Question

How to use induction on $p+q$ in functional equations

score 234 · Answer 1 · 2020-04-02 14:54:05

234

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How to use induction on $p+q$ in functional equations

Published on 02 Apr 2020 - 14:54

score 271 · Answer 2 · 2020-04-03 02:51:43

271

Views

Find all functions $f:\mathbb{R} \to [0, \infty)$such that $f(x^2 + y^2)=f(x^2 - y^2)+ f(2xy)$.

Published on 03 Apr 2020 - 2:51

score 308 · Answer 3 · 2020-04-03 08:14:49

308

Views

$f(x) = x + \int_{0}^{1} (xy^2 + x^2y) f(y)dy$. Find $f(x)$

Published on 03 Apr 2020 - 8:14

score 47 · Answer 4 · 2020-04-03 12:59:31

47

Views

Prove that the sum $f(\frac{a}{a + b}) + f(\frac{b}{a + b})$ can only be written as $g(a + b)$ for linear functions $f$

Published on 03 Apr 2020 - 12:59

score 76 · Answer 5 · 2020-04-03 22:45:37

76

Views

Functional equation with delay: $f(t)\pm f(t-\tau)=g$

Published on 03 Apr 2020 - 22:45

score 54 · Answer 6 · 2020-04-04 02:48:35

54

Views

Find solutions of $F(x)=\frac32 F\left(\sqrt[3]{x^2}-1\right)$

Published on 04 Apr 2020 - 2:48

score 1.5k · Answer 7 · 2020-04-04 14:52:36

1.5k

Views

Find all functions $f$ such that $f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$.

Published on 04 Apr 2020 - 14:52

score 94 · Answer 8 · 2026-03-25 17:26:58

94

Views

Does $|f(\sqrt{xy})| \le |\frac{f(x)+f(y)}{2}|$ imply $f$ is a logarithm?

Published on 25 Mar 2026 - 17:26

score 51 · Answer 9 · 2020-04-06 05:18:41

51

Views

Characterization of continuous functions with the property $f(x) = f\left(\frac{x}{1-x}\right)$

Published on 06 Apr 2020 - 5:18

score 174 · Answer 10 · 2020-04-06 06:26:53

174

Views

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

Published on 06 Apr 2020 - 6:26

score 265 · Answer 11 · 2020-04-06 12:35:44

265

Views

Solving functional equation $f(x)=3f(x+1)-3f(x+2)$

Published on 06 Apr 2020 - 12:35

score 1.6k · Answer 12 · 2020-04-06 19:33:27

1.6k

Views

A possible solution to $\sqrt {5-x}=5-x^2$ (without taking square from both sides)

Published on 06 Apr 2020 - 19:33

score 98 · Answer 13 · 2020-04-07 05:48:10

98

Views

find all continuous functions $f(x+y)+f(y+z)+f(z+x)=f(x)+f(y)+f(z)+f(x+y+z)$

Published on 07 Apr 2020 - 5:48

score 34 · Answer 14 · 2020-04-07 20:26:14

34

Views

Finding $f(x)$ where $x\in\mathbb{Z}$ such that $f(1)=2020$ and $\sum_{r=1}^{x}f(r)=x^2f(x)\forall x\gt 1$

Published on 07 Apr 2020 - 20:26

score 56 · Answer 15 · 2020-04-08 08:30:50

56

Views

Find the function(s) satisfying $f(x/y)=f(x)-f(y)$ and $f(e)=1$ for $x>0$.

Published on 08 Apr 2020 - 8:30

List Question

How to use induction on $p+q$ in functional equations

Find all functions $f:\mathbb{R} \to [0, \infty)$such that $f(x^2 + y^2)=f(x^2 - y^2)+ f(2xy)$.

$f(x) = x + \int_{0}^{1} (xy^2 + x^2y) f(y)dy$. Find $f(x)$

Prove that the sum $f(\frac{a}{a + b}) + f(\frac{b}{a + b})$ can only be written as $g(a + b)$ for linear functions $f$

Functional equation with delay: $f(t)\pm f(t-\tau)=g$

Find solutions of $F(x)=\frac32 F\left(\sqrt[3]{x^2}-1\right)$

Find all functions $f$ such that $f(xf(y)+y)+f(xy+x)=f(x+y)+2xy$.

Does $|f(\sqrt{xy})| \le |\frac{f(x)+f(y)}{2}|$ imply $f$ is a logarithm?

Characterization of continuous functions with the property $f(x) = f\left(\frac{x}{1-x}\right)$

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

Solving functional equation $f(x)=3f(x+1)-3f(x+2)$

A possible solution to $\sqrt {5-x}=5-x^2$ (without taking square from both sides)

find all continuous functions $f(x+y)+f(y+z)+f(z+x)=f(x)+f(y)+f(z)+f(x+y+z)$

Finding $f(x)$ where $x\in\mathbb{Z}$ such that $f(1)=2020$ and $\sum_{r=1}^{x}f(r)=x^2f(x)\forall x\gt 1$

Find the function(s) satisfying $f(x/y)=f(x)-f(y)$ and $f(e)=1$ for $x>0$.

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