关于非线性矩阵方程$X^s-A^*X^{-t}A=Q$的Hermitian正定解及其扰动估计

In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^＊X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples.     

On the Nonlinear Matrix Equation X+ A *f1（X）A +B*f2（X）B=Q

In this paper, nonlinear matrix equations of the form X + A*f1 (X)A + B*f2 (X)B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.     

ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X＋A^＊X^-2A=Q

Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q, where Q is a square Hermitian positive definite matrix and A＊ is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X＋A^＊X^-2A=Q. At last, we further generalize these results to the nonlinear matrix equation X＋A^＊X^-nA=Q, where n≥2 is a given positive integer.     

OPTIMIZATION APPROACH FOR THE MONGE-AMPèRE EQUATION

In this paper, we introduce and study a method for the numerical solution of the elliptic Monge-Ampère equation with Dirichlet boundary conditions. We formulate the Monge-Ampère equation as an optimization problem. The latter involves a Poisson Problem which is solved by the finite element Galerkin method and the minimum is computed by the conjugate gradient algorithm. We also present some numerical experiments.     

EXISTENCE OF A NONTRIVIAL SOLUTION FOR CHOQUARD'S EQUATION

In this article, the authors consider the existence of a nontrivial solution for the following equation: where q(x) satisfies some conditions. Using a Min-Max method, the authors prove that there is at least one nontrivial solution for the above equation.     

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.     

Logistic型方程周期解和概周期解的存在性

In this paper, we study the Logistic type equation x= a（t）x -b（t）x^2＋ e（t）. Under the assumptions that e（t） is small enough and a（t）, b（t） are contained in some positive intervals, we prove that this equation has a positive bounded solution which is stable. Moreover, this solution is a periodic solution if a（t）, b（t） and e（t） are periodic functions, and this solution is an almost periodic solution if a（t）, b（t） and e（t） are almost periodic functions.     

关于矩阵方程Xs+A*X-qA=I*（英文）

In Theorem 2.4 of the paper[Yang Yueting,The iterative method for solving nonlinear matrix equation X^s + A^*X^-tA = Q,Appl.Math.Comput., 188（2007）46-53],Yang showed that this equation has the maximal solution X_L. In this paper,we point out that Yang’s result is doubtful,study the existence of the Hermitian positive definite solutions,the maximal solution and the minimal solution for the case that Q = I,and give the algorithms for finding these special solutions.     

A NOTE ON COMPARISON THEOREM OF TWO-STAGE ITERATIVE METHOD FOR HERMITIAN POSITIVE DEFINITE LINEAR SYSTEMS

We discuss two-stage iterative methods for the solution of linear system Ax = b, and give a new proof of the comparison theorems of two-stage iterative method for an Hermitian positive definite matrix. Meanwhile, we put forward two new versions of well known comparison theorem and apply them to some examples.     

PERMANENCE AND PERIODIC SOLUTION FOR NONAUTONOMOUS COMPETITION-PREDATOR SYSTEM WITH TYPE II FUNCTIONAL RESPONSE

A nonautonomous multispecies competition-predator system with type Ⅱfunctional response is studied in this paper. It is proved that the system is per-manent under some appropriate conditions. Further, if the system is a periodic one, a set of easily verifiable suflficient conditions which guarantee the existence, uniqueness and global attractivity of a strictly positive periodic solution are ob-tained.     

Positive Definite Solutions of Operator Equations X m + A * X - n A = I

In this article, we discuss the positive definite solutions of the operator equation X m + A * X -n A = I , , on a Hilbert space. Necessary and sufficient conditions for the existence of a positive semi-definite solution are derived. Under some conditions, two effective iterative methods for getting a positive definite solution of this equation are proposed and error estimations are obtained.     

Positive Definite Solutions of Operator Equations X m + A * X − n A = I

In this article, we discuss the positive definite solutions of the operator equation X m + A * X m n A = I , , on a Hilbert space. Necessary and sufficient conditions for the existence of a positive semi-definite solution are derived. Under some conditions, two effective iterative methods for getting a positive definite solution of this equation are proposed and error estimations are obtained.     

一类广义Lyapunov矩阵方程的正定解

研究了双线性系统中的一类广义Lyapunov矩阵方程的正定解.基于混合单调算子不动点定理,给出新的存在正定解的充分条件,构造了求其正定解的不动点迭代方法,并给出了迭代误差估计公式.数值实验表明新方法是可行的.     

Thompson metric method for solving a class of nonlinear matrix equation

Based on the elegant properties of the Thompson metric, we prove that the general nonlinear matrix equation X^q-A^∗F(X)A=Q(q>1) always has a unique positive definite solution. An iterative method is proposed to compute the unique positive definite solution. We show that the iterative method is more effective as q increases. A perturbation bound for the unique positive definite solution is derived in the end.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.     

On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

On the Hermitian positive defnite solution of the nonlinear matrix equation X + A*X −1 A + B*X −1 B = I

In this paper, we study the matrix equation X + A*X ⁻¹ A + B*X ⁻¹ B = I, where A, B are square matrices, and obtain some conditions for the existence of the positive definite solution of this equation. Two iterative algorithms to find the positive definite solution are given. Some numerical results are reported to illustrate the effectiveness of the algorithms. This research supported by the National Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry of Education of China 20060532014.     

求解混合型Lyapunov矩阵方程的参数迭代法

采用参数迭代法求一类混合型Lyapunov矩阵方程A~TX XA B~TXB=C的对称解.在方程相容的条件下,给出了迭代法收敛的充要条件和一些充分条件,以及参数的选取方法.最后,利用数值算例对有关结果进行了验证.     

On the nonlinear matrix equation

Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, we prove that the nonlinear matrix equation always has a unique positive definite solution. A conjecture which is proposed in [X.G. Liu, H. Gao, On the positive definite solutions of the matrix equation X^s±A^TX^-tA=I_n, Linear Algebra Appl. 368 (2003) 83–97] is solved. Multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective.