Abstract :
In [P. Gerhardy, A quantitative version of Kirk’s fixed point theorem for asymptotic contractions, J.Math.
Anal. Appl. 316 (2006) 339–345], P. Gerhardy gives a quantitative version of Kirk’s fixed point theorem for
asymptotic contractions. This involves modifying the definition of an asymptotic contraction, subsuming the
old definition under the new one, and giving a bound, expressed in the relevant moduli and a bound on the
Picard iteration sequence, on how far one must go in the iteration sequence to at least once get close to the
fixed point. However, since the convergence to the fixed point needs not be monotone, this theorem does not
provide a full rate of convergence. We here give an explicit rate of convergence for the iteration sequence,
expressed in the relevant moduli and a bound on the sequence. We furthermore give a characterization of
asymptotic contractions on bounded, complete metric spaces, showing that they are exactly the mappings
for which every Picard iteration sequence converges to the same point with a rate of convergence which is
uniform in the starting point.
© 2006 Elsevier Inc. All rights reserved
