ ...
0
votes
0answers
23 views

Functional equation similar to Babbage's equation

I'm interested in the potential solutions $f: R_{+} \rightarrow R_{+}$ to the functional equation: $ \forall x \in R_{+}, \quad f(s(x) - f(x)) = ax $ where $a>0$ is a constant. $s: R_{+} ...
0
votes
0answers
57 views

Extension of the Cauchy functional equations

Let $f:(0,a)\to\mathbb{R}$ satisfying $$f(x+y)=f(x)+f(y)$$ for all $x,y,x+y\in(0,a)$, where $a$ is a positive real number. Show that there exists an additive function $A:\mathbb{R}\to\mathbb{R}$ such ...
0
votes
0answers
15 views

Solving a linear functional equation

Working with Green functions, I have found to solve the following equation $$ -\omega^2G(\omega)-m^2G(\omega)+\kappa\sum_{n=-\infty}^\infty b_nG(\omega-n\omega_0)=1 $$ where $m$, $\kappa$ and ...
0
votes
0answers
31 views

Does this problem have analytic/approximate analytic solution?

Define $F[i]=F[i-1]-F[i-1]^c$ with $F[0]=r>0,c\in(0,1)$. Define $R(r,c)=\min\{i:F[i]<1\}$. What is solution to $c\in(0,1)$ in $R(r,c)=r^{1/k}$ with some given $k>1$?
0
votes
0answers
25 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
0
votes
0answers
13 views

Closed form of chained Stationary Functions

Consider the following operator $$D_{(a-1)x,x}\left[ f \right] = \frac{f(ax) - f(x)}{(a-1)x} $$ It's pretty easy to see that a nontrival function $g(x)$ such that $$D_{(a-1)x,x}[g] = g$$ Is given ...
0
votes
0answers
14 views

Recurrence relation involving ordinary generating function

Let $f_1,f_2,\ldots$ be a given infinite sequence of functions. Define the sequence of functions $F_1,F_2,\ldots$ by the recurrence relation $$F_n(x)=f_n(x)\sum_{k=0}^\infty F_{n+1}(k)x^k$$ or ...
0
votes
0answers
31 views

show the following: (functional equations)

If f : R to R is a solution of the additive Cauchy functional equation, then show that f is either everywhere or nowhere zero.
0
votes
0answers
22 views

Periodic Function With A Given Functional Equation

Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant '$a$' the equation $f(x+a)=1/2+\sqrt{f(x)-(f(x))^2}$ holds for all $x$. Then prove that $f$ is ...
0
votes
0answers
50 views

Differential equation involving composition

I've been studying Euler's method to approximate a solution to a differential equation in an algorithm class. I faced a weird differential equation in a mathematics exercise, and I wanted to know if ...
0
votes
0answers
39 views

Prove $\lim\limits_{x \to +\infty} \frac{f_1(x)}{f_2(x)} = \text{Constant}$

Let $g(x)$ be a real-analytic strictly rising function for $x>0$. Define for $x>1$ two real analytic functions $f_1,f_2$ such that : $$f_1(x) - f_1(x-1) = f_1(g(x))$$ $$f_2'(x) = f_2(g(x))$$ ...
0
votes
0answers
7 views

find the distribution of 100 heart transplant patients at a low volume and high volume hospital using boxplot graph 0-40 mortality

Using a boxplot graph find the distribution show mortality rates within one year of 100 patients having heart transplants at various hospitals. The low volume hospital perform between 5 and 9 ...
0
votes
0answers
30 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which is a convolution kernel. I am interested in the following equation: $0=A+B~(\varphi*v)+C~v+D~v~(\varphi*v)$, where ...
0
votes
0answers
39 views

Existance of solution to integral Fredholm equation

I am a bit confused with the existence/non existence of a solution to the following equation: $$\int_{-T}^{T} R(t,\tau) x(\tau) d\tau = y(t), \forall t\in [-T,T]$$ where $y(t)=C$ (a constant ...
0
votes
0answers
11 views

Prove that a functional has an unique global minimun.

Consider the funcional $$ E(u)=\frac{1}{p}\int_{\Omega} |\nabla u|^pdx-\int_{\Omega}fudx. $$ Where $u \in W^{1,p}_{0}(\Omega)$, $\Omega\subset\mathbb{R}^n$ is a regular and bounded domanin, $p \in (1, ...
0
votes
0answers
17 views

Meta Euler Lagrange

Consider the following generalized standard Euler Lagrange Problem which is to maximize the quantity $$ \int_{x_1}^{x_2} L(x,y,y', y'' ... y^{(n)}) dx $$ It can be solved by first determining a ...
0
votes
0answers
43 views

Conjecture about $A f(x) = f(g(x)) + f(h(x))$

Let a given real $A$ satisfy $0 < A < 2$. Conjecture : For any real entire nonconstant $f(x)$ there exist real entire $g(x)$ and $h(x)$ such that $A f(x) = f(g(x)) + f(h(x))$ or $ A f(x) ...
0
votes
0answers
16 views

In which condition related to coefficients, some equation has a solution?

