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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1answer
56 views

Find all continuous functions $f:\mathbb{R}\to\mathbb{R}$ satisfying $\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}$

Find all continuous $f:\mathbb{R}\to\mathbb{R}$ satisfying $$\frac{f(x+3)}{3+f(x)}=\frac{4+x^2}{x^2}.$$ I believe the original question was $$\frac{f(x)}{3+f(x)}=\frac{4+x^2}{x^2},$$ which has a ...
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1answer
27 views

How to solve the delay algebraic equation $xf(x) + \alpha f(x - {x_0}) - \alpha f(x + {x_0}) = 0$?

In the process of solving a problem, I am faced with the problem of finding a non-zero function $f:\mathbb{R} \to \mathbb{R}$ which satisfies the equation $$xf(x) + \alpha f(x - {x_0}) - \alpha f(x + ...
14
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1answer
1k views

Prove that function is continuous without knowing the function explicitly

Let $f\colon \mathbb R^+\to\mathbb R$ be a function that satisfies the following conditions: $$\tag1 \lim_{x\to 1}f(x)=0 $$ $$\tag2f(x_1)+f(x_2)=f(x_1x_2)$$ Show that $f$ is continuous in its domain. ...
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0answers
26 views

Calculus of Variations for discrete functionals

Question: How does one determine the optimal function f that either maximize or minimizes: $$\int_{x_1}^{x_2} L\left(x, f, D_{h,x}[f ]\right) dx$$ Whereas: $$ D_{h,x}[f] = \frac{f(x + h) - ...
5
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0answers
107 views

$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$

I have recently met with this integral: $$\int_0^1 \frac{\ln(1+x^a)}{1+x}\, dx$$ I want to evaluate it in a closed form, if possible. 1st functional equation: $\displaystyle f(a)=\ln^2 2-f\left ( ...
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0answers
32 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 $k>1$?
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0answers
34 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 + ...
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1answer
27 views

Defining a rectangular prism using a formula and complex numbers.

I recently read that a line can be defined using the formula $$ A = O + dL $$ where $A$ represents any point on the line, $O$ represents the vector origin of the line, $L$ represents the direction ...
3
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0answers
393 views

What special role plays the function $\pi^{\frac x\pi}$ in analysis?

I have tried to redefine some special functions in the most "natural" way, that is the way which allows to simplify the relations the most. I would call these functions "parelementary". The ...
3
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4answers
140 views

A functional equation: $4f(x)^3 +f(3x)=3f(x)$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$4f(x)^{3}+f(3x)=3f(x)$$ I know of 2 functions that satisfy the equation but I do not know how to prove that they are the only ones. ...
7
votes
3answers
267 views

Does there exist a function such that $f(a)f(b)=f(a^2b^2)?$

Given $S=\{2,3,4,5,6,7,\cdots,n,\cdots,\} = \Bbb N_{>1}$, prove whether there exists a function $f:S\to S$, such that for any positive $a,b$: $$f(a)f(b)=f(a^2b^2),a\neq b?$$ This is 2015 ...
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2answers
40 views

Continuous, 1-periodic $f$ with $f(x+y) = f(x) f(y)$ for $x, y \in \mathbb{R}$

The problem is to find all $f : \mathbb{R} \to \mathbb{C}$ that is continuous, has $f(x) = f(x+1) \forall x$, and $$f(x+y) = f(x) f(y) \quad x, y \in \mathbb{R}$$ Plug in $y=0$, we find $f(x) = ...
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1answer
53 views

What function satisfies the following equation?

$$f(x)e^{-x}\Gamma(x/\pi)=f(\pi/2-x)e^{x-\pi/2}\Gamma(1/2-x/\pi)$$ I think it should be similar to Zeta function, but what is it exactly?
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1answer
58 views

Solving the functional equation $f(x)=f(f(x-p))+q$

I can see that $f(x) = x + (p-q)$ is a solution. Is this the only possible solution?
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3answers
285 views

Find all real functions that satisfy the functional equations $f(x+y) = f(x) + f(y)$ and $f(xy)=f(x)\,f(y)$ [closed]

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy the two functional equations $f(x + y) =f(x)+f(y)$ and $f(xy)=f(x)\,f(y)$.
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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 ...
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0answers
15 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 ...
2
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2answers
78 views

What is the formula and the name of the mathematical-phenomenon seen at the ending of “Around the World in Eighty Days”?

