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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4
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4answers
116 views

Problem in solving functional equation.

To find all functions $f$ which is a real function from $\Bbb R \to \Bbb R$ satisfying the relation $$f(x^2 + yf(x)) = xf(x+y)$$ It can be easily seen that the identity function $i.e.$ $f(x)=x$ and ...
0
votes
2answers
75 views

$f(xy)=\frac{f(x)+f(y)}{x+y}$ Prove that $f$ is identically equal to $0$

For all $x,y\in\mathbb{R}$. also $f : \mathbb{R} → \mathbb{R}$ and $x+y\not=0$. My attempt: I restated it as $a[x^2 y^2 (\frac{x}{y}+\frac{y}{x}-\frac{1}{y^2}-\frac{1}{x^2})] + ...
3
votes
3answers
117 views

Solving functional equation $f(x)f(y) = f(x+y)$

I'm having some trouble solving the following equation for $f: A \rightarrow B$ where $A \subseteq \mathbb{R}$ and $B \subseteq \mathbb{C}$ such as: $$f(x)f(y) = f(x+y) \quad \forall x,y \in A$$ ...
4
votes
0answers
37 views

Solution techniques for f'(x)=f(g(x))

I stumbled over this seemingly natural question and was surprised, that I couldn't find a satisfying answer. Differential equations of the type $f'(x)=g(f(x))$ are studied for all kind of classes of ...
4
votes
0answers
78 views

Improvement IMO 1988 $f(f(n))=n+1987$

The following problem was given at IMO 1987. Prove that there is no function f from the set of non-negative integers into itself such that $f(f(n)) = n + 1987$ for every $n$. So I tried to ...
0
votes
1answer
15 views

Calculation of Sigma (Multiple Sigmas)

When I was studying Game Theory, I came across this equation: $$F_i(q_1, \ldots, q_n) = \sum_{s_1 \in S_1} \: \ldots \: \sum_{s_n \in S_N} \big\{ \prod_{j=1}^n q_j(s_j) \big\} \: \: f_i(s_1, ...
1
vote
1answer
26 views

A functional equation over integers

I was working in a problem in number theory and I blocked over the problem : Given functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$, $g:\mathbb{Z^2}\rightarrow \mathbb{Z}$ and ...
7
votes
2answers
106 views

For all $x,y\in G$ we have: $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?

A group $G$ and a function $f:G\longrightarrow G$ are given and for all $x,y\in G$ satisfying $f(xf(y))=f(x)y$. Prove that $f$ is an isomorphism?
0
votes
1answer
23 views

Fixed Point Iteration $x = g(x)$ method for $y_1 = e ^{-x}$ and $y_2= \cos x$

The question reads as follows: Find the x and y coordinates of the intersection points by means of the $x = g(x)$ method. ( I believe they are referring to the Fixed Point Iteration method) The ...
5
votes
1answer
32 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
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
1answer
95 views
+100

Solutions to functional equation $ \gamma(s,t)=f(t \cdot g(s))+h(t) $

Let $$ \gamma(s,t)=f(t \cdot g(s))+h(t) $$ where $\gamma$ is a known function of $s \in \mathbb{R}$ and $t \in \mathbb{R}$ while $f$, $g$, and $h$ are unknown functions. Assume ...
10
votes
4answers
118 views

Polynomial equation $f(x)f(2x^2)=f(2x^3+x)$

Find all polynomials $f(x)$, for which $f(x)f(2x^2)=f(2x^3+x)$. I have no idea how to do it.
1
vote
0answers
31 views
+50

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
2
votes
1answer
36 views

Functional equation extended solution

The question is Find all functions $f:R \to R$ such that $$f(x+y)f(x-y)=(f(x)+f(y))^2-4x^2f(y)$$ Taking $x=y=0$, we get $f(0)^2=4f(0)^2 \implies f(0)=0$. Now take $x=y$ which immediately gives ...
2
votes
3answers
258 views

Solve the following functional equation

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ so that: a) $[f(x)+f(y)][f(x+2y)+f(y)]=[f(x+y)]^2+f(2y)f(y)$ b) for every real $a>b\ge 0$ we have $f(a)>f(b)$ As much as I know: ...
4
votes
2answers
110 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get ...
0
votes
1answer
30 views

Functional equation, probably involving discrete differentiating?

I want to know how to solve this problem on functions. Question: Find all functions $f: \mathbb{R}\rightarrow \mathbb{R}$ satisfying $$f(x+1)-f(x)=nf(x)$$ where $\mathbb{R}$ is set of real ...
0
votes
0answers
16 views

Commutative version of hyper operators.

