this is an interesting question. You will need to know a few properties about f(x), such as: is it monotonically increasing/decreasing and with a few assumptions like that, it's probably possible to prove the existence of a fixed point via general methods, if one exists
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I will have to pick some functions and some methods of fractional differentiation and experiment. The last differential I gaffed on. My sensation is that should fixed points exist, they will have /very well distributed not-so-local/ structures to them.
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