Halpern Iteration Research Articles

Strong convergence of modified Halpern’s iterations for a k-strictly pseudocontractive mapping

In this paper, we discuss three modified Halpern iterations as follows: I II III and obtained the strong convergence results of the iterations (I)-(III) for a k-strictly pseudocontractive mapping, where satisfies the conditions: (C1) and (C2) , respectively. The results presented in this work improve the corresponding ones announced by many other authors.

Strong convergence of a modified Halpern-type iteration for asymptotically quasi-∅-nonexpansive mappings

Abstract In this paper, the problem of modifying Halpern iteration for approximating a common fixed point of a family of asymptotically quasi- ∅-nonexpansive mappings is considered. Strong convergence theorems are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper mainly improve the corresponding results announced in [Y.J. Cho, X. Qin, S.M. Kang, Strong convergence of the modified Halpern- type iterative algorithms in Banach spaces, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17 (2009) 51-68].

Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one

In this paper, we deal with the Halpern iterative scheme for a strongly quasinonexpansive mapping in the setting of a complete geodesic space with curvature bounded above by one. Our result can be applied to the image recovery problem. We also consider the approximation of a fixed point of a nonexpansive mapping and obtain convergence theorems, one of which is a supplement of the result by Piatek with an additional sufficient condition.

New hybrid algorithm of modified Halpern iteration scheme for quasi-\phi-nearly asymptotically nonexpansive mappings in Banach spaces

New hybrid algorithm of modified Halpern iteration scheme for quasi-\phi-nearly asymptotically nonexpansive mappings in Banach spaces

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

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. 231:2525-2556, 2012), we extract uniform rates of metastability. This includes extracting rates of asymptotic regularity and replacing an ineffective argument that uses Banach limits.

Effective metastability of Halpern iterates in CAT(0) spaces

This paper provides an effective uniform rate of metastability (in the sense of Tao) on the strong convergence of Halpern iterations of nonexpansive mappings in CAT(0) spaces. The extraction of this rate from an ineffective proof due to Saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs. This methodology is applied here for the first time to a proof that uses Banach limits and hence makes a substantial reference to the axiom of choice.

On the computational content of convergence proofs via Banach limits

This paper addresses new developments in the ongoing proof mining programme, i.e. the use of tools from proof theory to extract effective quantitative information from prima facie ineffective proofs in analysis. Very recently, the current authors developed a method of extracting rates of metastability (as defined by Tao) from convergence proofs in nonlinear analysis that are based on Banach limits and so (for all that is known) rely on the axiom of choice. In this paper, we apply this method to a proof due to Shioji and Takahashi on the convergence of Halpern iterations in spaces X with a uniformly Gâteaux differentiable norm. We design a logical metatheorem guaranteeing the extractability of highly uniform rates of metastability under the stronger condition of the uniform smoothness of X. Combined with our method of eliminating Banach limits, this yields a full quantitative analysis of the proof by Shioji and Takahashi. We also give a sufficient condition for the computability of the rate of convergence of Halpern iterations.

Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points

In this paper, we consider an iteration process of Halpern's type for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points for a quasi-nonexpansive mapping with perturbation in a Hilbert space and then prove a strong convergence theorem for such iterations. Using this result, we obtain new strong convergence theorems in a Hilbert space. In particular, we solve partially an open problem posed by Kurokawa and Takahashi (Nonlinear Anal 73:1562---1568, 2010) concerning Halpern's iterations.

Strong convergence theorems by Halpern-Mann iterations for multi-valued relatively nonexpansive mappings in Banach spaces with applications

Abstract In this article, an iterative sequence for relatively nonexpansive multi-valued mapping by modifying Halpern and Mann's iterations is introduced, and then some strong convergence theorems are proved. At the end of the article some applications are given also. AMS Subject Classification: 47H09; 47H10; 49J25.

Strong Convergence of Hybrid Algorithm for Asymptotically Nonexpansive Mappings in Hilbert Spaces

The hybrid algorithms for constructing fixed points of nonlinear mappings have been studied extensively in recent years. The advantage of this methods is that one can prove strong convergence theorems while the traditional iteration methods just have weak convergence. In this paper, we propose two types of hybrid algorithm to find a common fixed point of a finite family of asymptotically nonexpansive mappings in Hilbert spaces. One is cyclic Mann′s iteration scheme, and the other is cyclic Halpern′s iteration scheme. We prove the strong convergence theorems for both iteration schemes.

