Nonlinear Ergodic Theorems Research Articles

Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications

Abstract In this article, we prove ergodic convergence for a sequence of nonexpansive mappings in Hadamard (complete CAT ( 0 ) {\rm{CAT}}\left(0) ) spaces. Our result extends the nonlinear ergodic theorem by Baillon for nonexpansive mappings from Hilbert spaces to Hadamard spaces. We further establish the standard nonlinear ergodic theorem and apply our results to the problem of finding a common fixed point of a countable family of nonexpansive mappings. Finally, we propose some applications of our results to solve convex optimization problems in Hadamard spaces.

Fixed point and nonlinear ergodic theorems for some new type semigroups of mappings

Fixed point and nonlinear ergodic theorems for some new type semigroups of mappings

Bounds for a nonlinear ergodic theorem for Banach spaces

AbstractWe extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.

A Mean Ergodic Theorem for Nonexpansive Mappings in Hadamard Spaces

In this paper, we prove a mean ergodic theorem for nonexpansive mappings in Hadamard (nonpositive curvature metric) spaces, which extends the Baillon nonlinear ergodic theorem. The main result shows that the sequence given by the Karcher means of iterations of a nonexpansive mapping with a nonempty fixed point set converges weakly to a fixed point of the mapping. This result also remains true for a 1-parameter continuous semigroup of contractions.

Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

Abstract We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ∣·∣, ⪯), T:C → C a monotone 1-Lipschitz mapping and x ⪯ T(x), then the sequence of averages 1 n ∑ i = 0 n − 1 T i ( x ) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T. As a consequence, it is shown that the sequence of Picard’s iteration {T n (x)} also converges weakly to a fixed point of T. The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

Attractive point and nonlinear ergodic theorems without convexity in reflexive Banach spaces

In this paper, we prove attractive point theorem for two commutative generic 2-generalized Bregman nonspreading mappings in reflexive Banach spaces. Also, a nonlinear ergodic theorem of Baillon type without convexity assumption on the aforementioned mappings is proved in the space. Our results improve and generalize many results announced recently in the literature.

MEAN ERGODIC THEOREMS FOR A SEQUENCE OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES

Let C be a closed convex subset of a Hilbert space and {Tn} a sequence of nonexpansive self-mappings of C. Then we consider the following iterative sequence {zn}: x1 = x ∈ C, xn+1 = Tnxn, and zn =1 /n n=1 xk for n ∈ . In this paper, we obtain a weak convergence theorem for such a sequence {zn}. Using our result, we get a nonlinear ergodic theorem which is a generalization of Baillon (2). Further we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings.

Fixed point, convergence and nonlinear ergodic theorems for some semigroups in Hilbert spaces

In this paper, we first introduce some semigroups of mappings called quasi-nonexpansive, nonspreading, hybrid, TJ-1, TJ-2, and generalized hybrid semigroup. Then, using the theory of invariant means, we prove fixed point theorems, weak convergence theorem of Mann’s type and generalized nonlinear ergodic theorem for such semigroups in a Hilbert space. Furthermore, we prove a strong convergence theorem of Halpern’s type for the proposed semigroups in a Hilbert space. The results presented in this paper mainly extend and improved some well-known results in the literature.

Application of the product net technique and Kadec\u2013Klee property to study nonlinear ergodic theorems and weak convergence theorems in uniformly convex multi-Banach spaces

Let Y be a uniformly convex multi-Banach space which has not a Frechet differentiable norm. We use the technique of product net to obtain the nonlinear ergodic theorems in Y. Finally, let the dual of uniformly convex multi-Banach space have the Kadec–Klee property, we instate the weak convergence theorem in the case of reversible semi-group.

Weak and strong convergence theorems and a nonlinear ergodic theorem for N-generalized hybrid mappings

In this paper, assuming an appropriate condition, we prove that [Formula: see text]-generalized hybrid mappings are demiclosed in Hilbert spaces. Using this fact, we prove a weak convergence theorem of Ishikawa type for these nonlinear mappings. Also, a strong convergence theorem of Halpern–Ishikawa type and a nonlinear ergodic theorem for [Formula: see text]-generalized hybrid mappings have been proven in Hilbert spaces.

Nonlinear ergodic theorems for amenable semigroups of nonexpansive mappings in Hadamard spaces

We generalize some theorems for continuous representation of an amenable semitopological semigroup S as nonexpansive maps on Hilbert spaces to complete CAT(0) spaces, i.e., Hadamard spaces. For proper Hadamard space, we prove the convergence of asymptotically invariant means of the bounded trajectories. In the nonproper case, adding a suitable condition on the means and assuming S is a locally compact Hausdorff semigroup, we prove $${\Delta}$$ -convergence of asymptotically invariant means of the bounded trajectories.

Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces

Recently, two retractions (projections), which are different from the metric projection and the sunny nonexpansive retraction in a Banach space, were found. In this article, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions.

