A uniform quantitative form of sequential weak compactness and Baillon’s nonlinear ergodic theorem
نویسنده
چکیده
We apply proof-theoretic techniques of ‘proof mining’ to obtain an effective uniform rate of metastability in the sense of Tao for Baillon’s famous nonlinear ergodic theorem in Hilbert space. In fact, we analyze a proof due to Brézis and Browder of Baillon’s theorem relative to the use of weak sequential compactness. Using previous results due to the author we show the existence of a bar recursive functional Ω∗ (using only lowest type bar recursion B0,1) providing a uniform quantitative version of weak compactness. Primitive recursively in this functional (and hence in T0 + B0,1) we then construct an explicit bound φ on for the metastable version of Baillon’s theorem. From the type level of φ and another result of the author it follows that φ is primitive recursive in the extended sense of Gödel’s T. In a subsequent paper also Ω∗ will be explicitly constructed leading to the refined complexity estimate φ ∈ T4.
منابع مشابه
On quantitative versions of theorems due to
This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theo...
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