Banach空间中一类映像的三重迭代法

利用赋范线性空间X的凸性模定义,以及凸性模的单调性及半紧性条件,研究了渐近非扩张映射不动点的三步迭代法.减弱了许多条件,从而推广了同类问题的某些结果.     

渐近非扩张型映射的不动点迭代和△-收敛定理

研究了一致凸双曲度量空间中渐近非扩张型映射的三步迭代,这种迭代包括了Ishikawa型迭代和Krasnoselski-Mann迭代作为特例.作为应用还得到了迭代算法在CAT(0)空间中的△-收敛定理,得到的结论推广并加强了以前的许多已知结果.     

集合的$\varOmega$-凸性及其基础性质[英文]

作为集合凸性概念的一种推广以及统一凸性与近似（nearly）凸性等概念的一种尝试, 本文引入集合$\varOmega$-凸性的概念, 并对$\varOmega$-凸集合的性质进行初步的研究. 另外, 本文还研究了一些常见变换与集合运算的保$\varOmega$-凸性质.     

一类条件不等式的控制证明与应用

通过判断相关函数的Schur凸性、Schur几何凸性和Schur调和凸性,证明并推广了一类条件不等式,并据此建立了某些单形不等式.     

Hahn-Banach定理在V-凸性模定义中的应用

先给出了V-凸性模的两个等价定义,并利用Hahn-Banach定理给出了它们的等价性.其次,在V-凸性模定义的基础上引进了广义V-凸性模的概念,并给出了其两个等价定义.     

关于随机一致凸性

致力于随机一致凸性概念的进一步探讨．首先,通过一个特殊的层次剖分指出对任意的随机赋范模而言随机凸性模都有良好定义,从而改进了近期的文献中许多已知的结果．然后,提出并研究了一种与随机一致凸性密切相关的新性质,从一个新的角度阐述了随机一致凸性的复杂性．     

有限个广义渐近非扩张映射具误差的复合隐迭代过程

在一致凸Banach空间研究了一个新的有限个广义渐近非扩张映射具误差的复合隐迭代过程.利用空间满足Opial条件和算子满足半紧性条件,我们证明了这个隐迭代过程强、弱收敛于有限个广义渐近非扩张映射的公共不动点.这些结果是目前所得成果的完善和推广.     

集值渐近拟非扩张映射的不动点收敛定理

研究一致凸Banach空间中集值渐近拟非扩张映射的关于有限步迭代序列逼近公共不动点的充分必要条件,并在此条件下,证明了该序列收敛到公共不动点的一些强收敛定理,所得结果是单值映射情形的推广和发展.     

一般化凸空间上的一个连续选择定理(英文)

本文研究一般化凸空间上的连续选择定理.利用在D■X的条件下,一般化凸空间(X,D;Γ)上Γ-凸子集的概念,得到了两类一般化凸空间之间,以及φ映射和Γ-凸映射之间的关系,并且得到了一个连续选择定理.本文推广了一般化凸空间上凸子集的概念.     

广义非凸变分不等式解的存在性与投影算法

本文运用Banach压缩映象原理和投影技巧研究一类新的广义非凸变分不等式问题解的存在唯一性,并在非凸集上建立一个逼近广义非凸变分不等式解的三步投影算法,在一定条件下证明了该投影算法所产生的迭代序列的收敛性.     

An Iteration Process for Nonlinear Mappings in Uniformly Convex Linear Metric Spaces

We obtain necessary conditions for convergence of the Cauchy Picard sequence of iterations for Tricomi mappings defined on a uniformly convex linear complete metric space.     

A reconsideration on convergence of three-step iterations for asymptotically nonexpansive mappings

We illustrate that the control conditions of the main convergence theorems of Yao and Noor [Convergence of three-step iterations for asymptotically nonexpansive mappings, Appl. Math. Comput. in press] are incorrect. We also provide new control conditions which are complementary to Nilsrakoo and Saejung’s results [W. Nilsrakoo, S. Saejung, A new three-step fixed point iteration scheme for asymptotically nonexpansive mappings, Appl. Math. Comput. 181 (2006) 1026–1034].     

