We extract rates of convergence and rates of metastability (in the sense of Tao) for convergence results regarding abstract Cauchy problems generated by φ-accretive at zero operators A : D(A)(⊆ X) → 2 X where X is a real Banach space, proved in [8], by proof-theoretic analysis of the proofs in [8] and having assumed a new, stronger notion of uniform accretivity at zero, yielding a new notion of modulus of accretivity. We compute the rate of metastability for the convergence of the solution of the abstract Cauchy problem generated by a uniformly accretive at zero operator to the unique zero of A via proof mining based on a result by the first author. Finally, we apply our results to a special class of Cauchy problems considered in [8]. This work is the first application of proof mining to the theory of partial differential equations.
We give explicit bounds on the computation of approximate common fixed points of one-parameter strongly continuous semigroups of nonexpansive mappings on a subset C of a general Banach space. Moreover, we provide the first explicit and highly uniform rate of convergence for an iterative procedure to compute such points for convex C. Our results are obtained by a logical analysis of the proof (proof mining) of a theorem by T. Suzuki.
We give an explicit, computable and uniform bound for the computation of approximate common fixed points of one-parameter nonexpansive semigroups on a subset C of a Banach space, by proof mining on a proof by Suzuki. The bound obtained here is different to the bound obtained in a very recent work by Kohlenbach and the author which had been derived by proof mining on the-completely differentproof of a generalized version of the particular theorem by Suzuki. We give an adaptation of a logical metatheorem by Gerhardy and Kohlenbach for the given mathematical context, illustrating how the extractability of a computable bound is guaranteed. For uniformly convex C , as a corollary to our result we moreover give a computable rate of asymptotic regularity with respect to Kuhfittig's classical iteration schema, by applying a theorem by Khan and Kohlenbach.
This is an overview of a formalization project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erd ős-Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all m ∈ N, ω ω −→ (ω ω , m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalization process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic.
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