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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23 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) ...
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1answer
38 views

Integral equation involving Planck radiation formula

I am stuck in solving the following integral equation: $$\sigma T^4=\pi\int_{\lambda_0}^{\lambda_1}d\lambda W_{\lambda,T}$$ where: ...
8
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1answer
90 views

Complicated real to real functional equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x)+1)^2$$ for all $x,y \in \mathbb{R}.$ So far I have proved that $f$ is bijective. How should I continue?
4
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1answer
51 views

A functional equation with no term outside functions

Find all $f:\mathbb{N} \rightarrow \mathbb{N}$ satisfying $$f(m-n+f(n))=f(m)+f(n)$$ for all $m,n \in \mathbb{N}.$ I have no idea about how to find them, because there are no terms outside of the ...
4
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1answer
57 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
3
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3answers
202 views

What function satisfies $f(x)+f(−x)=f(x^2)$?

What function satisfies $f(x)+f(−x)=f(x^2)$? $f(x)=0$ is obviously a solution to the above functional equation. We can assume f is continuous or differentiable or similar (if needed).
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1answer
60 views

Trigonometric functional equations. I need hints for this problem

Find all functions $ f,g : \mathbb R \to \mathbb R $ that satisfy the functional equation $$g(x)f(y) = f \left(\frac{x + y}{2}\right)^2-f\left(\frac{x-y}{2}\right)^2$$ for all $x,y \in \mathbb R$.
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2answers
83 views

Solving the functional equation $f(x + y) + g(x-y) = \lambda g(x) f(y)$

Let $\lambda$ be a nonzero real constant. Find all functions $f,g : \mathbb R \rightarrow \mathbb R$ that satisfy the functional equation for all $x,y \in\Bbb R$: $$f(x + y) + g(x-y) = \lambda ...
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1answer
78 views

Proof regarding a probability generating function (Poisson)

Let $f(s)$ be the probability generating function ($pgf$) of a non-negative, integer valued random variable. It is also given that $f(1-p+ps)f(p) = f(ps)$. Prove that $f(s) = e^{\lambda(s-1)}$ for ...
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0answers
49 views

Cauchy's function

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) ...
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1answer
36 views

Pexerized Dalembert funtional equation..

Let $\lambda$ be a nonzero real constant. Find all functions $f,g: \Bbb R \rightarrow \Bbb R$ that satisfy the functional equation $f(x+y)+g(x−y)=\lambda f(x)g(y)$. I try this : Let $y=0$ in the ...
2
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4answers
77 views

Solve: $f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x).f(\frac{1}{y})$

Here i have one functional equation: If $$f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x)\cdot f(\tfrac{1}{y})\text{ for all x},y\in\mathbb{R}-{0}$$ and $f(0) = \frac{1}{2}$ , then find the value of ...
2
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2answers
81 views

How do I prove this function is monotonic?

Let $f:\mathbb R\to \mathbb R$ be a function such that $f(x+y)=f(x)+f(y)$ and $f(xy)=f(x)f(y)$ for every $x,y\in \mathbb R$ and $f(1)=1$. In order to prove this function is 1-1, I just need to prove ...
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0answers
32 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 ...
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2answers
113 views

Functional equation $f'(x)=cf(x+1)$ has a solution if and only if $c\leq 1/e$

In Contests in Higher Mathematics: Miklos Schweitzer Competitions, 1962-1991 by Gabor J Szekely, problem F.57 there is the study of $f~:~[0,\infty)\to (0,\infty)$ such that: $\exists c>0, \forall ...
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33 views

Solving a particular Functional Differential Equation

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 ...
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1answer
64 views

Prove equation using taylor series

Given $f(x)$, knowing that $f'(x)$ and $f''(x)$ exist for every $0\leq x\leq1$, and provided I know that $f(0)=f(1)$ and that for each $0\leq x\leq1$, $|f''(x)|\leq A$, how can I prove that ...
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5answers
82 views

Suppose $f$ is a real function satisfying $f(x+f(x))$ = $4f(x)$ and $f(1) = 4$. Then the value of $f(21)$?

