On Properties of Solutions for a Class of Functional Equations Arising in Dynamic Programming

The existence, uniqueness, and iterative approximation of solutions for a class of functional equations arising in dynamic programming of multistage decision processes are discussed. Our results resolve in the affirmative an open problem posed in Ref. 1 and generalize important known results.     

Fixed Points and Asymptotic Behavior for Asymptotically Nonexpansive Type Semigroups in Banach Spaces

Let X be a Banach space with a weakly continuous duality map Jψ, C a non-empty weakly compact convex subset of X, and T = (T(t) : t ∈ S} an asymptotically nonexpansive type semigroup on C. In this paper, the inequality K ∩ F(T) ≠ (?) is characterized, where K is a subset of C and F(T) is the set of all common fixed points of T. Furthermore, it is shown that an almost-orbit     

渐近伪压缩映象的收敛性定理

在任意的实Banach空间中,证明了一致Lipschitzian和渐近伪压缩映象的具误差的Ishikawa和Mann迭代序列的一些收敛性定理.     

UNIFORM NORMAL STRUCTURE AND SOLUTIONS OF REICH’S OPEN QUESTION

IntroductionThroughout this paper we assume thatEis a real Banach space,E*is the dual space ofE,Dis a nonempty subset ofEandJ:E→2E*is the normalized duality mapping defined byJ(x)={f∈E*:〈x,f〉=‖x‖.‖f‖,‖f‖=‖x‖},x∈E,(1)where〈.,.〉denotes the ge     

Banach空间中Reich-Takahashi迭代法的强收敛定理

设E是具有一致正规结构的实Banach空间,其范数是一致Gateaux可微的;设D是E的非空有界闭凸子集,T:D→D是渐近非扩张映象.本文证明了,在一些适当的条件下,由修正的Reich-Takahashi迭代法(1.2)式所定义的序列{xn}强收敛到渐近非扩张映象的不动点,其中x0是D中一任给点,{αn},{β}是区间[0,1]中满足某些限制的实数列.     

渐近非扩展映象的具误差的Ishikawa迭代序列的新强收敛定理

使用新的证明方法，在去掉数列{an}单调递减的条件下，建立了一致凸Banach空间中的渐近非扩展映象不动点的具误差的Ishikawa迭代序列的新强收敛定理．其结果推广和改进了Schu，Rhoades及周海云等作者的相关结果．     

渐近非扩张映象的修正的Ishikawa迭代程序

本文研究一致凸Banach空间中关于渐近非扩张映象不动点的修正的Ishikawa迭代程序的强收敛性，本文结果统一，推广与改进了目前文献中的一些最新结果。     

The method of alternating projections and the method of subspace corrections in Hilbert space

A new identity is given in this paper for estimating the norm of the product of nonexpansive operators in Hilbert space. This identity can be applied for the design and analysis of the method of alternating projections and the method of subspace corrections. The method of alternating projections is an iterative algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces by alternatively computing the best approximations from the individual subspaces which make up the intersection. The method of subspace corrections is an iterative algorithm for finding the solution of a linear equation in a Hilbert space by approximately solving equations restricted on a number of closed subspaces which make up the entire space. The new identity given in the paper provides a sharpest possible estimate for the rate of convergence of these algorithms. It is also proved in the paper that the method of alternating projections is essentially equivalent to the method of subspace corrections. Some simple examples of multigrid and domain decomposition methods are given to illustrate the application of the new identity.

     

Finding best approximation pairs relative to two closed convex sets in Hilbert spaces

We consider the problem of finding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed AAR for averaged alternating reflections, is a special instance of an algorithm due to Lions and Mercier for finding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of AAR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the AAR method is shown to coincide with a special case of Spingarn's method of partial inverses.     

Banach空间中带误差的修改的Ishikawa迭代程序

本文研究在任意的实Banach空间中用带误差的修改的Ishikawa迭代序列来逼近一致Lipschitz的渐近伪压缩映象的不动点的问题.在去掉限制limn→∞βn=0之下,证明了张石生教授的结果(见文[1])仍真.另一方面,也把他的结果推广到了带误差的修改的Ishikawa迭代序列的情形.