矩阵多分裂迭代法的收敛条件

本文给出了求解非奇异线性方程组的矩阵多分裂并行迭代法的一些新的收敛结果.当系数矩阵单调和多分裂序列为弱正则分裂时,得到了几个与已有的收敛准则等价的条件,并且证明了异步迭代法在较弱条件下的收敛性.对于同步迭代,给出了与异步迭代不同且较为宽松的收敛条件.     

On the convergence of a class of outer approximation algorithms for convex programs

This paper presents a new class of outer approximation methods for solving general convex programs. The methods solve at each iteration a subproblem whose constraints contain the feasible set of the original problem. Moreover, the methods employ quadratic objective functions in the subproblems by adding a simple quadratic term to the objective function of the original problem, while other outer approximation methods usually use the original objective function itself throughout the iterations. By this modification, convergence of the methods can be proved under mild conditions. Furthermore, it is shown that generalized versions of the cut construction schemes in Kelley-Cheney-Goldstein's cutting plane method and Veinott's supporting hyperplane method can be incorporated with the present methods and a cut generated at each iteration need not be retained in the succeeding iterations.     

Demiclosed principle and convergence for modified three step iterative process with errors of non-Lipschitzian mappings

A demiclosed principle is proved for asymptotically nonexpansive mappings in the intermediate sense. Moreover, it is proved that the modified three-step iterative sequence converges weakly and strongly to common fixed points of three asymptotically nonexpansive mappings in the intermediate sense under certain conditions. The results of this paper improve and extend the corresponding results of [M.O. Osilike, S.C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32 (2000) 1181-1191; G.E. Kim, T.H. Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces, Comput. Math. Appl. 42 (2001) 1565-1570; B.L. Xu, M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) 444-453; K. Nammanee, S. Suantai, The modified Noor iterations with errors for non-Lipschitzian mappings in Banach spaces, Appl. Math. Comput. 187 (2007) 669-679; K. Nammanee, M.A. Noor, S. Suantai, Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 314 (2006) 320-334] and other corresponding known ones. On the other hand, we show the necessary and sufficient condition for the strong convergence of the modified three-step iterative sequence to some common fixed points of .     

On Generalizations of the Frank-Wolfe Theorem to Convex and Quasi-Convex Programmes

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or a quasi-convex polynomial function over the feasible set defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities, where the faithfully convex functions satisfy some mild assumption. The conditions are provided in the form of an algorithm, terminating after a finite number of iterations, the implementation of which requires the identification of implicit equality constraints in a homogeneous linear system. We prove that the optimal solution set of the considered problem is nonempty, this way extending the attainability result well known as the so-called Frank-Wolfe theorem. Finally we show that our extension of the Frank-Wolfe theorem immediately implies continuity of the solution set defined by the considered system of (quasi)convex inequalities.     

On the finite termination of the Douglas-Rachford method for the convex feasibility problem

In this paper we apply the Douglas–Rachford (DR) method to solve the problem of finding a point in the intersection of the interior of a closed convex cone and a closed convex set in an infinite-dimensional Hilbert space. For this purpose, we propose two variants of the DR method which can find a point in the intersection in a finite number of iterations. In order to analyse the finite termination of the methods, we use some properties of the metric projection and a result regarding the rate of convergence of fixed point iterations. As applications of the results, we propose the methods for solving the conic and semidefinite feasibility problems, which terminate at a solution in a finite number of iterations.     

Endpoint theory for set-valued nonlinear asymptotic contractions with respect to generalized pseudodistances in uniform spaces

In uniform spaces, inspired by ideas of Banach, Tarafdar and Yuan, we introduce the concepts of generalized pseudodistances and generalized gauge maps, for set-valued dynamic systems we define various nonlinear asymptotic contractions and contractions with respect to these pseudodistances and gauges, provide conditions on the iterates of these set-valued dynamic systems and present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints (stationary points) of these set-valued dynamic systems and conditions that each generalized sequence of iterations (in particular, each dynamic process) converges and the limit of a generalized sequence of iterations is an endpoint. The definitions, the results and the method are new for set-valued dynamic systems in uniform, locally convex and metric spaces and even for single-valued maps. The paper includes a number of various examples which show a fundamental difference between our results and those existing in the literature.     

Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings

In this paper, weak and strong convergence theorems of the modified Noor iterations with errors are established for asymptotically nonexpansive mappings in Banach spaces. The results obtained in this paper extend and improve the several recent results in this area.