Should I proceed with just putting the value of $f(1)=4$ in the first equation or there will be a different way of solving this ?
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1answer
60 views

Find all functions $f: \mathbb{Z} \to \mathbb{Z} $ such that for all $x,y, \in \mathbb{Z}$, $f(x-y+f(y))=f(x)+f(y)$. [closed]

Need help proving the following: Find all functions $f: \mathbb{Z} \to \mathbb{Z} $ such that for all $x,y, \in \mathbb{Z}$, $f(x-y+f(y))=f(x)+f(y)$.
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2answers
38 views

Functional equation $f(f(x))=2x$ on $\mathbb{Z}_{>0}$

I have this functional equation: $$f(f(x))=2x$$ with $f: \mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$. And I want to know if it is possible to list all solutions. I already know that ...
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5answers
129 views

$f(x)f(1/x)=f(x)+f(1/x)$

Find a function $f(x)$ such that: $$f(x)f(1/x)=f(x)+f(1/x)$$ with $f(4)=65$. I have tried to let $f(x)$ be a general polynomial: $$a_0+a_1x+a_2x^2+\ldots a_nx^n$$ which leaves $f(1/x)$ as: ...
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1answer
71 views

$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y)$ implies $f $ linear?

Let $X$ be a Banach space. $f:X \to X$ a continuous function. If we assume that $f$ satisfies the following convexity condition: $$\lambda f(x)+(1-\lambda)f(y)=f(\lambda x+(1-\lambda)y),$$ for all ...
2
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1answer
37 views

Solutions of $f : \Bbb Q^*_+ \to \Bbb Q$ such as $f(\tfrac1x)=f(x)$ and $f(x) = (1+\tfrac1x)f(x-1)$ for all $x$

This is an extension to : Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ What can be said about functions $f : \Bbb Q^*_+ \to \Bbb Q$ such as ...
3
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1answer
67 views

Functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$

How to find all functions $f: \Bbb Q_{+} \to \Bbb Z$ such that $f(\frac{1}{x}) = f(x)$ and $(x+1)f(x-1)= xf(x)$ for $x> 1$?
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1answer
80 views

Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$

Find all $ f : R\rightarrow R $ such that $f(x^2+f(y))=(f(x))^2+y, \forall\text{ x,y}\in R$ Thanks in advance!
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0answers
39 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 ...
2
votes
2answers
41 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
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0answers
33 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)}, ...
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1answer
20 views

How can one solve this equation in $Z^2$?!

Ho can one solve the egality $2x+3y=xy$ ? I have to find a value of $x$ in fonction of $y$ so ? I have to add somthing and substrate it I added -2xy then $2x(1+y)-3y(1+x)=0$ Here im suck Can ...
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2answers
24 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
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2answers
46 views

Deterministic condition for the nature of one real root of a cubic equation

A cubic equation $ax^3+bx^2+cx+d=0 \space$ where, $a\neq 0$ always has one real root. Is there any direct condition for determining the nature i.e. sign of one real root for sure? Is it possible ...
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1answer
48 views

How to solve a equation with special conditions?

I have this equation: $z = 11n + 13m$ Conditions: $z < 2015$ $z$, $n$ and $m$ must be natural numbers ($>0$). How many options are possible for $z$?
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1answer
81 views

Find the function satisfying the given condition

If $f(xy)=e^{xy-x-y}[e^y f(x) +e^x f(y)]$ and $f'(1)=e$. $f'$ denotes the derivative of function $f(x)$. Find $f(x)$. I could find that $f(0)=0$ and $f(1)=0$ and then found the derivative the got ...
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0answers
49 views

Find all differentiable functions $f$ such that $f(f(x))=f'(x)$ [duplicate]

Here is a problem I made up: Find all differentiable functions $f$ from the reals to the reals such that $f(f(x))=f'(x)$ for all real $x$.
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0answers
46 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 ...
4
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2answers
155 views

Solve differential equation $f'(z) = e^{-2} (f(z/e))^2$

I'm curious if there's a simple closed solution to the following DE and, if so, what it is. $$\begin{align} f'(z) &= e^{-2} (f(z/e))^2 \\ f(0) &= 1. \end{align}$$
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0answers
78 views

We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
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1answer
47 views

What are 'Regular Products'?

