Schauder Fixed Point conjecture deals with the existence of fixed points for certain types of operators on Banach spaces. It suggests that every non-expansive mapping of a non-empty convex, weakly compact subset of a Banach space into itself has a fixed point. The status of this conjecture may depend on the specific assumptions and settings.
You could start with reading : "Remarks on “some fixed point theorems for s-convex subsets in p-normed spaces” published in 2023. In this paper it is shown by examples that a published proof is not correct. From this paper one can learn a lot. But you have to use your own imagination and hence you have to play around with this stuff. It always helps to use Google Scholar.
A search with keywords "weak fixed point property" (which is the official name of the property you are interested in) and with "weak normal structure" (which is a widely used sufficient condition for this property) may give you a lot of information on the subject.
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