A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback...

The asymptotic behaviour for solutions of a difference equation $z_n=f(n,z_n)$, where the complex-valued function $f(n,z)$ is in some meaning close to a holomorphic function $h$, and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right-hand sides.

In the paper we consider the difference equation of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,n\in \mathbb{N}\left(n_0\right),$$ where $p,q:\mathbb{N}\left(n_0\right)\to {ℝ}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$$\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda }^{*}<1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l,$$ and $$1<{\lambda }_{*}\le p\left(n\right),\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l,$$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty$ with a weaker assumption on $q$ than the...

Asymptotic properties of solutions of the difference equation of the form $${\Delta }^m{x}_n=a_n\varphi (x_{{\tau }_1\left(n\right)},\cdots ,x_{{\tau }_k\left(n\right)})+b_n$$ are studied. Conditions under which every (every bounded) solution of the equation ${\Delta }^m{y}_n=b_n$ is asymptotically equivalent to some solution of the above equation are obtained.

Asymptotic properties of solutions of difference equation of the form $${\Delta }^m{x}_n=a_n{\varphi }_n\left(x_{\sigma \left(n\right)}\right)+b_n$$ are studied. Conditions under which every (every bounded) solution of the equation ${\Delta }^m{y}_n=b_n$ is asymptotically equivalent to some solution of the above equation are obtained. Moreover, the conditions under which every polynomial sequence of degree less than $m$ is asymptotically equivalent to some solution of the equation and every solution is asymptotically polynomial are obtained. The consequences of the existence of asymptotically...