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Inertias and ranks of some Hermitian matrix functions with applications

Xiang ZhangQing-Wen WangXin Liu — 2012

Open Mathematics

Let S be a given set consisting of some Hermitian matrices with the same size. We say that a matrix A ∈ S is maximal if A − W is positive semidefinite for every matrix W ∈ S. In this paper, we consider the maximal and minimal inertias and ranks of the Hermitian matrix function f(X,Y) = P − QXQ* − TYT*, where * means the conjugate and transpose of a matrix, P = P*, Q, T are known matrices and for X and Y Hermitian solutions to the consistent matrix equations AX =B and YC = D respectively. As applications,...

Positive periodic solution for ratio-dependent n -species discrete time system

Mei-Lan TangXin-Ge Liu — 2011

Applications of Mathematics

In this paper, sharp a priori estimate of the periodic solutions is obtained for the discrete analogue of the continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modelling the dynamics of the n - 1 competing preys and one predator having nonoverlapping generations. Based on more precise a priori estimate and the continuation theorem of the coincidence degree, an easily verifiable sufficient criterion of the existence of positive periodic solutions...

The zero distribution and uniqueness of difference-differential polynomials

Kai LiuXin-Ling LiuLian-Zhong Yang — 2013

Annales Polonici Mathematici

We consider the zero distribution of difference-differential polynomials of meromorphic functions and present some results which can be seen as the discrete analogues of the Hayman conjecture. In addition, we also investigate the uniqueness of difference-differential polynomials of entire functions sharing one common value. Our theorems improve some results of Luo and Lin [J. Math. Anal. Appl. 377 (2011), 441-449] and Liu, Liu and Cao [Appl. Math. J. Chinese Univ. 27 (2012), 94-104].

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