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Let $f$ be a differentiable real function. In many situations a solution of $f(x)=x$ can be found as limit of the recurrent sequence determined by some initial value and the recurrence $x_{n+1}=f(x_n)$.

To list only a few examples:

Of course, it can happen that the iterations do not converge. Examples of such functions are $1/x$ and $x^2$.

Fixed point iterations can be illustrated nicely with cobweb plots. The plots corresponding to functions I mentioned above are shown below. More similar illustration can be found here or elsewhere.

From the pictures it seems that the local behavior of $f'(x)$ might be enough to describe the behavior of the iterated sequence. (Whether it converges or not, whether the convergence is monotone, etc.) At least in the case that $|f'(x)|\ne1$ in some interval around the solution.

  • What can be said about the sequence given by $x_{n+1}=f(x_n)$ if we have information about the values of $f'(x)$ close to the fixed point of $f$?
  • Are there some references with proofs, related results, and so on.?

Illustration of Babylonian method for $f(x)=\frac12(x+\frac2x)$.

Iterations of f(x)=(x+2/x)/2

Illustration for iterations of $f(x)=1+\frac1x$

Iterations of f(x)=1+1/x

Iterations of $f(x)=\sqrt x$

Iteratrions of sqrt(x)

Iterations of $f(x)=\frac1x$

Iterations of f(x)=1/x

Iterations of $f(x)=x^2$

Iterations of f(x)=x^2

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  • $\begingroup$ The topic you are interested in is called "complex dynamics" and there are many related results. I could give you the link to a lecture script dealing with such questions but since it's in german I don't know if you can use it effectively. $\endgroup$ – Yaddle Aug 6 '16 at 15:06
  • $\begingroup$ @Yaddle The point of this site is to serve not only to the OP but to other users as well. So the link might be useful at least for those users who speak German. (And I do speak German a bit.) $\endgroup$ – Martin Sleziak Aug 6 '16 at 15:10
  • $\begingroup$ I will provide some links in an answer then :) $\endgroup$ – Yaddle Aug 6 '16 at 15:12
  • $\begingroup$ For $x_{n+1}=f(x_n)$ it can be approximated by $\frac{dx}{dn} = f(x) - x$ see math.stackexchange.com/questions/1865370/… $\endgroup$ – arthur Aug 7 '16 at 3:12
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Here are some links to the scripts of two lectures dealing with complex dynamics written by Walter Bergweiler:

http://analysis.math.uni-kiel.de/vorlesungen/kompdyn1.12/Dynamik1.pdf (german/basics)

http://analysis.math.uni-kiel.de/vorlesungen/compdyn2.16/ComplexDynamics2.pdf (english/advanced)

I hope it helps you :)

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See the article on recurrence relations / difference equations here:

https://en.wikipedia.org/wiki/Recurrence_relation#Stability_of_nonlinear_first-order_recurrences

You will find the results/proofs you are interested in in any introductory textbook on difference equations.

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