Oscillation and global attractivity in a discrete survival red blood cells model

I. Kubiaczyk¹ ; S. H. Saker²

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland
² Department of Mathematics Faculty of Science Mansoura University Mansoura, 35516, Egypt

Applicationes Mathematicae, Tome 30 (2003) no. 4, pp. 441-449

Résumé

We consider the discrete survival red blood cells model \begin{equation*} N_{n+1}-N_{n}=-\delta _{n}N_{n}+P_{n}e^{-aN_{n-k}}, \tag{$\ast$}\end{equation*} where $\delta _{n}$ and $P_{n}$ are positive sequences. In the autonomous case we show that $(\ast )$ has a unique positive steady state $N^{\ast }$, we establish some sufficient conditions for oscillation of all positive solutions about $N^{\ast }$, and when $k=1$ we give a sufficient condition for $N^{\ast }$ to be globally asymptotically stable. In the nonatonomous case, assuming that there exists a positive solution $\{ N_{n}^{\ast }\} ,$ we present necessary and sufficient conditions for oscillation of all positive solutions of $(\ast )$ about $\{ N_{n}^{\ast }\} $. Our results can be considered as discrete analogues of the recent results by Saker and Agarwal [12] and solve an open problem posed by Kocic and Ladas [8].

Détail

DOI : 10.4064/am30-4-6

Keywords: consider discrete survival red blood cells model begin equation* n delta an n k tag ast end equation* where delta positive sequences autonomous ast has unique positive steady state ast establish sufficient conditions oscillation positive solutions about ast sufficient condition ast globally asymptotically stable nonatonomous assuming there exists positive solution ast present necessary sufficient conditions oscillation positive solutions ast about ast results considered discrete analogues recent results saker agarwal solve problem posed kocic ladas