Spoiler in brief for those who don't know the ending yet: At the end, Phileas Fogg alongside with his companions realize that they have arrived back to London a day earlier than expected due fact the ...
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3answers
1k views

If $ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x $, then $f(x)=x$

Let $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ be a function which has the following property: $$ f(x \cdot f(y) + f(x)) = y \cdot f(x) + x \;,\; \forall \; x, y \in \mathbb{Q} $$ Prove that $ f(x) = ...
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6answers
168 views

Functional equation $ f(x)+f(x+1)=x$

What functions satisfy $f(x)+f(x+1)=x$? I tried but I do not know if my answer is correct. $f(x)=y$ $y+f(x+1)=x$ $f(x+1)=x-y$ $f(x)=x-1-y$ $2y=x-1$ $f(x)=(x-1)/2$
6
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2answers
204 views

Find $f(x)$ satisfy $f(2x)=2f(x)+x$

I would appreciate if somebody could help me with the following problem: Find $f(x)$, given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous at $x=0$, and ...
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1answer
32 views

monotonic function. I need to show ots linear

If a real additive function f is monotonic, then it is linear. I need to show that the monotonic function f that satisfies caushy additives functional equation is linear
1
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1answer
78 views

Show that f is linear

Let $f : \mathbb R \to \mathbb R$ be a solution of the additive Cauchy functional equation satisfying the condition $$f(x) = x^2 f(1/x)\quad \forall x \in \mathbb R\setminus \{0\}.$$ Then show that ...
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1answer
69 views

analitycs solutions to the equation $f'(x)=f(x)f(x-1)$

As the title says I'm serching for functions ($C^n$ or analitycs $f$) that satisfies $f'(x)=f(x)f(x-1)$ some details: I've come at this equation after looking for a function $g$ satisfying for ...
1
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0answers
38 views

Generalized Riesz theorem of operator value function

I am reading a book Gustafson-Rao's Numerical Range and come across a problem that I really don't understand. In theorem 2.1-2 of the book, it asserts that for an operator valued function ...
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0answers
45 views

What problems are related with the following type of FDE with delay?

Consider the following class of functional differential equations with delay: $$\begin{align} \frac{du}{dt} &= F(x,t,u(x,t),u_{t,x}), & (x,t) &\in [a,b] \times [0,T] \\ u(x,t) &= ...
7
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0answers
102 views

Looking for all sequences such that $a_i^2+a_j^2=a_k^2+a_l^2$ whenever $i^2+j^2=k^2+l^2$

I'm working in a difficult functional equation, and I have reduced the problem to the following question ($\mathbb{N}$ denotes the set of non negative integers $0,1,2,3,4\cdots$) Question: Can we ...
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2answers
81 views

Show that $f$ is a Cauchy function

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a solution of the functional equation $$|f(x + y)| = |f(x)| + |f(y)| \quad \forall x,y \in\mathbb{R}.$$ Show that $f$ is an additive function.
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1answer
78 views

Find the functions

Find all the functions $ f : \mathbb{Q} \rightarrow \mathbb{Q} $ with the following property: $$ f(x + 3f(y)) = f(x) + f(y) + 2y, \: \forall x, y \in \mathbb{Q} $$
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2answers
91 views

functional equations.. I need hints for this problem

Find all functions $f : \mathbb{R} → \mathbb{R}$ that satisfy the functional equation $f(x + y) = f(x) + f(y) + xy$, for all $x,y ∈ \mathbb{R}$.
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3answers
580 views

Do there exist functions $f$ such that $f(f(x))=x^2-x+1$ for every $x$?

