Convergence in Norm of Projection Regularized Krasnoselski-Mann Iterations for Fixed Points of Cutters

In this article, we introduce a projection regularized Krasnoselski-Mann iteration for cutters. The proposed algorithm ensures the strong convergence of the generated sequence toward the least norm element of the set of fixed points of the cutter. It is verified that the projection regularized Krasnoselski-Mann iteration converges locally faster than the regularized Krasnoselski-Mann iteration introduced by Maingé and Maruster [11 P. E. Maingé and S. Maruster ( 2011 ). Convergence in norm of modified Krasnoselski-Mann iterations for fixed points of demicontractive mappings . Appl. Math. Comput. 217 : 9864 – 9874 .[Crossref], [Web of Science ®] , [Google Scholar]]. Furthermore, we present projection regularized Krasnoselski-Mann iterations for quasi-nonexpansive and nonexpansive mappings in Hilbert spaces.     

Further investigation into split common fixed point problem for demicontractive operators

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.     

General algorithms for split common fixed point problem of demicontractive mappings

In the first part of this paper, we present a new general algorithm for solving the split common fixed point problem for an infinite family of demicontractive mappings. We establish strong convergence of the algorithm in an infinite dimensional Hilbert space. As applications, we consider algorithms for split variational inequality problem and split common null point problem. In the second part of this paper, we present a new algorithm and strong convergence theorem for approximation of solutions of split equality fixed point problems for an infinite family of demicontractive mappings. Our results improve and generalize some recent results in the literature.     

Convergence theorems for common fixed points of asymptotically quasi-nonexpansive mappings in convex metric spaces

In this paper, we consider an implicit iteration process to approximate the common fixed points of two finite families of asymptotically quasi-nonexpansive mappings in convex metric spaces. As a consequence of our result, we obtain some related convergence theorems. Our results generalize some recent results of Khan and Ahmed [4], Khan et al. [6], Sun [12], Wittmann [14] and Xu and Ori [15].     

Approximating common fixed points of nearly asymptotically nonexpansive mappings

We use an iteration scheme to approximate common fixed points of nearly asymptotically nonexpansive mappings.We generalize corresponding theorems of [1] to the case of two nearly asymptotically nonexpansive mappings and those of [9] not only to a larger class of mappings but also with better rate of convergence.     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications, 2008, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and many others.     

Realization of the hybrid method for Mann iterations

The Mann iterations for nonexpansive mappings have only weak convergence even in a Hilbert space H. In order to overcome this weakness, Nakajo and Takahashi proposed the hybrid method for Mann’s iteration process:

     

Strong convergence of averaging iterations of nonexpansive nonself-mappings

Let C be a closed convex subset of Hilbert space H, T a nonexpansive nonself-mapping from C into H, and x₀,x,y₀,y elements of C. In this paper, we study the convergence of the two sequences generated by

     

Random iterations, fixed points and invariant CRF-horospheres in complex Banach spaces

For holomorphic noncontractive maps on (not necessarily bounded) domains in complex Banach spaces, we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations. In particular, we show that the problem, concerning the existence of the horospheres determined by Carathéodory-Reiffen-Finsler pseudometrics defined on unbounded domains, has the solution and we prove new results of type of Julia's lemma and Wolff's theorem.     

(渐近)非扩张映象的不动点的迭代逼近

Let E be a uniformly convex Banach space which satisfies Opial‘s condition or has aFrechet differentiable norm,and C be a bounded closed convex subset of E. If T： C→C is(asymptotically)nonexpansive,then the modified Ishikawa iteration process defined by     

Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces

The aim of this work is to propose implicit and explicit viscosity-like methods for finding specific common fixed points of infinite countable families of nonexpansive self-mappings in Hilbert spaces. Two numerical approaches to solving this problem are considered: an implicit anchor-like algorithm and a nonimplicit one. The considered methods appear to be of practical interests from the numerical point of view and strong convergence results are proved.     

Iterative methods for the computation of fixed points of demicontractive mappings

This paper surveys some of the main convergence properties of the Mann-type iteration for the demicontractive mappings. Some variants of the Mann iteration that ensure the strong convergence, like the (CQ) algorithm and a variant for the asymptotically demicontractive mappings are also considered. The usual framework of our study is a (real) Hilbert space and only to a certain extent some particular Banach spaces. Historical aspects are pointed out and some applications for the convex feasibility problem are discussed.     

自反Banach空间中非扩张映像的强收敛定理

在Banach空间框架下考虑了一个关于无限族非扩张映射的一般迭代方法,此结论改进和推广了他人的许多结论.     

Existence of fixed points for some convex operators and applications to multi-point boundary value problems

Existence results of fixed points for some convex operators are given by means of fixed point theorem of cone expansion and compression, then they are applied to nonlinear multi-point boundary value problems.     

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak^? continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping JΦ with gauge ?. Let f be an α-contraction and {Tn} a sequence of nonexpansive mappings, we study the strong convergence of explicit iterative schemes

(1)     

Fixed points, selections and common fixed points for nonexpansive-type mappings

We study the existence of fixed points in the context of uniformly convex geodesic metric spaces, hyperconvex spaces and Banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity. Besides, we use the fixed point theorems proved here to give common fixed point results for commuting mappings.     

Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems

In this paper, we propose a new composite iterative method for finding a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of nonexpansive mappings. Our results improve and extend the corresponding ones announced by many others.     

Convergence theorems of fixed points for asymptotically hemi-pseudocontractive mappings

In this paper, the concept of asymptotically hemi-pseudocontractive mapping in Banach space, which is a proper generalization of asymptotically pseudocontractive mapping with nonempty fixed point set as shown by an example, is introduced. Some sufficient and necessary conditions on the strong convergence of the modified Ishikawa and the modified Mann iteration sequences with mean errors for uniformly L-Lipshitzian asymptotically hemi-pseudocontractive mappings are presented.     

Fixed points for weakly inward mappings in Banach spaces

S. Hu and Y. Sun [S. Hu, Y. Sun, Fixed point index for weakly inward mappings, J. Math. Anal. Appl. 172 (1993) 266-273] defined the fixed point index for weakly inward mappings, investigated its properties and studied the fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we continue to investigate boundary conditions, under which the fixed point index for the completely continuous and weakly inward mapping, denoted by i(A,Ω,P), is equal to 1 or 0. Correspondingly, we can obtain some new fixed point theorems of the completely continuous and weakly inward mappings and existence theorems of solutions for the equations Ax=μx, which extend many famous theorems such as Leray-Schauder's theorem, Rothe's two theorems, Krasnoselskii's theorem, Altman's theorem, Petryshyn's theorem, etc., to the case of weakly inward mappings. In addition, our conclusions and methods are different from the ones in many recent works.     

Regularized and inertial algorithms for common fixed points of nonlinear operators

This paper deals with a general fixed point iteration for computing a point in some nonempty closed and convex solution set included in the common fixed point set of a sequence of mappings on a real Hilbert space. The proposed method combines two strategies: viscosity approximations (regularization) and inertial type extrapolation. The first strategy is known to ensure the strong convergence of some successive approximation methods, while the second one is intended to speed up the convergence process. Under classical conditions on the operators and the parameters, we prove that the sequence of iterates generated by our scheme converges strongly to the element of minimal norm in the solution set. This algorithm works, for instance, for approximating common fixed points of infinite families of demicontractive mappings, including the classes of quasi-nonexpansive operators and strictly pseudocontractive ones.