On the super fixed point property in product spaces

We prove that if F is a finite-dimensional Banach space and X has the super fixed point property for nonexpansive mappings, then F⊕X has the super fixed point property with respect to a large class of norms including all lp norms, 1?p<∞. This provides a solution to the “super-version” of the problem of Khamsi (1989).     

Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces

In this paper, we propose three different kinds of iteration schemes to compute the approximate solutions of variational inequalities in the setting of Banach spaces. First, we suggest Mann-type steepest-descent iterative algorithm, which is based on two well-known methods: Mann iterative method and steepest-descent method. Second, we introduce modified hybrid steepest-descent iterative algorithm. Third, we propose modified hybrid steepest-descent iterative algorithm by using the resolvent operator. For the first two cases, we prove the convergence of sequences generated by the proposed algorithms to a solution of a variational inequality in the setting of Banach spaces. For the third case, we prove the convergence of the iterative sequence generated by the proposed algorithm to a zero of an operator, which is also a solution of a variational inequality.     

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.     

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.     

Strong Convergence of Composite Iteration Process for Nonexpansive Semigroups

It is to study the strong convergence of a composite iteration process to a common fixed point for two nonexpansive semigroups, which is a solution of certain variational inequality under certain conditions. The results of this article extend the corresponding recent results by some others.     

-class algorithms for pseudocontractions and -strict pseudocontractions in Hilbert spaces

In this paper we study iterative algorithms for finding a common element of the set of fixed points of κ-strict pseudocontractions or finding a solution of a variational inequality problem for a monotone, Lipschitz continuous mapping. The last problem being related to finding fixed points of pseudocontractions. These algorithms were already studied in [G.L. Acedo, H.-K. Xu, Iterative methods for strict pseudo-contractions in hilbert spaces, Nonlinear Analysis 67 (2007) 2258–2271] and [N. Nadezhkina, W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and lipschitz-continuous monotone mappings, SIAM Journal on Optimization 16 (4) (2006) 1230–1241] but our aim here is to provide the links between these known algorithms and the general framework of -class algorithms studied in [H.H. Bauschke, P.L. Combettes, A weak-to-strong convergence principle for fejér-monotone methods in hilbert spaces, Mathematics of Operations Research 26 (2) (2001) 248–264].     

Viscosity methods for zeroes of accretive operators

This paper deals with the general iteration method , for calculating a particular zero of A, an m-accretive operator in a Banach space X, T_n being a sequence of nonexpansive self-mappings in X. Under suitable conditions on the parameters and X, we state strong and weak convergence results of (x_n). We also show how to compute a common zero of two m-accretive operators in X.     

The set of common fixed points of a one-parameter continuous semigroup of mappings is

In this paper we prove the following theorem: Let be a one-parameter continuous semigroup of mappings on a subset of a Banach space . The set of all fixed points of is denoted by for each . Then

holds. Using this theorem, we discuss convergence theorems to a common fixed point of .

     

关于三个非扩张映射收敛于其公共不动点的具误差的三重迭代法

在一致凸Banach空间中,研究了关于三个不同非扩张映射的具误差的三步迭代算法,并在一定条件下证明了此迭代序列收敛于其公共不动点的弱强收敛定理.