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This is the second part of the question Fixed point: linear operators. Here I would like to ask you about the general case.

A lot of concepts can be described or even defined as a solution of a fixpoint equation of the form $\mathcal Ax = x$ for an operator $\mathcal A:X\to X$. Due to the form of this equation, we can formulate it for an arbitrary set $X$. There are several results on the existence of the solution for such an equation, say Kakutani fixed point theorem, but what about the finding of this points?

The main question here is the methods for finding fixpoints of nonlinear maps, starting from nonlinear operators acting on Banach spaces and going to set-valued maps like in Kakutani theorem. This is mostly the reference request.

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Here are a few, chosen from a broad span of the literature:

Construction of fixed points of nonlinear mappings in Hilbert space (Browder, Petryshyn) doi.org/10.1016/0022-247X(67)90085-6

Fixed points and iteration of a nonexpansive mapping in a Banach space (Ishikawa) doi.org/10.1090/S0002-9939-1976-0412909-X

Fixed points by a new iteration method (Ishikawa) doi.org/10.1090/S0002-9939-1974-0336469-5

An example concerning fixed points (Genel, Lindenstrauss) doi.org/10.1007/BF02757276

Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions (Kim, Xu) doi.org/10.1016/j.na.2007.02.029