Approximation of Nearest Common Fixed Point of Nonexpansive Mappings in Hilbert Spaces

The purpose of this paper is to study the convergence problem of the iteration scheme xn＋l = λn＋1y ＋ （1 - λn＋1）Tn＋1xn for a family of infinitely many nonexpansive mappings T1, T2,... in a Hilbert space. It is proved that under suitable conditions this iteration scheme converges strongly to the nearest common fixed point of this family of nonexpansive mappings. The results presented in this paper extend and improve some recent results.     

Hilbert空间中的K-框架

K-框架是框架理论的一种推广.K-框架可以用于重构Hilbert空间中有界线性算子值域内的元素.本文首先研究了K-框架与框架理论的关系,得到了紧K-框架成为框架当且仅当有界线性算子K是满的,给出了有界线性算子K具有闭值域的K-框架的一个充要条件.并利用有界线性算子K和合成算子构造K-框架,讨论在一定扰动条件下K-框架的稳定性.     

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces $\varLambda^{p}_{u}(w)$ , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L ^p (u) and Muckenhoupt weights A _p , and the theory on classical Lorentz spaces Λ ^p (w) and Ariño-Muckenhoupt weights B _p .     

最短距离与费马点

本文讨论到两个定点和一条定直线距离之和最短问题,证明了该点的存在性并给出该点的具体做法.     

Scalar Derivatives in Hilbert Spaces

Computational formulae for scalar derivatives of mappings and pairs of mappings in the direction of a set will be given. These formulae are very important for explicitly verifying conditions of existence for fixed point theorems, surjectivity theorems, integral equations, variational inequalities and complementarity problems under additional differentiability conditions. To emphasize this idea at the end of the paper we give an application to complementarity problems. Some theorems which extend the correspondence between monotone mappings and scalar derivatives from Euclidean spaces to Hilbert spaces will also be given.     

Hilbert空间中的紧K-框架

K-框架是框架的一种推广.本文在Hilbert空间将紧框架推广到K-框架上,引入紧K-框架的概念.通过紧K-框架的算子K和合成算子给出紧K-框架的算子刻画,并利用紧K-框架的算子K给出紧K-框架成为紧框架的一个充要条件.还讨论紧K-框架的构造以及两个紧K-框架集的包含与涉及的算子K的相互关系.     

Uncertainty Principles in Hilbert Spaces

In this article we provide several generalizations of inequalities bounding the commutator of two linear operators acting on a Hilbert space which relate to the Heisenberg uncertainty principle and time/frequency analysis of periodic functions. We develop conditions that ensure these inequalities are sharp and apply our results to concrete examples of importance in the literature.     

Penalty Algorithms in Hilbert Spaces

We analyze the classical penalty algorithm for nonlinear programming in Hilbert spaces and obtain global convergence results, as well as asymptotic superlinear convergence order. These convergence results generalize similar results obtained for finite-dimensional problems. Moreover, the nature of the algorithms allows us to solve the unconstrained subproblems in finite-dimensional spaces.     

Feedback stabilizability in Hilbert Spaces

This paper is divided in four sections. In the first one, we recall the main notions related to the infinite dimensional control system such as the controllability and stability notions. The second section is devoted to a brief survey of the literature relevant to this paper. In the third section, we concentrate on the weak stabilizability of semigroups which are similar to contraction semigroups. The result presented here, which is believed to be new is that: IfA generates a semigroup which is similar to aC ₀ contraction semigroup, the system is weakly stabilizable if and only if the weakly unstable states of the system are approximately controllable. Finally, in the last section, we present two counterexamples showing the limitations of the weak stabilizability approach and discuss its possible extensions in view of some other fundamental problems.This research was supported in part by the United States Air Force Office of Scientific Research, Applied Mathematics Division, under Grant 73-2492     

Hilbert空间中广义平衡问题和不动点问题的粘滞逼近法

在Hilbert空间,我们用粘滞逼近法建立了一迭代序列来逼近两个集合的公共点,这两个集合分别是广义平衡问题的解集和渐进非扩张映射的不动点集.我们表明这一迭代序列强收敛到这两个集合的公共点,而且这一公共点还是一变分不等式的解.用这一结果,还研究了三个强收敛问题和优化问题.     

Iterative Methods for Equilibrium and Fixed Point Problems for Nonexpansive Semigroups in Hilbert Spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup (T(s)) _s≥0 in Hilbert spaces. We prove that both approaches converge strongly to a common element z of the set of the equilibrium points and the set of common fixed points of (T(s)) _s≥0. Such common element z is the unique solution of a variational inequality, which is the optimality condition for a minimization problem.     

Finite-Dimensional Bicomplex Hilbert Spaces

This paper is a detailed study of finite-dimensional modules defined on bicomplex numbers. A number of results are proved on bicomplex square matrices, linear operators, orthogonal bases, self-adjoint operators and Hilbert spaces, including the spectral decomposition theorem. Applications to concepts relevant to quantum mechanics, like the evolution operator, are pointed out.     

空间H中G-框架的扰动

本文运用算子理论方法，讨论了Hilbert 空间$H$中$g$-框架和$g$-框架算子的性质; 并且研究了$g$-框架的扰动，给出了一些有意义的结果.     

Hilbert空间中的K-g-框架的性质

在复Hilbert空间中定义了K-g-框架与K-g-Riesz基,探讨K-g-框架与g-框架的一些性质差别,并给出了K-g-Riesz基的等价刻画.利用分析与框架理论上的方法和技巧,研究了复Hilbert空间中K-g-框架与K-g-Riesz基扰动的稳定性,得到了K-g-框架与K-g-Riesz基满足扰动稳定性的充分条件.     

A Strongly Convergent Combined Relaxation Method in Hilbert Spaces

We consider a combined relaxation method for variational inequalities in a Hilbert space setting. Methods of this class are known to solve finite-dimensional variational inequalities under mild monotonicity type assumptions, whereas in Hilbert space strong monotonicity is the standard assumption for strong convergence. Here, we relax this condition and show strong convergence of such a method, when strong monotonicity holds only on a subspace of finite co-dimension. Thus, the method applies to semi-coercive unilateral boundary value problems in mathematical physics.     

实Hilbert空间中点到有限余维子空间的距离问题

本文讨论实Ｈｉｌｂｅｒｔ空间中一点到有限余维子空间的距离，将一点到超平面的距离公式推广到一点到余维ｎ子空间Ｍ以及仿射集Ｑ的距离公式，并对Ｍ或Ｑ闭的情形表示出最佳逼近元。     

Hilbert空间中求解带约束的分裂公共不动点问题的三种迭代算法

该文的目的是研究带约束的分裂公共不动点问题.建立和分析了求解带约束的分裂公共不动点问题的三种新的迭代算法.建立了三种迭代算法的强收敛性结果,这些结果改进并推广了某些作者的相关结论.