My purpose is to know in which condition related to the coefficients $c>0, \, n\geq 2, 0<p<1,\, a>c\, \, \, \text{and} \, \, b>0$, this equation $$ F(x)= -c x^{n+p} -bx^n + (a-c)x^p ...
0
votes
0answers
18 views

Solving functional equation $b(x)=\int b(xy)f(y)dy$

I want to prove that given a real-valued smooth function $f$, the set of functions $b$ solving $b(x)=\int_0^{\infty} b(xy)f(y)dy$ is given by linear combinations of $x^{\sigma}$ where $\sigma$ is a ...
0
votes
0answers
30 views

Find all functions: $f:C\rightarrow C$

Find all functions $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))$ Extension: Find all $f:C\rightarrow C$ such that $f(x)=(x-f(0))(x-f(\pi))(x-f(-\pi))(x-f(2\pi))(x-f(-2\pi))\cdots ...
0
votes
0answers
34 views

Does there exist a solution of that system of functional equations?

Does there exist a non-constant rational function $f(x)$ (i. e. a ratio of two polynomials in $x$ over the reals) which simultaneously satisfies $f(x)=f(1-x)$ and $f(x)=f\left(\frac 1 x \right)$ on ...
0
votes
0answers
16 views

Extract independent paramets

I have a 2-variable function which depends also on a number of parameters (6 to be exact) $f(x,y; c1, c2, c3, .. c6)$. The explicit form is quite complicated so I will not give it here. It suffices to ...
0
votes
0answers
21 views

unknown function: calculation of coefficients in series expansion up to a given degree

I am trying to solve an functional equation of unknown $h\mapsto h(x)$ ($x\in\mathbb{R}$, in the neighbourhood of $0$): $$\mathcal{F}(h)=\mathcal{G}(h) \qquad (*)$$ (assume $\mathcal{F}$ and ...
0
votes
0answers
28 views

Existence of a function satisfying all the given infima

Given a function $f : \mathbb{R} \to \mathbb{R}$, we can compute its infimum over $A$ for all the Borel measurable $A \subset \mathbb{R}$. I am wondering when we can deduce in the other direction, ...
0
votes
0answers
19 views

obtain the continuous form of a master equation

I have the following master equation: $$ \partial_tP(x,y,t)=-[x+(1-x-y)]P(x,y,t)\\ +(x+\epsilon)P(x+\epsilon,y-\epsilon,t)\\ +(1-x-y+\epsilon)P(x,y-\epsilon,t) $$ where $\epsilon$ is a small number, ...
0
votes
0answers
29 views

An equation with a nested function

I'm trying to find the function $\eta(x)$ such that $\eta(x) F(\eta(x))-G(\eta(x)) = \eta(x)H(x) - G(x)$ but I have no idea how to go about it, or where to look. Thanks for the inputs. All ...
0
votes
0answers
45 views

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$?

What functions satisfy $\int_{-\infty}^{\infty} f(x,y)dx=\int_{-\infty}^{\infty} g(x,y)dx$ for all $y\in \mathbb{R}$? Under what conditions would this imply that $f(x,y)=g(x,y)$?
0
votes
0answers
20 views

Methods of showing that a non-trivial solution exists for a functional equation?

I am just looking at a one variable functional equation, I won't put it down here because it is university related, and I keep thinking that 0 is the only possible solution so it got me wondering how ...
0
votes
0answers
14 views

Measuring availability of a service

I have a task to measure availability of some IT-components. The problem is that I need to create a equation for measuring this. For now the availability is going to be measured through life-cycle ...
0
votes
0answers
48 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
0
votes
0answers
33 views

A functional equation for a Dirichlet serie

I'm looking for a functional equation for the following Dirichlet serie $$\varphi(s)=\sum_{n=1}^{+\infty}{\frac{2 \cos(2n \pi q)}{n^s}}$$ where $q$ is a rational number. Any help ? Thank you !
0
votes
0answers
22 views

Logarithmic functional equation on positive rational numbers

Let $\Bbb Q^+$ be the set of positive rationals and $f:\Bbb Q^+ \to \Bbb R$ be a continuous function satisfying $$ f(rs)=f(r)+f(s) $$ for all $r, s\in \Bbb Q^+$. Then $f$ has only the form $$ f(r)=c ...
0
votes
0answers
53 views

Does there exist a nontrivial cumulative distribution function $F$ on $\mathbb{R}_+$ for which $(F^2)'= (F')^{\ast k}$ for some $k > 0$?

This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ...
0
votes
0answers
53 views

Non-constant solution of $f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$ in rings

The answers to this question prove that over an integral domain the functional equation $$f(x+y+z)=f(x)g(y)+f(y)g(z)+f(z)g(x)$$ has only solutions where one of $f$ or $g$ is constant. The proofs make ...
0
votes
0answers
51 views

Integral Functional Equation

If $f(x) = 1 + \int_{x/c}^1 f(t) dt$, find $f(x)$ in terms of $c$. This problem was inspired by trying to find the expected value of the longest increasing subsequence $a_1, a_2, a_3, \cdots, a_n$ ...
0
votes
0answers
52 views

Follow up to previous functional equation question.

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ s.t $$f(x^2+f(y))=(x-y)^2f(x+y)$$ Putting $x=y$ yields $f(x^2+f(x))=0$. (1) Putting $y=-x$ yields $f(x^2+f(-x))=(2x)^2f(0)$. (2) By (1) $(2x)^2f(0)=0$ ...
0
votes
0answers
684 views

An example of the Sequential Quadratic Programming (SQP)

Given an objective function $f(x,y)=(x+6)^2+(y-10)^2$, we want to minimize the function $f$ under a constrained condition $xy-5\leq0$. Obviously it is a constrained optimization, a good algorithm to ...