CONVERGENT AND DIVERGENT SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR DIFFERENCE SYSTEM

In this paper, we are concerned with the existence of convergent or divergent solutions of two-dimensional nonlinear difference system of the form (?)xn+1=anxn+bnf(yn), yn=cnyn-1+dng(xn). We classify their solutions according to asymptotic behavior and give some sufficient and necessary conditions for the existence of solutions of such classes by using the method of the fixed point theorem. We also give an example and show how the results can he applied to certain difference systems.     

ON AMODIFIED NEWTON''''S METHOD AND CONVERGENCE

In this paper we discuss the convergence of a modified Newton's method presented by A. Ostrowski [1] and J.F. Traub [2], which has quadratic convergence order but reduces one evaluation of the derivative at every two steps compared with Newton's method. A convergence theorem is established by using a weak condition a≤3-2(2~(1/2)) and a sharp error estimate is given about the iterative sequence.     

STABILITY AND SUPER CONVERGENCE ANALYSIS OF ADI-FDTD FOR THE 2D MAXWELL EQUATIONS IN A LOSSY MEDIUM

Several new energy identities of the two dimensional(2D) Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved.These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction implicit finite difference time domain method for the 2D Maxwell equations(2D-ADI-FDTD).It is proved that 2D-ADI-FDTD is approximately energy conserved,unconditionally stable and second order convergent in the discrete L2 and H1 norms,which implies that 2D-ADI-FDTD is super convergent.By this super convergence,it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate.It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H1 norm.Experimental results to confirm the theoretical analysis on stability,convergence and energy conservation are presented.     

解变分不等式问题的混合方法

By using Fukushima‘s differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993. In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty, polyhedral,closed and convex is proposed. Armijo-type line search and trust region strategies as well as Fukushima‘s differentiable merit function are incorporated into the method. It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al. ,the method reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experiences show the efficiency of the proposed method.     

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.     

QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

Let f : U(x0) (?) E→F be a. C1 map and f'(X0) be the Prechet derivative of /fat X0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f'(x0) is double splitting and R(f'(x))∩N(T0 ) = {0} near X0. However, in infinite dimensional Banach spaces, f'(x0) is not always double splitting (i.e., the generalized inverse of f'(xo) does not always exist), but its bounded outer inverse of f'(x0) always exists. Only using the C1 map f and the outer inverse T0# of f'(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if X0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.     

HOMOCENTRIC CONVERGENCE BALL OF THE SECANT METHOD

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper.For every convergence theorem,a convergence ball is respectively introduced,where the hypothesis conditions of the corresponding theorem can be satisfied.Since all of these convergence balls have the same center x~*,they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.     

Ambarzumyan Theorems for Dirac Operators

We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential.The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators,which are subject to separation boundary conditions or periodic(semi-periodic)boundary conditions,and some analogs of Ambarzumyan's theorem are obtained.The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators,which are the second power of Dirac operators.     

A Constrained Optimization Approach for LCP

In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x,y) = 0 by using the famous NCP function-Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K-T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point, However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.     

Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev's Method

The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_2~((m,0)) by Sobolev's method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas.Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W_2~((m,0)) is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue Dm(hβ) of the differential operator (d~(2m))/ (dx~(2m))-1 is constructed. Further, Sobolev's method of construction of optimal quadrature formulas in the space W_2~((m,0)), which based on the discrete analogue D_m(hβ), is described. Next, for m = 1 and m = 3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained. Finally, at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W_2~((m,0)) for the cases m = 1 and m = 3 are presented.     

渐近非扩张非自映射的收敛定理(英文)

设E是实的一致凸Banach空间,K是E的一个非空闭凸集,P是E到K上的非扩张的保核收缩映射.设T1,T2,T3:K→E分别是具有数列{hn},{ln},{kn}[1,∞)的渐近非扩张非自映射,使得sum (hn-1) from n=1 to ∞<∞,sum ((ln-1)) from n=1 to ∞<∞及sum (n=1(kn-1) from n=1 to ∞<∞,且F=F(T1)∩F(T2)∩F(T3)={x∈K:T1x=T2x=T3x}≠Ф.定义迭代序列{xn}:x1∈K,xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-βn)xn+βnT2(PT2)n-1zn),zn=P((1-γn)xn+γnT3(PT3)n-1xn),其中{αn},{βn},{γn}[ε,1-ε],ε是大于零的实数.(i)如果T1,T2,T3中有一个是全连续的或者半紧的,则{xn}强收敛于某一点q∈F;(ii)如果E具有Frechet可微范数或者满足Opial’s条件或者E的对偶空间E~*具有Kadec-Klee性质,则{xn}弱收敛于某一点q∈F.     

逼近非扩张非自映像不动点（英文）

设E是一致凸Banach空间,K是E中非空闭凸集且是一个非扩张收缩核,T：K→E是具非空不动点集F（T）：={x∈K：Tx=x}的非扩张映像.设{α_n},{β_n},{γ_n},{α′_n},{β′_n},{γ′_n}是[0,1]中实数列满足α_n＋β_n＋γ_n=α′_n＋γ′_n＋γ′_n=1,对任意初值x_1∈K,定义{x_n}如下（ⅰ）如果对偶空间E＊具有Kadec-Klee性质,那么{x_n}弱收敛于T的某不动点x＊∈F（T）;（ⅱ）若T满足（A）条件,那么{x_n}强收敛于T的某不动点x＊∈F（T）.     

Banach空间非扩张映像不动点强收敛定理

设E是具弱序列连续对偶映像自反Banach空间, C是E中闭凸集, T:C→ C是具非空不动点集F(T)的非扩张映像.给定u∈ C,对任意初值x0∈ C,实数列{αn}n∞=0,{βn}∞n=0∈ (0,1),满足如下条件:(i)sum from n=α to ∞α_n=∞, α_n→0;(ii)β_n∈[0,α) for some α∈(0,1);(iii)sun for n=α to ∞|α_(n-1) α_n|<∞,sum from n=α|β_(n-1)-β_n|<∞设{x_n}_(n_1)~∞是由下式定义的迭代序列:{y_n=β_nx_n (1-β_n)Tx_n x_(n 1)=α_nu (1-α_n)y_n Then {x_n}_(n=1)~∞则{x_n}_(n=1)~∞强收敛于T的某不动点.     

序列覆盖的闭映射保持可度量性

设f:X→Y是连续的满映射. f称为序列覆盖映射,若{y})是Y中的收敛序列,则存在X中的收敛序列{xn},使得每一xn∈f-1(yn);f称为1序列覆盖映射,若对于每-y∈Y,存在x∈f-1(y),使得如果{yn}是Y中收敛于点y的序列,则有X中收敛于点x的序列{xn},使得每一xn∈f-1(yn).本文研究度量空间序列覆盖的闭映射之构造,否定地回答了Topology and its Applications上提出的一个问题.     

扰动压缩条件的点列收敛性问题

研究完备度量空间X中满足ρ(xn,xn+1)≤Lρ(xn-1,xn)+εn的点列{xn}收敛性问题,其中L∈(0,1)为常数,εn非负是无穷小量称为扰动.文中的主要结论是:点列{xn}的收敛性由扰动εn决定,即当幂级数sum from n=1 to ∞εnxn的收敛半径R>1/L时,点列{xn}收敛.特别地,当R>1时,点列收敛;而R=1时,{xn}敛散性不能确定.     

Approximation Solutions of Nonlinear Strongly Accretive Operator Equations by Ishikawa Iteration Procedure with Errors

Let 1＜ρ≤2,E be a real ρ-uniformly smooth Banach space and T:E→E be a continuous and strongly accretive operator.The purpose of this paper is to investigate the problem of approximating solutions to the equation Tx=f by the Ishikawa iteration procedure with errors (?) where x_0 ∈ E,{u_n},{υ_n}are bounded sequences in E and{α_n},{b_n},{c_n},{a_n~'},{b_n~'},{c_n~'} are real sequences in[0,1].Under the assumption of the condition 0＜α≤b_n c_n,An≥0, it is shown that the iterative sequence{x_n}converges strongly to the unique solution of the equation Tx=f.Furthermore,under no assumption of the condition(?)(b_n~' c_n~')=0,it is also shown that{x_n}converges strongly to the unique solution of Tx=f.     

伪压缩族公共不动点的复合隐格式迭代逼近问题

设H是一实Hillber空间,K是H之一非空间凸子集,设{Ti}Ni=1是N个Lipschitz伪压缩映象使得F=∩Ni=1F(Ti)≠0,其中F(Ti)={x∈K:Tix=x}并且{αn}n∞=1,{βn}∞n=1[0,1]是满足如下条件的实序列(i)∑∞n=1(1-αn)2= ∞;(ii)limn→∞(1-αn)=0;(iii)∑∞n=1(1-βn)< ∞;(iv)(1-αn)L2<1,n1;(v)αn(1-βn)2 αn[βn L(1-βn)]2<1,其中L1是{Ti}iN=1的公共Lipschitz常数,对于x0∈K,设{xn}n∞=1是由下列定义的复合隐格式迭代xn=αnxn-1 (1-αn)Tnyn,yn=βnxn (1-βn)Tnxn,其中Tn=TnmodN,则(i)limn→∞‖xn-p‖存在,对于所有的p∈F;(ii)limn→∞d(xn,f)存在,其中d(xn,F)=infp∈F‖xn-p‖;(iii)liminfn→∞‖xn-Tnxn‖=0.本文的结果推广并且改进H-K.Xu和R.G.Ori在2001年的结果和Osilike在2004年的结果,并且在这篇文章中,主要的证明方法也不同与H-K.Xu和Osilike的方法.     

具误差隐格式迭代逼近严格伪压缩映像族公共不动点

设K是实Banach空间E中非空闭凸集, {Ti}i=1N是N个具公共不动点集F的严格伪压缩映像, {an}(?)[0,1]是实数列, {un}(?)K是序列,且满足下面条件设X0∈K,{xn}由下式定义xn=αnxn-1 (1-αn)Tnxn-un-1,n≥1其中Tn=TnmodN,则有下面结论(i)limn→∞‖xn-p‖存在,对所有P∈F; (ii)limn→∞d(xn,F)存在,当d(xn,F)=infp∈F‖xn-p‖; (iii)liminfn→∞‖xn-Tnxn‖=0．文中另一个结果是,如果{xn}(?){1-2-n,1},则{xn}收敛．文中结果改进与扩展了Osilike(2004)最近的结果,证明方法也不同．     

广义复合隐迭代格式逼近非扩张映像公共不动点

提出并使用如下广义复合隐迭代格式逼近非扩张映像族{Ti}Ni=1公共不动点:{xn=αnxn-1 (1-αn)Tnyn,yn=rnxn snxn-1 tnTnxn wnTnxn-1,rn sn tn wn=1,{αn},{rn},{sn},{tn},{wn}∈[0,1],这里Tn=TnmodN.该文提出的广义复合隐迭代格式包含了目前多种迭代格式,因此,所得强弱收敛定理推广及发展了Mann,Ishikawa,XuandOri,等许多作者的结果.