An equation with a nested function

Asked Jul 10 '14 at 23:04

Active Jul 10 '14 at 23:14

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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 functions can be assumed to be continuous, non-zero, non-decreasing on a real interval.

edited Jul 10 '14 at 23:14

asked Jul 10 '14 at 23:04

ste_kwr

Do we know anything about F, G and H? – DavidButlerUofA Jul 10 '14 at 23:06
Thanks for pointing out, edited to include. – ste_kwr Jul 10 '14 at 23:07
What are $F,G$ and $H$? – Ethan Splaver Jul 10 '14 at 23:08
1

Can you be more specific? If you take $F$, $G$ and $H$ to be identically zero then $\eta$ can be any function you like. – Shawn O'Hare Jul 10 '14 at 23:09
They are non-zero monotone non-decreasing functions on an interval $[a,b]$. There is not much else I know about them. – ste_kwr Jul 10 '14 at 23:13
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Take $F(x)=G(x)=H(x)=x$ then solve $\eta^2(x)-(x+1)\eta(x)+x=0$ – Brika Jul 10 '14 at 23:30
Where has this problem come from? If it's from a book, can you put a photo of the page itself? If not, where has it come from? – DavidButlerUofA Jul 10 '14 at 23:37
It is something I formulated, and does not come from a book. It is part of an elaborate setting where I am trying to solve for the social welfare maximizing function. – ste_kwr Jul 10 '14 at 23:41

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