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In chapter 1 on fixed points, the book says we can find fixed points of certain functions using 

f(x) = f(f(x)) = f(f(f(x))) ....

What are those functions?

It doesn't work for y = 2y when i rewrite it as y = y/2 it works

Does y need to get smaller everytime? Or are there any general attributes that a function has to have to find fixed points by that method?

What conditions it should satisfy to work?

vinothkr
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2 Answers2

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According to the Banach fixed-point theorem, such a point exists iff the mapping (function) is a contraction. That means that, for example, y=2x doesn't have fixed point and y = 0,999... * x has. In general, if f maps [a,b] to [a,b], then |f(x) - f(y)| should be equal to c * |x - y| for some 0 <= c < 1 (for all x, y from [a, b]).

tkroman
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Say you have:

f(x) = sin(x)

then x = 0 is a fixed point of the function since:

f(0) = sin(0) = 0
f(f(0)) = sin(sin(0)) = sin(0) = 0

Not every point along x is a fixed point of sin, only 0 is.

Different functions have different fixed points, if at all. You can find more on fixed points of functions at Wikidpedia

R Sahu
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  • I get what fixed points are. But my question is more of how for some functions we are able to find fixed points and for some function even if they have fixed points we are not able to find them through that method – vinothkr Apr 20 '14 at 18:16