Mann Iteration Research Articles

A Note on Implicit and Explicit Mann Iterative Processes for Asymptotically ϕ‐Strongly Pseudocontractive Mappings

In this paper we prove the equivalence and the strong convergence of an explicit Mann iterative process and a modified implicit iterative process for asymptotically ϕ‐strongly pseudocontractive mappings in a uniformly smooth Banach space.

Implicit Mann Type Iteration Method Involving Strictly Hemicontractive Mappings in Banach Spaces

We proved that the modified implicit Mann iteration process can be applied to approximate the fixed point of strictly hemicontractive mappings in smooth Banach spaces.

An extragradient-like approximation method for variational inequalities and fixed point problems

AbstractThe purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings.1991 MSC: 47H09; 47J20.

Strong convergence of the Mann iteration for [formula omitted]-demicontractive mappings

The paper deals with strong convergence properties of the Mann iteration. A new class of demicontractive mappings (called α - demicontractive) is introduced for which the strong convergence of the computed sequence is assured. The paper presents also an overview of relevant contributions of the last two decades, concerning strong convergence for Mann-type iteration of demicontractive mappings.

Strong convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258–2271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140–1152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400–2411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372–379 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336–346 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334–1345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835–1842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77–88 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766–776 (2007)].

Effective rates of convergence for Lipschitzian pseudocontractive mappings in general Banach spaces

This paper gives an explicit and effective rate of convergence for an asymptotic regularity result ‖ T x n − x n ‖ → 0 due to Chidume and Zegeye in 2004 [14] where ( x n ) is a certain perturbed Krasnoselski–Mann iteration schema for Lipschitz pseudocontractive self-mappings T of closed and convex subsets of a real Banach space. We also give a qualitative strengthening of the theorem by Chidume and Zegeye, by weakening the assumption of the existence of a fixed point. For the bounded case, our bound is polynomial in the data involved.

Convergence in norm of modified Krasnoselski–Mann iterations for fixed points of demicontractive mappings

This paper deals with fixed points methods related to the general class of demicontractive mappings (including the well-known classes of nonexpansive and quasi-nonexpansive mappings) in Hilbert spaces . Specifically, we point out some historical aspects concerning the concept of demicontactivity and we investigate a regularized variant of the Krasnoselski-Mann iteration that can be alternatively regarded as a simplified form of the inertial iteration (P-E. Maingé, J. Math. Anal. Appl. 344 (2008) 876–887) with non-constant relaxation factors. These two methods ensure the strong convergence of the generated sequence towards the least norm element of the set of fixed-points of demicontractive mappings. However, for convergence, our method does not require anymore the knowledge of some constant related to the involved demicontractive operator. A new and simpler proof is also proposed for its convergence even when involving non-constant relaxation factors. We point out the simplicity of this algorithm (at least from computational point of view) in comparison with other existing methods. We also present some numerical experiments concerning a convex feasibility problem, experiments that emphasize the characteristics of the considered algorithm comparing with a classical cyclic projection-type iteration.

Weak and Strong Convergence Theorems for Positively Homogenuous Nonexpansive Mappings in Banach Spaces

Our purpose in this paper is first to prove a weak convergence theorem by Mann's iteration for positively homogeneous nonexpansive mappings in a Banach space. Further, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove a strong convergence theorem for such mappings. From two results, we obtain weak and strong convergence theorems for linear contractive mappings in a Banach space. These results are new even if the mappings are linear and contractive.

Almost Stability of Modified Iteration Method with Errors for a Fixed Point of Uniformaly L- Lipschitzian

      In this  paper, we prove strong convergence theorem of modified Mann iteration sequence with errors for uniformaly L- Lipschitzian mapping in arbitrary Banach space. Our results improve and gernalize the recent results Osilike , Xu and Xie and many others.

Viscosity methods for common solutions for equilibrium and hierarchical fixed point problems

Implicit and explicit viscosity methods for finding common solutions of equilibrium and hierarchical fixed points are presented. These methods are used to solve systems of equilibrium problems and variational inequalities where the involving operators are complements of nonexpansive mappings. The results here are situated on the lines of the research of the corresponding results of Moudafi [Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Probl. 23 (2007), pp. 1635–1640; Weak convergence theorems for nonexpansive mappings and equilibrium problems, to appear in JNCA], Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl. Art ID 95453 (2006), 10 pp.; Strong convergence of an iterative method for hierarchical fixed point problems, Pac. J. Optim. 3 (2007), pp. 529–538; Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, to appear in JNCA], Yao and Liou [Weak and strong convergence of Krasnosel'skiĭ–Mann iteration for hierarchical fixed point problems, Inverse Probl. 24 (2008), 015015 8 pp.], S. Takahashi and W. Takahashi [Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2006), pp. 506–515], Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, preprint.], Combettes and Hirstoaga [Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), pp. 117–136] and Plubtieng and Pumbaeang [A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (2007), pp. 455–469.].

Strong Convergence Theorems for Variational Inequalities and Fixed Point Problems of Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz continuous mapping. We introduce a modified hybrid Mann iterative scheme with perturbed mapping which is based on well-known CQ method, Mann iteration method and hybrid (or outer approximation) method. We establish a strong convergence theorem for three sequences generated by this modified hybrid Mann iterative scheme with perturbed mapping. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings.

