This paper investigates positivity and asymptotic stability for a class of timescale-type differential–difference equation with time-varying delay. When the delay is bounded, a necessary and sufficient condition is obtained based on timescale theory and comparison principle. While for the case of unbounded delay, a sufficient criterion is acquired. Further, for the corresponding continuous-time system with unbounded delay, a necessary and sufficient condition is secured under an extra constraint. This work relaxes a previous assumption on the unbounded time-varying delay and also generalizes and deepens some existing results.

中文翻译：

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时间尺度上具有时变延迟的耦合微分-差分方程的正性和稳定性

本文研究了一类具有时变延迟的时标型微分差分方程的正性和渐近稳定性。当延迟有界时，根据时间尺度理论和比较原理得到一个充要条件。而对于无界时延的情况，则获得了充分的判据。进一步地，对于相应的无界时延连续时间系统，在额外约束下保证了一个充分必要条件。这项工作放宽了先前对无界时变延迟的假设，并概括和深化了一些现有结果。