On the continuity of the function describing the times of meeting impulsive set and its application

Sanyi Tang; Wenhong Pang; Sanyi Tang; Wenhong Pang

doi:10.3934/mbe.2017072

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1399-1406. doi: 10.3934/mbe.2017072

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On the continuity of the function describing the times of meeting impulsive set and its application

Sanyi Tang ^,,
Wenhong Pang

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China

Received: 25 April 2016 Accepted: 19 September 2016 Published: 01 October 2017
MSC : Primary: 34A37; Secondary: 47N60

The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.
- Impulsive semi-dynamical system,
- Poincaré map,
- continuity,
- state-dependent feedback control,
- counter example
Citation: Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1399-1406. doi: 10.3934/mbe.2017072

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Abstract

The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.

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