scholarly journals A quantitative mean ergodic theorem for uniformly convex Banach spaces

2009 ◽  
Vol 29 (6) ◽  
pp. 1907-1915 ◽  
Author(s):  
U. KOHLENBACH ◽  
L. LEUŞTEAN
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Hilbert Spaces ◽  
Local Stability ◽  
Uniformly Convex ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Mean Ergodic Theorem ◽  
Uniformly Convex Banach Spaces ◽  
The Mean

AbstractWe provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of Tao of the mean ergodic theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad et al [Local stability of ergodic averages. Trans. Amer. Math. Soc. to appear] and Tao [Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys.28(2) (2008), 657–688].

scholarly journals Common fixed point iterations for a class of multi-valued mappings in $$\mathbf {CAT}(0)$$ spaces

2020 ◽  
Vol 9 (3) ◽  
pp. 681-690
Author(s):  
Khairul Saleh ◽  
Hafiz Fukhar-ud-din
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Finite Family ◽  
Uniformly Convex ◽  
Uniformly Convex Banach Spaces ◽  
Fixed Point Iterations

Abstract In this work, we propose an iterative scheme to approach common fixed point(s) of a finite family of generalized multi-valued nonexpansive mappings in a CAT(0) space. We establish and prove convergence theorems for the algorithm. The results are new and interesting in the theory of $$CAT\left( 0\right) $$ C A T 0 spaces and are the analogues of corresponding ones in uniformly convex Banach spaces and Hilbert spaces.

The implicit midpoint rule for nonexpansive mappings in 2-uniformly convex hyperbolic spaces

2019 ◽  
Vol 26 (1/2) ◽  
pp. 95-105
Author(s):  
H. Fukhar-ud-din ◽  
A.R. Khan
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Hyperbolic Spaces ◽  
Uniformly Convex ◽  
Convergence Theorems ◽  
Implicit Midpoint Rule ◽  
Midpoint Rule ◽  
Uniformly Convex Banach Spaces ◽  
Special Cases

The purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in 2- uniformly convex hyperbolic spaces and study its convergence. Strong and △-convergence theorems based on this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convex Banach spaces, CAT(0) spaces and Hilbert spaces as special cases.

scholarly journals Powers of sequences and convergence of ergodic averages

2009 ◽  
Vol 30 (5) ◽  
pp. 1431-1456 ◽  
Author(s):  
N. FRANTZIKINAKIS ◽  
M. JOHNSON ◽  
E. LESIGNE ◽  
M. WIERDL
Keyword(s):  
Ergodic Theorem ◽  
Measurable Function ◽  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Bounded Measurable Function ◽  
Mean Ergodic Theorem ◽  
The Mean ◽  
Good For

AbstractA sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

scholarly journals Concentration analysis in Banach spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550038 ◽  
Author(s):  
Sergio Solimini ◽  
Cyril Tintarev
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Uniformly Convex ◽  
Opial Condition ◽  
Uniformly Convex Banach Spaces ◽  
Weak Star ◽  
Compactness Arguments

The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach–Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of [Formula: see text]-convergence by Lim [Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) 179–182] instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and [Formula: see text]-spaces, but not in [Formula: see text], [Formula: see text]. [Formula: see text]-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies the connection of [Formula: see text]-convergence with the Brezis–Lieb lemma and gives a version of the latter without an assumption of convergence a.e.

scholarly journals The finitary content of sunny nonexpansive retractions

2020 ◽  
Vol 23 (01) ◽  
pp. 1950093
Author(s):  
Ulrich Kohlenbach ◽  
Andrei Sipoş
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Point Of View ◽  
Uniformly Convex ◽  
Pseudocontractive Mappings ◽  
Proof Mining ◽  
Restricted Form ◽  
Uniformly Convex Banach Spaces ◽  
Order Arithmetic ◽  
Uniformly Continuous

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 mappings in Banach spaces which are uniformly convex and uniformly smooth, i.e. a slightly restricted form of the classical result of Reich. This is made possible by the existence of a modulus of uniqueness specific to uniformly convex Banach spaces and by the arithmetization of the use of the limit superior. The metastable convergence can thus be proved in a system which has the same provably total functions as first-order arithmetic and therefore one may interpret the resulting proof in Gödel’s system [Formula: see text] of higher-type functionals. The witness so obtained is then majorized (in the sense of Howard) in order to produce the final bound, which is shown to be definable in the subsystem [Formula: see text]. This piece of information is further used to obtain rates of metastability to results which were previously only analyzed from the point of view of proof mining in the context of Hilbert spaces, i.e. the convergence of the iterative schemas of Halpern and Bruck.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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