As I understand it, addition and multiplication are defined on the reals as having identity elements 0 and 1 and being commutative and associative. Multiplication is also distributive over addition. ...
15
votes
4answers
265 views

Find $f''(x)$ if $f\circ f'(x) = 4x^2 + 3$

Can you tell me the solution of this question? If: $f\circ f'(x)=4 x^2 +3$ then what is $f''(x)$? This was a question in math test which I just took yesterday. One function satisfying the ...
2
votes
0answers
42 views

Is the product rule for logarithms an if-and-only-if statement?

If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$ My question is, Is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ ...
3
votes
2answers
106 views

Solutions of the functional equation $f(2x) = \frac{f(x)+x}{2}$

How can I solve the following functional equation? $$f(2x) = \frac{f(x)+x}{2},$$ for $x \in \mathbb{R}$ with $f$ being a continuous function.
0
votes
1answer
48 views

Solutions of the functional equation $f(x) + f(qx) = 0$

How can I find the solutions of $$f(x) + f(qx) = 0,$$ where $q \in \mathbb{Q}, q\neq1, x \in \mathbb{R}$, with $f$ being a continuous function?
0
votes
1answer
11 views

Criteria for elevating a rational FE solution to reals

I have a functional equation which I have solved for rational numbers. Now to 'elevate' my solution to reals, I use the given fact that f is continuous. I could have done the same process, if I knew ...
2
votes
3answers
47 views

Solve the funtional equation $f(xf(y)+x)=xy+f(x)$

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ so that $f(xf(y)+x)=xy+f(x)$. If you put $x=1$ it's easy to prove that f is injective. Now putting $y=0$ you can get that $f(0)=0$. $y=\frac{-f(x)}{x}$ ...
-3
votes
1answer
91 views

Find $f(x)$ such that $f(x+y)+f(x)=2f(x-y)+2f(y)$ [closed]

Problem: Let $f(x): \mathbb{R} \to \mathbb{R}$ such that: $$f(x+y)+f(x)=2f(x-y)+2f(y) \ \ \ \forall x,y \in \mathbb{R}$$ This is a problem in my analytics exam, I can't find it if $f$ is not a ...
2
votes
2answers
31 views

To find an extremal of a given functional

I have to find extremal of following : $\int_0^1 [(y')^2 + 12 xy] dx$ with $y(0) = 0$ and $y(1) = 1$. I applied the Euler's equation $\frac{\partial F}{\partial y} - \frac{d}{dx}(\frac{\partial ...
1
vote
0answers
22 views

the family of functions with alternating derivatives in different orders

I'm wondering whether there's a name for the family of functions with alternating derivatives in different orders. Formally, $f(x)\in C^\infty:\mathbb{R^+}\rightarrow\mathbb{R}$. For all x, ...
0
votes
0answers
15 views

A functional equation involving convolution

Let's say I have a smooth function $\varphi:\mathbb R \to \mathbb R$ which could be either: a convolution kernel or a difference of two convolution kernels: $\varphi=\varphi_1-\varphi_2$. I am ...
1
vote
1answer
23 views

Is $Tf=x_0+\int_0^tf(t)(1-f)\,dt$ a Lipschitz function?

Consider the Banach space $C[0,1]$ with the uniform norm and the operator given by $(T(f))x=x_0+\int_0^xf(t)(1-f(t))\,dt$ for $x\in[0,1]$ and $x_0\geq0$. I want to show that there is a number ...
0
votes
1answer
10 views

Proving that $(Sf)(x)=1+\int_0^x t^2f(t)\,dt$ is $K$-Lipschitz

I want to show that $S:C[0,1]\to C[0,1]$ where $$(Sf)(x) = 1+\int_{0}^{x} t^2\cdot f(t) \,dt$$ is a Lipschitz map for $x\in[0,1]$. There I use the Euclidean norm on $\mathbb{R}$ and the uniform norm ...
2
votes
1answer
28 views

Existence solution pendulum equation using the contraction principle

How can I prove that the differential equation $\ddot{x}(t)=-(g/\ell)\sin (x(t))$ must have a solution by using the contraction principle (by Banach). The numbers $g$ and $\ell$ are fixed constants ...
1
vote
1answer
44 views

Is there a $f\in C[0,1]$ such that $f(x)=\frac12 \sin (f(1-x))$

Is there a $f\in (C[0,1],\lVert\cdot\rVert_{\infty})$ such that $f(x)=\frac12 \sin (f(1-x))$? I feel like you need to apply Banach´s fixed point theorem, which means it suffices to prove that ...
2
votes
2answers
43 views

Is $T:C[0,1]\to C[0,1]: Tf(x)=\frac12 x f(x^2)+1$ a contraction?

