Recent zbMATH articles in MSC 34A37https://www.zbmath.org/atom/cc/34A372021-04-16T16:22:00+00:00WerkzeugBendixson criterion in impulsive systems.https://www.zbmath.org/1456.340112021-04-16T16:22:00+00:00"Ding, Changming"https://www.zbmath.org/authors/?q=ai:ding.changming"Duan, Zhipeng"https://www.zbmath.org/authors/?q=ai:duan.zhipeng"Pan, Shiyao"https://www.zbmath.org/authors/?q=ai:pan.shiyaoConsider the sufficiently smooth planar vector field \(F(x,y)\). Under the assumption that the divergence of \(F\) has constant sign in some simply connected region \(U\subset\mathbb{R}^2\) and is not identically zero in any subregion of \(U\), then the criterion of Bendixson says that there is no closed orbit of \(F\) entirely located in \(U\). The authors recall basic definitions and properties of planar impulsive systems. They present a simple example showing that the Bendixson criterion does not hold for planar impulsive systems in its original form. Introducing the concept of a periodic orbit of order \(k\), where \(k\) is related to the number of jumps, they prove an extended Bendixson criterion which excludes the existence of periodic orbits of order \(k \leq 2\).
Reviewer: Klaus R. Schneider (Berlin)Forced oscillation of sublinear impulsive differential equations via nonprincipal solution.https://www.zbmath.org/1456.340292021-04-16T16:22:00+00:00"Mostepha, Naceri"https://www.zbmath.org/authors/?q=ai:mostepha.naceri"Ă–zbekler, Abdullah"https://www.zbmath.org/authors/?q=ai:ozbekler.abdullahSummary: In this paper, we give new oscillation criteria for forced sublinear impulsive differential equations of the form
\[
\begin{cases}
(r(t)x^\prime)^\prime + q(t)|x|^{\gamma - 1} x = f(t), &t \neq \theta_i;\\
\Delta r(t) x^\prime + q_i|x|^{\gamma-1}x = f_i, &t= \theta_i,
\end{cases}
\]
where \(\gamma \in (0,1)\), under the assumption that associated homogenous linear equation
\[
\begin{cases}
(r(t)z^\prime)^\prime + q(t)z = 0, &t \neq \theta_i;\\
\Delta r (t)z^\prime + q_i z = 0, &t = \theta_i
\end{cases}
\]
is nonoscillatory.Dynamics and optimal control of a Monod-Haldane predator-prey system with mixed harvesting.https://www.zbmath.org/1456.921222021-04-16T16:22:00+00:00"Liu, Xinxin"https://www.zbmath.org/authors/?q=ai:liu.xinxin"Huang, Qingdao"https://www.zbmath.org/authors/?q=ai:huang.qingdao