Continuous maps that are asymptotic pointwise contractions under local uniform convergence on complete metric spaces with bounded orbits have iterates converging to a unique fixed point.
Schauder, Der Fixpunktsatz in Funktionsr¨ aumen, Studia Math.2(1930), 171–181
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Fixed Points of Asymptotic Pointwise Contractions under Local Uniform Convergence
Continuous maps that are asymptotic pointwise contractions under local uniform convergence on complete metric spaces with bounded orbits have iterates converging to a unique fixed point.