Symmetric positive solutions for second order boundary value problems with integral boundary conditions on time scales

This paper investigates the existence of symmetric positive solutions for a class of nonlinear boundary value problem of second order dynamic equations with integral boundary conditions on time scales. Under suitable conditions, the existence of symmetric positive solution is established by using monotone iterative technique     

THE SYMMETRIC POSITIVE SOLUTIONS OF $2n$-ORDER BOUNDARY VALUE PROBLEMS ON TIME SCALES

In this paper, we are concerned with the symmetric positive solutions of a 2n-order boundary value problems on time scales. By using induction principle, the symmetric form of the Green''s function is established. In order to construct a necessary and sufficient condition for the existence result, the method of iterative technique will be used. As an application, an example is given to illustrate our main result.     

Saddle point methods,and alogorithms,for non-symmetric linear equations ∗

This paper describes methods for solving non-singular, non-symmetric linear equations whose symmetric part is positive definite. First, the solutions are characterized as saddle points of a convex-concave function. The associated primal and dual variational principles provide quadratic, strictly convex, functions whose minima are the solutions of the original equation and which generalize the energy function for symmetric problems.

Direct iterative methods for finding the saddle point are then developed and analyzed. A globally convergent algorithm for finding the saddle points is described. We show that requiring conjugacy of successive search directions with respect to the symmetric part of the equation is a poor strategy.     

Computable error bounds and estimates for the conjugate gradient method

The conjugate gradient method is one of the most popular iterative methods for computing approximate solutions of linear systems of equations with a symmetric positive definite matrix A. It is generally desirable to terminate the iterations as soon as a sufficiently accurate approximate solution has been computed. This paper discusses known and new methods for computing bounds or estimates of the A-norm of the error in the approximate solutions generated by the conjugate gradient method.     

Half thresholding eigenvalue algorithm for semidefinite matrix completion

The semidefinite matrix completion(SMC) problem is to recover a low-rank positive semidefinite matrix from a small subset of its entries. It is well known but NP-hard in general. We first show that under some cases, SMC problem and S1/2relaxation model share a unique solution. Then we prove that the global optimal solutions of S1/2regularization model are fixed points of a symmetric matrix half thresholding operator. We give an iterative scheme for solving S1/2regularization model and state convergence analysis of the iterative sequence.Through the optimal regularization parameter setting together with truncation techniques, we develop an HTE algorithm for S1/2regularization model, and numerical experiments confirm the efficiency and robustness of the proposed algorithm.     

Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators acting on an ordered finite dimensional Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well us in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequalities) or forward affine equations (inequalities).     

EXISTENCE AND ITERATION OF POSITIVE SYMMETRIC SOLUTIONS TO A MULTI-POINT BOUNDARY VALUE PROBLEM

In this paper,we consider the existence of symmetric solutions to a nonlinear second order multi-point boundary value problem,and establish corresponding iterative schemes based on the monotone iterative method.     

Existence and iteration of monotone positive solutions for an elastic beam equation with a corner

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with a corner. The boundary conditions mean that the beam is embedded at one end and fastened with a sliding clamp at the other end. By applying monotone iterative techniques, we not only obtain the existence of positive solutions, but also establish iterative schemes for approximating the solutions.     

Projected iterative algorithms for complex symmetric systems arising in magnetized multicomponent transport

We investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport. The matrices of the corresponding linear systems are symmetric with a positive semi-definite real part and an imaginary part with a compatible nullspace. We discuss well posedness, the symmetry of generalized inverses and Cholesky methods. We investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field.     

A gradient based iterative algorithm for solving model updating problems of gyroscopic systems

Model updating should be correlated with experimental data to ensure that it models the dynamics of the real structure and the updated model predicts the dynamics of a structure more accurately. Considering that the iterative methods for model updating have aroused little public attention, this paper studies an iterative algorithm for quadratic model updating problems which can incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the mass, gyroscopic and stiffness matrices that closely match the experimental modal data. By this method, the best approximation symmetric and skew-symmetric solutions can be obtained by choosing a convergence factor. Numerical example shows that the introduced iterative algorithm is quite efficient.     

一般Lidstone边值问题的n个正解的存在性

考察了含所有偶数阶导数的一般Lidstone边值问题的正解和对称正解.通过 选择合适的Banach空间和锥,对该问题建立了n个正解或者对称正解的存在性,其 中n是一个任意的自然数.基本工具是等价范数和锥拉伸与锥压缩型的Krasnosel'skii 不动点定理. 结论的主要条件是局部的.换言之,如果非线性项f在某些有界集上的 "高度"是适当的,则该问题可以具有n个正解.     

