We give an improved version of a theorem of Kincses and Totik concerning fixed points of a very general class of mappings of contractive type. By isolating the requirements on the mapping, specifically on the contractivity condition in question, we give an extension of the theorem from the compact case to the setting of arbitrary metric spaces. We also supply numerical information concerning the convergence of the Picard iteration sequence to the fixed point.
Using the uniformity features of the Cauchy rate exhibited we in addition show that any continuous selfmapping on a compact metric space satisfying one of the conditions (1)-(50) treated by B.E. Rhoades in the paper [B.E. Rhoades, A comparison of various definitions of contractive mappings, Transactions of the American Mathematical Society 226 (1977), 257-290] is an asymptotic contraction in the sense of Kirk.
The results were derived with the help of techniques and insights from proof mining.
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