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# A proof mining case study on the unit interval

## March 18, 2021 @ 10:00 am - 12:00 pm

**Title: **A proof mining case study on the unit interval

**Speaker: **Andrei Sipoș (University of Bucharest & IMAR)

**Abstract: **

In 1991, Borwein and Borwein proved [1] the following: if L>0, f : [0,1] → [0,1] is L-Lipschitz, (x

_{n}), (t_{n}) ⊆ [0,1] are such that for all n, x_{n+1}=(1-t_{n})x_{n}+t_{n}f(x_{n}) and there is a δ>0 such that for all n, t_{n}≤(2-δ)/(L+1), then the sequence (x_{n}) converges.The relevant fact here is that the main argument used in their proof is of a kind that hasn’t been analyzed yet from the point of view of proof mining, and thus it may serve as an illustrative new case study. We shall present our work [2] on the proof, showing how to extract a uniform and computable rate of metastability for the above family of sequences.

**References**:

[1] D. Borwein, J. Borwein, Fixed point iterations for real functions.

*J. Math. Anal. Appl.*157, no. 1, 112–126, 1991.[2] A. Sipoș, Rates of metastability for iterations on the unit interval. arXiv:2008.03934 [math.CA], 2020.