## 42 Citations

### Effective metastability for modified Halpern iterations in CAT(0) spaces

- Mathematics
- 2012

We examine convergence results for modified Halpern iterations due to Cuntavepanit and Panyanak (Fixed Point Theory Appl. 2011:869458, 2011). Following Kohlenbach and Leuştean (Adv. Math.…

### Quantitative results for Halpern iterations of nonexpansive mappings

- Mathematics
- 2013

We give a rate of metastability for Halpern’s iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth Banach spaces, extracted from a proof due…

### Strong convergence for the alternating Halpern-Mann iteration in CAT(0) spaces

- Mathematics
- 2021

In this paper we consider, in the general context of CAT(0) spaces, an iterative schema which alternates between Halpern and Krasnoselskii-Mann style iterations. We prove, under suitable conditions,…

### A Rate of Metastability for the Halpern Type Proximal Point Algorithm

- Mathematics, Computer ScienceNumerical Functional Analysis and Optimization
- 2021

Using proof-theoretical techniques, a proof by Hong-Kun Xu regarding a result of strong convergence for the Halpern type proximal point algorithm is analyzed and a rate of metastability and also of asymptotic regularity for the iteration are obtained.

### The finitary content of sunny nonexpansive retractions

- MathematicsCommunications in Contemporary Mathematics
- 2020

We use techniques of proof mining to extract a uniform rate of metastability (in the sense of Tao) for the strong convergence of approximants to fixed points of uniformly continuous pseudocontractive…

### On quantitative metastability for accretive operators

- Mathematics
- 2022

Kohlenbach and the author have extracted a rate of metastability for approximate curves associated to continuous pseudocontractive self-mappings in Banach spaces which are uniformly convex and…

### Addendum to “Effective metastability of Halpern iterates in CAT(0) spaces” [Adv. Math. 231 (5) (2012) 2526–2556]

- Mathematics
- 2014

### A new metastable convergence criterion and an application in the theory of uniformly convex Banach spaces

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

### On the computational content of convergence proofs via Banach limits

- MathematicsPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2012

A logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X is designed, combined with the method of eliminating Banach limits, which yields a full quantitative analysis of the proof by Shioji and Takahashi.

## References

SHOWING 1-10 OF 57 REFERENCES

### Rates of Asymptotic Regularity for Halpern Iterations of Nonexpansive Mappings

- MathematicsJ. Univers. Comput. Sci.
- 2007

Effective rates of asymptotic regularity are given for the Halpern iterations of nonexpansive self-mappings of nonempty convex sets in normed spaces.

### Halpern's Iteration in CAT(0) Spaces

- Mathematics
- 2009

Motivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A…

### Reich’s problem concerning Halpern’s convergence

- Mathematics
- 2009

Abstract.We discuss Halpern’s convergence for nonexpansive mappings in Hilbert spaces. We prove that one of the conditions in [R. Wittmann, Approximation of fixed points of nonexpansive mappings,…

### On the computational content of convergence proofs via Banach limits

- MathematicsPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2012

A logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X is designed, combined with the method of eliminating Banach limits, which yields a full quantitative analysis of the proof by Shioji and Takahashi.

### On quantitative versions of theorems due to

- Mathematics
- 2010

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…

### Krasnoselskii's iteration process in hyperbolic space

- Mathematics
- 1982

A well-known iteration scheme due to Krasnoselskii for approximation of fixed points of nonexpansive mappings in Banach spaces is extended to a wider class of spaces. This class includes convex…

### On the interpretation of non-finitist proofs–Part II

- MathematicsJournal of Symbolic Logic
- 1952

Since the Hilbert substitution method has been applied by Ackermann to the formalism Z of number theory in [3], we shall give the no-counter-example interpretation for extensions (c) of Z, e.g. Zμ.…