Iterative Approximation of Fixed Points for Asymptotically Strict Pseudocontractive Type Mappings in the Intermediate Sense

We introduced the concept of an asymptotically $\kappa$-strict pseudocontractive type mapping in the intermediate sense which is not necessarily Lipschitzian. We proved that the modified Mann iteration process: $x_{n+1} = (1-\alpha_n) x_n + \alpha_n T^n x_n$, $\forall n \geq 1$ where $\{\alpha_n\}$ is a sequence in $(0,1)$ with $\delta \leq \alpha_n \leq 1 - \kappa - \delta$ for $\delta \in (0,1)$ converges weakly to a fixed point of an asymptotically $\kappa$-strict pseudocontractive type mapping $T$ in the intermediate sense. Furthermore, a CQ method which generates a strongly convergent sequence for this class of mappings is proposed and strong convergence result for this CQ method is established.

Equivalence problem of Mann and Ishikawa Iterations in Banach Spaces

In this paper,we prove some equivalence theorems between convergence and stability of modified Mann and modified Ishikawa iterative sequence with errors for uniformly Lipschitz strongly successively pseudo-contractive mappings in Banach spaces by using different methods,under the condition on the iteration para-meters limsupβ_nk/L(L+1).Our results prove and extend the results that need limβ_n=0 and initial points of two different iteration are the same one.

The Equivalence of Mann and Implicit Mann Iterations for Uniformly Pseudocontractive Mappings in Uniformly Smooth Banach Spaces

In this paper, suppose is an arbitrary uniformly smooth real Banach space, and is a nonempty closed convex subset of . Let be a generalized Lipschitzian and uniformly pseudocontractive self-map with . Suppose that , are defined by Mann iteration and implicit Mann iteration respectively, with the iterative parameter satisfying certain conditions. Then the above two iterations that converge strongly to fixed point of are equivalent.

Strong convergence theorems by Halpern–Mann iterations for relatively nonexpansive mappings in Banach spaces

In this paper, we modify Halpern and Mann’s iterations for finding a fixed point of a relatively nonexpansive mapping in a Banach space. Consequently, a strong convergence theorem for a nonspreading mapping is deduced. Using a concept of duality theorems, we also obtain analogue results for certain generalized nonexpansive and generalized nonexpansive type mappings. Finally, we discuss two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a Banach space.

On the convergence of hybrid projection algorithms for asymptotically quasi-ϕ-nonexpansive mappings

Using the Mann iteration in an infinite-dimensional Hilbert space, only weak convergence theorems are obtained even for nonexpansive mappings. The purpose of this paper is to modify the Mann iteration and prove the strong convergence theorems without any compactness assumption for asymptotically quasi- ϕ -nonexpansive mappings. Moreover, strong convergence theorems are also established in a uniformly smooth and strictly convex Banach space with the Kadec–Klee property.

Weak and strong convergence theorems for positively homogenuous nonexpansive mappings in Banach spaces

Our purpose in this paper is first to prove a weak convergence theorem by Mann's iteration for positively homogeneous nonexpansive mappings in a Banach space. Further, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove a strong convergence theorem for such mappings. From two results, we obtain weak and strong convergence theorems for linear contractive mappings in a Banach space. These results are new even if the mappings are linear and contractive.

The Equivalence between the Mann and Ishikawa Iterations for Generalized Contraction Mappings in a Cone

In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping in cone subset of a real Banach space is discussed.

一般粘滞迭代方法及其应用 General Viscosity Iterative Methods and Applications

本文研究内容的核心之一在于，针对非扩张映射 的不动点问题，我们深刻阐明几种求解其不动点的迭代方法的相互关系。结合目前该类问题研究重点的转变，即从不动点的存在性和唯一性转变到如何构造有效迭代方法。我们系统阐述了Moudafi提出的粘滞迭代算法的发展演变过程，为以后的工作者提供参考。对于该类方法的具体应用进行深入的总结是本文的另一重心，主要围绕Mann迭代算法的修正及关于变分不等式以及均衡问题的联立求解获得相关结论。 The main core of this work is to clarify the profound relationships between several kinds of iterative methods for a fixed point of a given nonexpansive mapping.Concerning about the fact that the research focus has changed from the existence and uniqueness of the fixed piont to how to construct effective iterative methods.We begin with the viscosity iterative algorithm proposed by Moudafi.In order to provide a reference for future workers,we elaborate the development and evolution. The paper also deeply summarizes concrete applications of such methods.We focus not only on the modification processes of Mann iteration,but also concern about relevant conclusions about the variational inequalities and equilibrium problems.

Existence of common fixed points via modified mann iteration in convex metric spaces and an invariant approximation result

We prove the existence of common fixed points for two selfmaps $T$ and $f$ of a convex metric space $X$ via the convergence of modified Mann iteration where $T$ is a nonlinear $f$-weakly contractive selfmap of $X$ and range of $f$ is complete. An invariant approximation result is also proved.