My question is on the existence (or not) of a function $f:\mathbb{R}\to\mathbb{R}$ which satisfy the equation: $$f(f(x))=x^2-x+1 \text{ for every }x\in\mathbb{R}$$ Supposing that such a map do exist ...
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0answers
53 views

A matrix version of Fredholm integral equation of the second kind

Fredholm integral equation are often encountered in physics. I have to deal with the following Fredholm matrix equation \begin{equation} \big[ L (x,y) \big]_{ij} = \big[ A (x,y) \big]_{ij} + \sum_{k} ...
3
votes
3answers
120 views

Find $f(2)$ if $f$ satisfies $2f(x)-3f(\frac1x)=x^2$

The following expression is given, and we are asked to find $f(2)$. \begin{equation} 2f(x)-3f\left(\frac{1}{x}\right) =x^2 \end{equation} Does a unique and well defined answer exist? Why? and what ...
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4answers
100 views

Determine all functions (functional equation) [closed]

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the functional equation $$f(x + y) + f(z) = f(x) + f(y + z)$$ for all $x, y, z \in \mathbb{R}$.
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0answers
33 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.
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2answers
164 views

Find value of a functional equation

Find $f(x)$ such that $$2 f(n) + \frac{1}{3}f\left(\frac{1}{n}\right) = 12.$$ Can anybody suggest me a way to solve this kind of functional equations?
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1answer
35 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 ...
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1answer
35 views

How to solve a set of equations where the unknowns are a function and some parameters?

I'd like to know how to solve something like this: $$\begin{eqnarray} f(f(x_2)-f(x_1)) & = & 27.5\\ f(f(x_3)-f(x_1)) & = & 21.6\\ f(f(x_4)-f(x_1)) & = & 15.1\\ f(f(x_5)-f(x_1)) ...
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1answer
26 views

Checking that a defined map is satisfied by some given condition

Suppose I have a group homomorphism $f: (\mathbb{R}^{2}, +) \rightarrow (\mathbb{R}, +)$ defined by $$f(x,y)= f(\frac{x+y}{2}, \frac{x+y}{2})+f(\frac{x-y}{2}, \frac{y-x}{2}) \text{,}$$ where $x, y ...
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0answers
58 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 ...
1
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1answer
28 views

Functional equation over $f(x) = \int_0^{ax}f(t)dt + g(x)$

Let $a\in(-1,1)$ and $g\in C^{\infty}(\mathbb{R}, \mathbb{R})$. Let $S(a, g)$ the set of all f such that : $$f(x) = \int_0^{ax}f(t)dt + g(x)$$ The first part was to show that : $$S(a, 0) = ...
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1answer
48 views

Technique to compositive functional equation

What is function $f,g:\Bbb R^+\rightarrow\Bbb R$ sought that satisfies $$\forall x\in\Bbb N,\,f_{(r)}(x)=\underbrace{f(f(\dots(f(f(x)))\dots))}_{r \,\mathsf{times}}=2^{(\log x)^c}$$ $$\forall ...
3
votes
2answers
121 views

Find all real functions $f:\mathbb {R} \to \mathbb {R}$ satisfying the equation $f(x^2+y.f(x))=x.f(x+y)$

Find all real functions $f:\mathbb {R} \to \mathbb {R}$ satisfying the equation $f(x^2+y.f(x))=x.f(x+y)$ My attempt - Clearly $f(0)=0$ Putting $x^2=x,y.f(x)=1$, we have $f(x+1)=x.f(x+y)$. Now ...
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votes
0answers
41 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 : $$f_1(x) - f_1(x-1) = f_1(g(x))$$ $$f_2'(x) = f_2(g(x))$$ ...
3
votes
1answer
100 views

Functional equation defined over non-negative real numbers

I'm new to this forum and I don't know how to write mathematical symbols. I have the following functional equation: $f$ defined on $[0, +\infty)$ with values in $[0, +\infty)$ $f$ is bijective and ...
2
votes
4answers
453 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
95 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
167 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
49 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 ...
17
votes
3answers
544 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 ...