Nonlinear ergodic theorems and weak convergence theorems for reversible semigroup of asymptotically nonexpansive mappings in Banach spaces

Abstract In this paper, we provide the nonlinear ergodic theorems and weak convergence theorems for almost orbits of a reversible semigroup of asymptotically nonexpansive mappings in a uniformly convex Banach space without assuming that X has a Fréchet differentiable norm. Since almost orbits in this paper are not almost asymptotically isometric, new methods have to be introduced and used for the proofs. Our main results include many well-known results as special cases and are new even for reversible semigroup of nonexpansive mappings. MSC:47H20.

On multiparameter Chacon’s type ergodic theorems

The purpose of this paper is to deal with generalizations of ratio ergodic theorems due to R.V. Chacon, G. Baxter, and K. Jacobs. We prove two weighted generalizations of the Chacon ergodic theorem and the Jacobs random ergodic theorem. L. Sucheston has formulated a general principle yielding simultaneous proofs of many almost everywhere multiparameter convergence theorems. This principle will allow us to derive multiparameter Chacon’s type ergodic theorems for positive linear contractions on L1. The advantage is that we can inquire further into the problem of improving the multiparameter Chacon-Ornstein ergodic theorem due to Frangos and Sucheston. A multiparameter generalization of the Dunford-Schwartz ergodic theorem is also obtained. In addition, our consideration comes to the a.e. convergence for sectorially restricted averages in the commutative case, as in the Brunel-Dunford-Schwartz theorem. Moreover, we establish two Chacon’s type nonlinear ergodic theorems for the nonlinear sums of affine operators on L1.

ATTRACTIVE POINT THEOREMS AND ERGODIC THEOREMS FOR NONLINEAR MAPPINGS IN HILBERT SPACES

In this paper, using Banach limits, we study attractive points and fixed points of nonlinear mappings in Hilbert spaces. Then we obtain attractive point theorems and fixed point theorems for nonlinear mappings in Hilbert spaces. Using these results, we finally prove a nonlinear ergodic theorem for 2-generalized hybrid mappings in Hilbert spaces.

A quantitative nonlinear strong ergodic theorem for Hilbert spaces

We give a quantitative version of a strong nonlinear ergodic theorem for (a class of possibly even discontinuous) selfmappings of an arbitrary subset of a Hilbert space due to R. Wittmann and outline how the existence of uniform bounds in such quantitative formulations of ergodic theorems can be proved by means of a general logical metatheorem. In particular these bounds depend neither on the operator nor on the initial point. Furthermore, we extract such uniform bounds in our quantitative formulation of Wittmannʼs theorem, implicitly using the proof-theoretic techniques on which the metatheorem is based. However, we present our result and its proof in analytic terms without any reference to logic as such. Our bounds turn out to involve nested iterations of relatively low computational complexity. While in theory these kind of iterations ought to be expected, so far this seems to be the first occurrence of such a nested use observed in practice.

A UNIFORM QUANTITATIVE FORM OF SEQUENTIAL WEAK COMPACTNESS AND BAILLON'S NONLINEAR ERGODIC THEOREM

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.

ATTRACTIVE POINT THEOREMS AND ERGODIC THEOREMS FOR NONLINEAR MAPPINGS IN HILBERT SPACES

In this paper, using Banach limits, we study attractive points and fixed points of nonlinear mappings in Hilbert spaces. Then we obtain attractive point theorems and fixed point theorems for nonlinear mappings in Hilbert spaces. Using these results, we finally prove a nonlinear ergodic theorem for 2-generalized hybrid mappings in Hilbert spaces.

Gödel functional interpretation and weak compactness

In recent years, proof theoretic transformations (so-called proof interpretations) that are based on extensions of monotone forms of Gödel’s famous functional (‘Dialectica’) interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for bounded closed and convex subsets of an abstract (not necessarily separable) Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω∗ definable from bar recursion (in the sense of Spector) of lowest type. While a case study on the analysis of strong convergence results (due to Browder and Wittmann resp.) that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an explicit bound on a metastable (in the sense of T. Tao) version of this theorem that is primitive recursive relative to (a somewhat restricted form of) Ω∗.In the current paper we for the first time give the construction of Ω∗ (in the form needed for the unwinding of Baillon’s theorem). As a corollary to the fine analysis of the use of bar recursion in this construction we obtain that Ω∗ elevates arguments in Tn at most to resulting functionals in Tn+2 (here Tn is the fragment of Gödel’s T with primitive recursion restricted to the type level n). In particular, one can conclude from this that our bound on Baillon’s theorem is at least definable in T4.

On quantitative versions of theorems due to F.E. Browder and R. Wittmann

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 theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder's original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann's proof follows a similar line of reasoning and we adapt our analysis of Browder's proof to get a quantitative version of Wittmann's theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder's theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon's nonlinear ergodic theorem.