Speaker: Pedro Pinto (Technische Universitat Darmstadt).
Abstract: Proof mining is a research program that employs proof theoretical tools to obtain additional information from mathematical results, [1]. Its techniques have been applied successfully to many areas of Mathematics with special focus on Nonlinear Analysis. This presentation reports ongoing joint work with Ulrich Kohlenbach.
Two well-known strongly convergent algorithms in Fixed Point Theory are due to Browder and Halpern. Their original convergence results were generalized in several different ways. First introduced by Moudafi, the viscosity approximation method is one such generalization in which the anchor point of the iteration is replaced with a strictly contracting mapping. In [2], Suzuki showed that the convergence of the generalized viscosity version of these algorithms (with more general Meir-Keeler contractions) can be reduced to the convergence of the original iterations.
Another extensively studied algorithm is the Krasnosel'skii-Mann iteration. This iteration has many useful properties (e.g. it is Fejér monotone), but in general is only weakly convergent. A hybrid version of the Krasnosel'skii-Mann iteration with the viscosity method was recently studied in [3] and shown to be strongly convergent.
In this talk, we will discuss the quantitative analysis of these results. We start by recalling the relevant iterations and useful quantitative notions. We will point out the main obstacles in the analysis of Suzuki's theorems, and explain how they were overcome to obtain rates of metastability and Cauchy rates. Some applications of our results are given. In a second part, we discuss a certain notion of uniform accretive operators (discussed also in [3]), which allowed for an extraction of a modulus of uniqueness in uniformly convex Banach spaces. It is well known that in this situation (together with a rate of asymptotic regularity) it is possible to obtain Cauchy rates. We illustrate this with some applications. We conclude with some remarks regarding the additional generality of our work and its connection with the original results.
[1] Ulrich Kohlenbach. Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. Springer-Verlag Berlin Heidelberg, 2008.
[2] Tomonari Suzuki. Moudafi 's viscosity approximations with Meir-Keeler contractions. Journal of Mathematical Analysis and Applications, 325(1):342-352, 2007.
[3] Hong-Kun Xu, Najla Altwaijry and Souhail Chebbi. Strong convergence of Mann's iteration process in Banach spaces. Mathematics 2020, 8, 954; doi:10.3390/math8060954.
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