I am a non-mathematician (a physicist) who wants to find a continuous function on $[0,1]$ that satisfies the relation $f(x)=\frac12 x f(x^2)+1$ with $0\le x\le 1$. I need it for one of my models. ...
3
votes
0answers
47 views

Problem in Putnum competition? [closed]

Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$ is a continuous function and $f(2x^2 -1)=2xf(x)$ for all $x\in \mathbb{R}$. Prove $f(x)=0\,\,\text{for all} \, x\in [-1,1].$
12
votes
2answers
140 views

Functions satisfying $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$

Find all functions $f$ such that $f:\mathbb{N}\rightarrow\ \mathbb{N}$ and $f(f(n))+f(n+1)=n+2$ Let us plug in $n=1$ $f(f(1))+f(2)=3$ Since the function is from $\mathbb{N}$ to $\mathbb{N}$, ...
2
votes
1answer
34 views

Is this a valid equivalence between the classes of Differential Equations?

Consider the general first order Linear Ordinary Differential Equation: $$ \frac{dy}{dx} = A(x,y) = \frac{F(x,y)}{G(x,y)}$$ This equation is characteristic equation of the Partial Differential ...
1
vote
1answer
38 views

Entire $f,g$ such that $f(f(z)) = p(g(z))$

Let $p(z)$ be a given polynomial of degree $3$. How to find nonconstant entire functions $f,g$ such that $f(f(z)) = p(g(z))$ or prove they do not exist ? I considered the related equation $f(f(z)) ...
8
votes
1answer
110 views

Functional Equation $f(f'(x))=-f(x)$

Assume $f:\mathbb{R}_{>0}\to\mathbb{R}$ is differentiable and satisfies $\forall x>0:f(f'(x))=-f(x)$. What is $f(x)$? I know that $f(x)=\ln x$ is a solution, but I don't know if there is ...
2
votes
0answers
43 views

Functions that hasn't any root

we say that a function like $f:X \to X$ has root if exists a function like $g:X\to X$ that for every $x \in X$: $$f(x) = g(g(x))$$ what is a necessary and sufficient condition for $f$ that it has a ...
2
votes
2answers
76 views

Continuous function with differentiation

Let $f: \mathbb{R} \to \mathbb{R} $ be a continuous function with $f(x+y) = f(x) +f(y),\forall x,y\in \mathbb{R}.$ Find $\frac {df} {dx} $, if it is exist?
3
votes
2answers
76 views

Find all real functions so that $f(xf(y)+f(x))=f(yf(x))+x$

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ so that $f(xf(y)+f(x))=f(yf(x))+x$ $f(x)=\pm x$ should be the only solution. It's easy to get that $f(f(0))=f(0)$.
4
votes
1answer
61 views

Is distributivity sufficient to define composition?

Function Composition has the property of distributivity: $$(f\star g)\circ h = (f\circ h)\star(g\circ h)\;\forall f,g,\star \in\{+,-,\times,\div\}$$ I was wondering if these properties uniquely ...
2
votes
2answers
100 views

When can we take that $f(1)=1$?

I have been doing some functional equations and in some of them they just say " WLOG let $f(1)=1$ ", but I don't get why they can do that... Can someone please help me? I can't find the example of ...
8
votes
3answers
102 views

Solve the following functional equation $f(xf(y))+f(yf(x))=2xy$

Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $f(xf(y))+f(yf(x)=2xy$. By putting $x=y=0$ we get $f(0)=0$ and by putting $x=y=1$ we get $f(f(1))=1$. Let $y=f(1)\Rightarrow ...
6
votes
0answers
84 views

Is there a function $f(x)$ such that $f(f(x))=e^x$? [duplicate]

Is there a function which would be a "functional square root" of the exponential function? I.e., function $f(x)$ such that: \begin{equation} f(f(x))\equiv f^2(x)=e^x \end{equation} If not, what is ...
0
votes
0answers
27 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 ...
2
votes
0answers
46 views

All $f$ such that $(\exists k)(f^\prime(x) = f(x+k))$ [duplicate]

I was wondering if there is a general way to solve the functional equation $$(\exists k)(f^\prime(x) = f(x+k))$$ I know that this is true for certain functions: $$(e^{cx})^\prime = e^{c(x+\frac{\ln ...
0
votes
0answers
10 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, ...
2
votes
0answers
33 views

Don't understand why this generating function needs to be taken to the power $-1$

I'm given the following recursive formula for a sequence: $u_{n+1}=u_n+\sum_{k=1}^{n-1}u_ku_{n-k}\ (n\ge1),\ u_0=0,\ u_1=1$ Since $u_0=0$ we can rewrite: $u_{n+1}=u_n+\sum_{k=0}^{n}u_ku_{n-k},\ ...