Qualitative analysis of delay differential equations with non-instantaneous impulses: existence and controllability

Shengke Xu; Yinuo Wang; Kaibo Shi; Tao Liu; Shengke Xu; Yinuo Wang; Kaibo Shi; Tao Liu

doi:10.3934/math.2026620

AIMS Mathematics

2026, Volume 11, Issue 5: 15056-15073. doi: 10.3934/math.2026620

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Research article

Qualitative analysis of delay differential equations with non-instantaneous impulses: existence and controllability

1.
College of Electronic Information and Electrical Engineering, Chengdu University, Sichuan 610106, China
2.
Chengdu Jinye Technology Co., Ltd, Sichuan 610106, China

Received: 18 March 2026 Revised: 28 April 2026 Accepted: 07 May 2026 Published: 29 May 2026
MSC : 34A12, 34A37, 34B37

In this paper, we mainly studied the non-instantaneous impulsive evolution equations with state-dependent delay under integral boundary conditions, and the evolution process of the non-instantaneous impulsive system was given. First, sufficient conditions were presented for the problem to have at least one solution when the related semigroup is compact. Second, the case when the associated semigroup is equicontinuous was presented by utilizing the theory of a semigroup. Third, the controllability of the considered problem was acquired via fixed point theory. In the end, examples were given to demonstrate the validity of the obtained results.
- non-instantaneous impulsive equations,
- state-dependent delay,
- integral boundary conditions,
- fixed point theorem,
- existence result,
- controllability
Citation: Shengke Xu, Yinuo Wang, Kaibo Shi, Tao Liu. Qualitative analysis of delay differential equations with non-instantaneous impulses: existence and controllability[J]. AIMS Mathematics, 2026, 11(5): 15056-15073. doi: 10.3934/math.2026620

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Abstract

In this paper, we mainly studied the non-instantaneous impulsive evolution equations with state-dependent delay under integral boundary conditions, and the evolution process of the non-instantaneous impulsive system was given. First, sufficient conditions were presented for the problem to have at least one solution when the related semigroup is compact. Second, the case when the associated semigroup is equicontinuous was presented by utilizing the theory of a semigroup. Third, the controllability of the considered problem was acquired via fixed point theory. In the end, examples were given to demonstrate the validity of the obtained results.

References

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