Strong Convergence Theorems for Modifying Halpern Iterations for Quasi‐ϕ‐Asymptotically Nonexpansive Multivalued Mapping in Banach Spaces with Applications

An iterative sequence for quasi‐ϕ‐asymptotically nonexpansive multivalued mapping for modifying Halpern′s iterations is introduced. Under suitable conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in the work by Chang et al. 2011.

Hybrid Mann–Halpern iteration methods for nonexpansive mappings and semigroups

In this paper, we introduce some new iteration methods based on the hybrid method in mathematical programming, the Mann’s iterative method and the Halpern’s method for finding a fixed point of a nonexpansive mapping and a common fixed point of a nonexpansive semigroup Hilbert spaces.

Halpern iteration in CAT(κ) spaces

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., Lopez, G., Martin-Marquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.

Strong Convergence of Modified Halpern Iterations in CAT(0) Spaces

AbstractStrong convergence theorems are established for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces. Our results extend and improve the recent ones announced by Kim and Xu (2005), Hu (2008), Song and Chen (2008), Saejung (2010), and many others.

Strong Convergence Theorems for a Countable Family of Nonexpansive Mappings in Convex Metric Spaces

We introduce a new modified Halpern iteration for a countable infinite family of nonexpansive mappings {Tn} in convex metric spaces. We prove that the sequence {xn} generated by the proposed iteration is an approximating fixed point sequence of a nonexpansive mapping when {Tn} satisfies the AKTT‐condition, and strong convergence theorems of the proposed iteration to a common fixed point of a countable infinite family of nonexpansive mappings in CAT(0) spaces are established under AKTT‐condition and the SZ‐condition. We also generalize the concept of W‐mapping for a countable infinite family of nonexpansive mappings from a Banach space setting to a convex metric space and give some properties concerning the common fixed point set of this family in convex metric spaces. Moreover, by using the concept of W‐mappings, we give an example of a sequence of nonexpansive mappings defined on a convex metric space which satisfies the AKTT‐condition. Our results generalize and refine many known results in the current literature.

A Generalization of Suzuki′s Lemma

Let {zn}, {wn}, and {vn} be bounded sequences in a metric space of hyperbolic type (X, d), and let {αn} be a sequence in [0, 1] with 0 < lim inf n αn ≤ lim sup n αn < 1. If zn+1 = αnwn ⊕ (1 − αn)vn for all n ∈ ℕ, lim n d(zn, vn) = 0, and lim sup n (d(wn+1, wn) − d(zn+1, zn)) ≤ 0, then lim n d(wn, zn) = 0. This is a generalization of Lemma 2.2 in (T. Suzuki, 2005). As a consequence, we obtain strong convergence theorems for the modified Halpern iterations of nonexpansive mappings in CAT(0) spaces.

Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces

The purpose of the paper is to introduce modified Halpern and Ishikawa iteration for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in Banach spaces. We also consider two strong convergence theorems for relatively nonexpansive mappings with some proper restriction.

Halpern’s iteration in Banach spaces

We prove strong convergence theorems for a sequence which is generated by Halpern’s iteration. We also apply our result for finding zeros of an accretive operator. Our result improves the recent result of Aoyama et al. [K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007) 2350–2360] by removing some assumptions on the parameters. Finally we discuss the new sufficient condition studied by Song [Y. Song, A new sufficient condition for the strong convergence of Halpern type iterations. Appl. Math. Comput. 198 (2) (2008) 721–728; Y. Song, New strong convergence theorems for nonexpansive nonself-mappings without boundary conditions. Comput. Math. Appl. 56 (6) (2008) 1473–1478] and correct the main result of Song and Chai [Y. Song, X. Chai, Halpern iteration for firmly type nonexpansive mappings, Nonlinear Anal. 71 (10) (2009) 4500–4506].

Equivalent theorems of the convergence between Ishikawa–Halpern iteration and viscosity approximation method

The aim of this paper is to establish four groups of equivalent theorems of convergence between Ishikawa–Halpern iteration and viscosity approximation method, respectively. Furthermore, the authors consider the viscosity approximation method with weakly contractive mapping. The results improve and extend the results announced by many others.

A Note on Strong Convergence of a Modified Halpern's Iteration for Nonexpansive Mappings

In the paper by Hu in 2008, the author proved a strong convergence result for nonexpansive mappings using a modified Halpern's iteration algorithm. Unfortunately, the case does not guarantee the strong convergence of the sequence . In this note, we provide a counter-example to the theorem.