SYMMETRIC POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATIONS (AX,XB)=(C,D) AND AXB=C

The symmetric positive definite solutions of matrix equations (AX,XB)=(C,D) and AXB=C are considered in this paper. Necessary and sufficient conditions for the matrix equations to have symmetric positive definite solutions are derived using the singular value and the generalized singular value decompositions. The expressions for the general symmetric positive definite solutions are given when certain conditions hold.

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.

Symmetric Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem

In this paper, we consider the following second order three-point boundary value problem u″（t）＋a（t）f（u（t））=0,0〈t〈1,u（0）-u（1）=0,u＇（0）-u＇（1）=u（1/2）,where a ： （0, 1） → [0, ∞） is symmetric on （0, 1） and may be singular at t = 0 and t = 1, f ： [0, ∞） → [O, ∞） is continuous. By using Krasnoselskii＇s fixed point theorem ia a cone, we get some existence results of positive solutions for the problem. The associated Green＇s function for the three-point boundary value problem is also given.

Three Symmetric Positive Solutions for Second-order Nonlocal Boundary Value Problems

Using the Leggett-Williams fixed point theorem, we will obtain at least three symmetric positive solutions to the second-order nonlocal boundary value problem of the form u″（t）＋g（t）f（t,u（t））=0,0〈t〈1,u（0）=u（1）=∫01m（s）u（s）ds. where m ∈ L1[0 1], g ： （0, 1）→ [0, ∞） is continuous, symmetric on （0, 1） and maybe singular at t = 0 and t = 1, f： [0, 1] × [0, ∞） → [0, ∞） is continuous and f（-, x） is symmetric on [0, 1] for all x∈ [0, ∞）.

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.

ON EIGENVALUE BOUNDS AND ITERATION METHODS FOR DISCRETE ALGEBRAIC RICCATI EQUATIONS

We derive new and tight bounds about the eigenvalues and certain sums of the eigenvalues for the unique symmetric positive definite solutions of the discrete algebraic Riccati equations. These bounds considerably improve the existing ones and treat the cases that have been not discussed in the literature. Besides, they also result in completions for the available bounds about the extremal eigenvalues and the traces of the solutions of the discrete algebraic Riccati equations. We study the fixed-point iteration methods for com- puting the symmetric positive definite solutions of the discrete algebraic Riccati equations and establish their general convergence theory. By making use of the Schulz iteration to partially avoid computing the matrix inversions, we present effective variants of the fixed-point iterations, prove their monotone convergence and estimate their asymptotic convergence rates. Numerical results show that the modified fixed-point iteration methods are feasible and effective solvers for computing the symmetric positive definite solutions of the discrete algebraic Riccati equations.

DIRICHLET BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER QUASI-LINEAR DIFFERENTIAL EQUATIONS WITH CHANGING SIGN NONLINEARITIES

This paper is concerned with the existence of positive solutions of two-point Dirichlet singular and nonsingular boundary problems for second-order quasi-linear differential equations with changing sign nonlinearities.