On the United Theory of the Family of Euler-Halley Type Methods with Cubical Convergence in Banach Spaces

The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the L-average, and a united convergence theory with its applications is presented.     

On the United Theory of the Family of Euler—Haller Type Methods with Cubical Convergence in Banach Spaces

The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the L-average,and a united convergence theory with its applications is presented.     

The Equivalence of Ishikawa-Mann and Multistep Iterations in Banach Space

Let E be a real Banach space and T be a continuous Φ-strongly accretive operator. By using a new analytical method, it is proved that the convergence of Mann, Ishikawa and three-step iterations are equivalent to the convergence of multistep iteration. The results of this paper extend the results of Rhoades and Soltuz in some aspects.     

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.     

Convergence analysis for the Secant method based on new recurrence relations

A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation.It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous.The convergence conditions differ from some existing ones and are easily satisfied.The results of the paper are justified by numerical examples that cannot be handled by earlier works.     

Convergence analysis for the Secant method based on new recurrence relations

A new convergence theorem for the Secant method in Banach spaces based on new recurrence relations is established for approximating a solution of a nonlinear operator equation. It is assumed that the divided difference of order one of the nonlinear operator is Lipschitz continuous. The convergence conditions differ from some existing ones and are easily satisfied. The results of the paper are justified by numerical examples that cannot be handled by earlier works.     

An Inertial Semi-forward-reflected-backward Splitting and Its Application

Inertial methods play a vital role in accelerating the convergence speed of optimization algorithms. This work is concerned with an inertial semi-forward-reflected-backward splitting algorithm of approaching the solution of sum of a maximally monotone operator, a cocoercive operator and a monotone-Lipschitz continuous operator. The theoretical convergence properties of the proposed iterative algorithm are also presented under mild conditions. More importantly, we use an adaptive stepsize rule in...     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

A superlinearly and quadratically convergent SQP type feasible method for constrained optimization

A new SQP type feasible method for inequality constrained optimization is presented, it is a combination of a master algorithm and an auxiliary algorithm which is taken only in finite iterations. The directions of the master algorithm are generated by only one quadratic programming, and its step-size is always one, the directions of the auxiliary algorithm are new “secondorder“ feasible descent. Under suitable assumptions, the algorithm is proved to possess global and strong convergence, superlinear and quadratic convergence.     

Inner iterations in the shift-invert residual Arnoldi method and the Jacobi-Davidson method

We establish a general convergence theory of the Shift-Invert Residual Arnoldi(SIRA)method for computing a simple eigenvalue nearest to a given targetσand the associated eigenvector.In SIRA,a subspace expansion vector at each step is obtained by solving a certain inner linear system.We prove that the inexact SIRA method mimics the exact SIRA well,i.e.,the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with low or modest accuracy during outer iterations.Based on the theory,we design practical stopping criteria for inner solves.Our analysis is on one step expansion of subspace and the approach applies to the Jacobi-Davidson(JD)method with the fixed targetσas well,and a similar general convergence theory is obtained for it.Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.     

ON REMES ALGORITHA FOR THE BEST CHEBYSHEV APPROXIMATION FROM VARISOLVENT FAMILY

In this paper, Remes algorithm is applied to compute the numerical solution of the best chebyshev approximation from varisolvent family. Feasibility and convergence of the algorithm are discussed carefully.     

广义的Lupas—Baskakov积分算子的逼近性质

In this paper,we shall define the improved Lupas-Baskakov integral operators,and study the pointwise convergence properties of this operator.We shall also prove our estimates are essentially best possible.     

A SQP METHOD FOR MINIMIZING A CLASS OF NONSMOOTH FUNCTIONS

In this paper,we present a successive quadratic programming(SQP)method for minimizing a class of nonsmooth functions,which are the sum of a convex function and a nonsmooth composite function.The method generates new iterations by using the Armijo-type line search technique after having found the search directions.Global convergence property is established under mild assumptions.Numerical results are also offered.     

PERTURBATIONS OF DEFINITIZABLE OPERATORS IN A SPACE WITH AN INDEFINITE METRIC

Perturbations of definitizable operators in Krein space are studied in this paper. First, the convergence of resolvents and spectral functions is discussed if a sequence of definitizable operators converges in a general sense. Second, for the operational calculus relating to continuous functions, various convergences of operator functions are studied. At last, the relation for the convergence of the sequence of resolvents and that of one-parameter unitary groups is studied. The main theorems of this paper can be regarded as the generalization of the results for self-adjoint operators in Hilbert space,     

Hilbert空间框架下严格渐近伪压缩映像的强收敛定理

The Mann iterations have no strong convergence even for nonexpansive mappings in Hilbert spaces. The aim of this paper is to propose a modification of the Mann iterations for strictly asymptotically pseudocontractive maps in Hilbert spaces to have strong convergence. Our results extend those of Kim, Xu, Nakajo, Takahashi and many others.     

Properties of a family of merit functions and a merit function method for the NCP

A family of merit functions are proposed, which are the generalization of several existing merit functions. A number of favorable properties of the proposed merit functions are established. By using these properties, a merit function method for solving nonlinear complementarity problem is investigated, and the global convergence of the proposed algorithm is proved under some standard assumptions. Some preliminary numerical results are given.     

Block monotone iterative algorithms for variational inequalities with nonlinear operators

Some block iterative methods for solving variational inequalities with nonlinear operators are proposed. Monotone convergence of the algorithms is obtained. Some comparison theorems are also established. Compared with the research work in given by Pao in 1995 for nonlinear equations and research work in given by Zeng and Zhou in 2002 for elliptic variational inequalities, the algorithms proposed in this paper are independent of the boundedness of the derivatives of the nonlinear operator.     

WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

In this paper,we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces.Then we study the weak convergence of the two iterative sequences.Our results improve and extend the results announced by many others.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

OPTIMAL CONVERGENCE RATES OF LANDAU EQUATION WITH EXTERNAL FORCING IN THE WHOLE SPACE

In this paper, we combine the method of constructing the compensating function introduced by Kawashima and the standard energy method for the study on the Landau equation with external forcing. Both the global existence of solutions near the time asymptotic states which are local Maxwellians and the optimal convergence rates are obtained. The method used here has its own advantage for this kind of studies because it does not involve the spectrum analysis of the corresponding linearized operator.