On the computational content of convergence proofs via Banach limits

Abstract

This paper addresses new developments in the ongoing proof mining programme, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method of extracting rates of metastability (as defined by Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all that is known) rely on the axiom of choice. In this paper, we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X . Combined with our method of eliminating Banach limits, this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.

Quantitative results for Halpern iterations of nonexpansive mappings

“…Definition 1.5. (See Kohlenbach and Leuştean [9].) Let X be a Banach space with a uniformly continuous duality selection mapping J.…”

“…This motivates the following Definition 1.4. (See Kohlenbach and Leuştean [9].) A Banach space X together with a mapping J : X → X * satisfying (i) x, Jx = x 2 = Jx 2 for all x ∈ X and (ii) J is norm-to-norm uniformly continuous on bounded subsets of X is called a Banach space with a uniformly continuous duality selection mapping.…”

Rates of convergence for iterative solutions of equations involving set-valued accretive operators

“…Although the assumptions on A being uniformly continuous and, respectively, on X being uniformly smooth are very different, it turns out they both follow as instances of the same technical lemma. The rates of convergence we extract in these cases then also depend (in addition to Θ) on moduli of uniform continuity for A and, respectively, for the duality mapping of X, where in the latter case such a modulus can be computed in terms of a modulus of uniform smoothness for X (see [14]). The various forms of strong (quasi-)accretivity used in the aforementioned results are all covered by mostly more restrictive versions of our concept of uniform accretivity at zero (note that [1] uses uniform accretivity to denote a concept which is much more restrictive than our notion of uniform accretivity at zero even when we drop the restriction 'at zero' as it corresponds to ψstrong accretivity as defined in Definition 2.3.…”

Fluctuations, effective learnability and metastability in analysis

“…[7,29,30,27,31,32]) and which we can explain now for the first time in terms of the logical structure of the given proof.…”