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Logical aspects of rates of convergence in metric spaces. (English) Zbl 1184.03055

Summary: In this paper we develop a method for finding, under general conditions, explicit and highly uniform rates of convergence for the Picard iteration sequences for selfmaps on bounded metric spaces from ineffective proofs of convergence to a unique fixed point. We are able to extract full rates of convergence by extending the use of a logical metatheorem recently proved by Kohlenbach. This metatheorem could earlier be used to extract such computable rates of convergence only in cases where the selfmappings are also nonexpansive. In recent case studies we were able to find such explicit rates of convergence in two concrete cases. without assuming the selfmappings in question to be nonexpansive. Our novel method now provides an explanation in logical terms for these findings. This amounts, loosely speaking, to general conditions under which we can transform, in this specific setting, a \(\forall \exists \forall \)-sentence into a \(\forall \exists \)-sentence via an argument involving product spaces. This reduction in logical complexity allows us to use the existing machinery to extract quantitative bounds of the sort we need.

MSC:

03F10 Functionals in proof theory
03F20 Complexity of proofs
03F35 Second- and higher-order arithmetic and fragments
54E35 Metric spaces, metrizability
54H25 Fixed-point and coincidence theorems (topological aspects)
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