Existence and uniqueness of elliptic periodic solutions of the Brillouin electron beam focusing system

We find conditions for the existence of an elliptic periodic solution of a singular equation that governs the motion of a magnetically focused axially symmetric electron beam with Brillouin flow by using a monotone iterative scheme starting from a couple of upper and lower solutions on the reversed order. Also, a uniqueness result is proved by using Brouwer degree and index of solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

无穷区间上一类分数阶微分方程正解的显式迭代序列

By applying the monotone iterative method,this study develops two explicit monotone iterative sequences for approximating the minimal and maximal positive solutions.At the same time,by applying the Banach fixed-point theory,an explicit iterative sequence and error estimate for approximating the unique positive solution is obtained.Some examples are given to illustrate the application of the results.     

Banach空间半直线上非线性二阶微分方程组正解的存在性

In this paper, the cone theory and MSnch fixed point theorem combined with the monotone iterative technique are used to investigate the positive solutions for a class of systems of nonlinear singular differential equations with multi-point boundary value conditions on the half line in a Banach space. The conditions for the existence of positive solutions are formulated. In addition, an explicit iterative approximation of the solution is also derived.     

Finite iterative algorithm for the symmetric periodic least squares solutions of a class of periodic Sylvester matrix equations

Numerical Algorithms - The present work proposes a finite iterative algorithm to find the least squares solutions of periodic matrix equations over symmetric ξ-periodic matrices. By this...     

Infinite Element Approximation to Axial Symmetric Stokes Flow

We considered in [1] the finite element approximation to axial symmetric Stokes flow in a bounded domain. The problem for the flow passing an obstacle in an unbounded domain is also frequently encountered. In this paper, we are going to give approximate solutions for this problem by an approach stated in [2]. An iterative method is used to calculate the combined stiffness matrix.     

Symmetric patterns in the Dirichlet problem for a two-cell cubic autocatalytor reaction model

By using an approach developed by one of the authors, approximate solutions of the soft periodic boundary conditions for a two-cell reaction diffusion model have been obtained. The system is considered with reactant A and autocatalyst B. The reaction is taken cubic in the autocatalyst in the two-cell with linear exchange through A. The formal exact solution is obtained which is symmetric with respect to the mid-point of the container. Approximate solutions are found through the Picard iterative sequence of solutions constructed after the exact one. It is found that the solution obtained is not unique. When the initial conditions are periodic, the most dominant modes initiate to traveling waves in systems with moderate size. Symmetric configurations forming a parabolic one for large time are observed. In systems of large size, spatially symmetric chaos are produced which are stationary in time. Furthermore, it is found the symmetric pattern formation hold irrespective of the condition of linear instability against small spatial disturbance.     

广义耦合Sylvester矩阵方程的对称-反对称最小二乘解

本文主要研究极小残差问题‖(A1XB1+C1YD1A2XB2+C2YD2)-(M1M2)‖=min关于X对称-Y反对称解的迭代算法.本文首先给出等价于极小残差问题的规范方程,然后,提出求解此规范方程的对称-反对称解的迭代算法.在不考虑舍入误差的情况下,任取一个初始的对称-反对称矩阵对(X0,Y0),该算法都可以在有限步内求得该极小残差问题的对称-反对称解.最后讨论该问题的极小范数对称-反对称解.     

Existence and iteration of positive solutions for high-order fractional differential equations with integral conditions on a half-line

In this paper, existence and iteration of positive solutions for a class of high-order fractional differential equations with integral conditions on a half-line is investigated. By means of a monotone iterative technique, we obtain not only the existence of positive solutions for the problems, but also establish iterative schemes for approximating the solutions.     

Tangential frequency filtering decompositions for symmetric matrices

Summary. The tangential frequency filtering decomposition (TFFD) is introduced. The convergence theory of an iterative scheme based on the TFFD for symmetric matrices is the focus of this paper. The existence of the TFFD and the convergence of the induced iterative algorithm is shown for symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are proven for a smaller class of matrices. Using this framework, the convergence independent of the number of unknowns is shown for Wittum's frequency filtering decomposition. Some characteristic properties of the TFFD are illustrated and results of several numerical experiments are presented. Received April 1, 1996 / Revised version July 4, 1996