Exponential estimates of solutions of difference equations with continuous time

H. Péics, University of Novi Sad, Subotica, Serbia and Montenegro

E. J. Qualitative Theory of Diff. Equ., Proc. 7'th Coll. Qualitative Theory of Diff. Equ., No. 17. (2004), pp. 1-12.

Abstract: In this paper we study the scalar difference equation with continuous time of the form
\[x(t)=a(t)x(t-1)+b(t)x(p(t)),\]
where $a,b:[t_0,\infty)\to {\bf R}$ are given real functions for $t_0>0$ and $p:[t_0,\infty)\to {\bf R}$ is a given function such that $p(t)\le t$, $\lim_{t\to\infty}p(t)=\infty$. Using the method of characteristic equation we obtain an exponential estimate of solutions of this equation which can be applied to the difference equations with constant delay and to the case $t-p_2\le p(t)\le t-p_1$ for real numbers $1<p_1\le p_2$.
We generalize the main result to the equation with several delays of the form
\[x(t)=a(t)x(t-1)+\sum_{i=1}^mb_i(t)x(p_i(t)),\]
where $a,b_i:[t_0,\infty)\to {\bf R}$ are given functions for $i=1,2,...,m$, and $p_i:[t_0,\infty)\to {\bf R}$ are given such that $p_i(t)\le t$, $\lim_{t\to\infty}p_i(t)=\infty$ for $i=1,2,...,m$.
We apply the obtained result to particular cases such as $p_i(t)=t-p_i$ for $i=1,2,...,m$, where $1\le p_1<p_2<...<p_m$ are real numbers.

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