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.     

关于线性正算子逼近的若干结果

In this paper we consider the degree of approximation of continuous function and intergrable function on the space with dimension m≥2 by the sequences of positive linear operators. Besides, we prove the Kolovkin type theorem for point-wise convergence of the sequences of positive linear operators.     

OSCILLATION AND GLOBAL ATTRACTIVITY OF IMPULSIVE PERIODIC DELAY RESPIRATORY DYNAMICS MODEL

This paper studies the nonlinear delay impulsive respiratory dynamics model. The model describes the sudden changes of the concentration of CO2 in the blood of the mammal. It is proved that the model has a unique positive periodic solution. Some sufficient conditions for oscillation of all positive solutions about the positive periodic solution are established and also some sufficient conditions for the global attractivity of the periodic solution are obtained.     

带$p$-Laplacian的四阶四点奇异边值问题的正解

In this paper, we consider the existence of positive solutions for the singular fourthorder four point boundary value problem with p-Laplacian operator. By using the fixed point theorem of cone expansion and compression, the existence of multiple positive solutions is obtained.     

NON-CONSTANT POSITIVE STEADY-STATES OF A PREDATOR-PREY-MUTUALIST MODEL

In this paper, the authors deal with the non-constant positive steady-states of a predator-prey-mutualist model with homogeneous Neumann boundary condition. They first give a priori estimates (positive upper and lower bounds) of positive steady-states, and then study the non-existence, the global existence and bifurcation of non-constant positive steady-states as some parameters are varied. Finally the asymptotic behavior of such solutions as d3→∞ is discussed.     

Existence and uniqueness of positive solutions of semilinear elliptic equations

This paper is devoted to the study of existence,uniqueness and non-degeneracy of positive solutions of semi-linear elliptic equations.A necessary and sufficient condition for the existence of positive solutions to problems is given.We prove that if the uniqueness and non-degeneracy results are valid for positive solutions of a class of semi-linear elliptic equations,then they are still valid when one perturbs the differential operator a little bit.As consequences,some uniqueness results of positive solutions under the domain perturbation are also obtained.     

奇异微分方程的广义Sturm--Liouville边值问题正解的存在性

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of generalized Sturm-Liouville boundary value problems for a nonlinear singular differential equation with a parameter. Some sufficient conditions for the existence of positive solutions are established. In the last section, an example is presented to illustrate the applications of our main results.     

Asymptotic Behavior on a Competition Model Arising from an Unstirred Chemostat

This paper is concerned with the unstirred chemostat model with two-species and one non-reproducingresource.The global attractivity of the positive steady-state solutions of the original system is established.Moreover,the effects of the growth rate on the unique positive equilibrium of the single population model arestudied.     

Positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional response.The main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional response.Here,a positive solution corresponds to a coexistence state of the model.Firstly,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter plane.In addition,we present the uniqueness of positive solutions in one dimension case.Secondly,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.     

A CONDITION OF THE EXISTENCE OF STABLE POSITIVE STEADY-STATE SOLUTIONS FOR A ONE PREDATOR TWO PREY SYSTEM

One predator two prey system is a research topic which has both the theoretical and practical values.This paper provides a natural condition of the existence of stable pcsitive steady-state solutions for the one predator two prey system.Under this conditon we study the existence of the positive steady-state solutions at vicinity of the triple eigenvalue by implicit function theorem,discuss the positive stable solution problem bifureated from the semi-trivial solutions containing two positive components with the help of bifurcation and perturbation methods.     

Nonsingularity,positive definiteness,and positive invertibility under fixed-point data rounding

For a real square matrix A and an integer d ? 0, let A _(d) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A _(d) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number α(d), computed solely from A _(d) (not from A), such that the following alternative holdsif d > α(d), then nonsingularity (positive definiteness, positive invertibility) of A _(d) implies the same property for A if d < α(d) and A _(d) is nonsingular (positive definite, positive invertible), then there exists a matrix A′ with A′_(d) = A _(d) which does not have the respective property.For nonsingularity and positive definiteness the formula for α(d) is the same and involves computation of the NP-hard norm ‖ · ‖_∞,1; for positive invertibility α(d) is given by an easily computable formula.     

A framework for positive dependence

This paper presents, for bivariate distributions, a unified framework for studying and relating three basic concepts of positive dependence. These three concepts are positive dependence orderings, positive dependence properties and measures of positive dependence. The latter two concepts are formally defined and their properties discussed. Interrelationships among these three concepts are given, and numerous examples are presented.Supported by the National Science Foundation under Grant DMS-8301361.Supported by the Air Force Office of Scientific Research under Contract 84-0113. Reproduction in whole or part is permitted for any purpose of the United States Government.     

COMPARISON CRITERIA OF POSITIVESOLUTIONS FOR A NEUTRAL DIFFERENCEEQUATION WITH POSITIVE AND NEGATIVE COEFFICIENTS

1.IntroductionInthispaper,weareconcernedwithaclassofneutraltypeflorenceequationswithpositiveandnegativecoefficielltsoftheformwhere',TandaarepositiveintegerssuchthatT2a){r'}:,isarealsequence3{p'}:,and{qn}Zoarenonnegativesequences.Similarequationshaver...     

Conditionally positive definite dot product kernels

This paper deals with conditionally positive definite kernels on Euclidean spaces. The focus here is on dot product kernels, that is, those depending on the inner product of the variables. Among the results, we include some properties relating conditional positive definiteness and standard convolution in the line and also results related to the characterization of the conditionally positive definite dot product kernels with respect to finite-dimensional polynomial spaces. We also introduce and characterize two large classes of strictly conditionally positive definite dot product kernels.     

子空间上的对称正定及对称半正定阵的左右特征值反问题

文[1][2][3]中讨论AX＝B的对称阵逆特征值问题，文[4][5][6]中讨论了半正定阵的逆特征值问题。本文讨论了空间了子空间上的对称正定及对称半正定阵的左右特征值反问题，给出了解存在的充分条件及解的表达式。     

On totally positive sequences and functions

Let a>0 be a fixed number. A function f:R→R is said to be a-shift-generating (a-SG) if for every x∈R, is a totally positive sequence and it does not coincide with a sequence of the form , where A?0 and λ>0. In this paper, we describe all a-SG functions and obtain a new characterization of totally positive functions in the terms of a-SG functions. In addition, using characteristic properties of a-SG functions, we generalize the famous Jacobian identity in theory of elliptic functions.     

Positivity and positive definiteness in generalized function algebras

The definitions of positivity and positive definiteness are extended to generalized function algebras in coherence with the corresponding notions for distributions. Versions of Bochner's theorem for a positive definite Colombeau generalized function are given.     

Irreducible positive linear maps on operator algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given.

     

矩阵正定性的进一步推广

矩阵的正定性在很多领域中都有广泛的应用,其定义得到了一系列的推广.进一步推广了矩阵的正定性,给出了更广义的正定矩阵的定义,并得到了它的若干性质.     

Robust stability of positive linear systems in Banach spaces

In this paper, we study the stability radii of positive linear systems with delays with respect to various classes of perturbations in infinite dimensional spaces. It is shown that the positive, real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by a simple example.