When looking at the functional equation for the Riemann zeta function, I came across the statement: For $s$ an even positive integer, the product $\sin{(\frac{\pi s}{2})}\Gamma({1-s})$ is regular. ...
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0answers
17 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": $|g(x)|<1, \forall x$: $$f(n+1)=f(n) + \log(1 + g(f(n)))$$ ...
5
votes
1answer
90 views

Find $f(x) $ given that: $f'(x)=\frac{f(x)-x}{f(x)+x}$ [closed]

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 differentiable function, and ...
7
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4answers
170 views

$f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is continuous , and $f(x+1)+f(x)=x^2$

I would appreciate if somebody could help me with the following problem: Find $f(x)$ ($f(x)$ is not Polynomial function), given that: $f \colon \mathbb{R} \rightarrow \mathbb{R}$, $f$ is ...
4
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0answers
103 views

Integer functional equation $f(f(f(n)))=f(n+1)+1$

Can you find all functions $f:\mathbb N\rightarrow\mathbb N$ satisfying the functional equation $$ f(f(f(n)))=f(n+1)+1 $$
3
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2answers
151 views

Solving functional equation for all real numbers.

The functional equation to be solved is $ f(x+y) +f(x)f(y)=f(x)+f(y)+f(xy)$. Domain:Reals,Codomain:Reals.I found about 4 possible solutions to the equation but ran into a fundamental problem with all ...
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0answers
61 views

Iterative (functional) roots of integer functions (functions on $\mathbb{Z}$)

A function $g:A\to A$ is called a $k$-th iterative root of another function $f:A\to A$ ($A$ an arbitrary set and $k\in\mathbb{N}$) iff $f=g^k$, where $g^k(x)=g\circ g\circ\ldots\circ g(x)=g(g(\ldots ...
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0answers
42 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_{+} ...
1
vote
1answer
23 views

Extension of the additive Cauchy functional equation

Let $f\colon (0,\alpha)\to \def\R{\mathbf R}\R$ satisfy $f(x + y)=f(x)+f(y)$ for all $x,y,x + y \in (0,\alpha)$, where $\alpha$ is a positive real number. Show that there exists an additive function ...
1
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1answer
28 views

hyperplane in $L^2$

Consider $R^2$ valued functions $f,g \in L^2([0,1],\mu, R^2)$ where $f=(f_1,f_2)$,$g=(g_1,g_2)$ and $\sqrt{\langle f, g\rangle}=\int_{[0,1]} (f_1(x)g_1(x)+f_2(x)g_2(x)) d\mu(x)$ Suppose for a given ...
2
votes
1answer
72 views

What are all pairs of functions f and g so that $f(x)f(y) = g(x+y)$?

It can be shown, and is a problem in Rudin's Principles of Mathematical Analysis (Chapter 8), that when $f$ is continuous, and $f(x)f(y) = f(x+y)$, $f$ is a function of the form $e^{cx}$. Must this ...
2
votes
4answers
85 views

Solving for $f(2004)$ in a given functional equation

Given that $$f(1)=2005$$ and $$f(1)+f(2)+...f(n) = n^{2}f(n)$$ for all $n>1$. Determine the value of $f(2004)$. My progress: I first substituted $n-1$ into the equation to get ...
2
votes
1answer
44 views

A function $\psi(z)$ satisfies the following functional equation…

I was given the following question to solve: "Given a function $\psi_0(z)$ satisfies the following functional equation: $\psi_0(z+1)=\frac{1}{z}+\psi_0(z)$ Prove that ...