The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem

Zihan Li; Xiao-Bao Shu; Fei Xu; Zihan Li; Xiao-Bao Shu; Fei Xu

doi:10.3934/math.2020398

AIMS Mathematics

2020, Volume 5, Issue 6: 6189-6210. doi: 10.3934/math.2020398

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Research article

The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem

1.
College of Mathematics, Hunan University, Changsha, Hunan 410082, PR China
2.
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

Received: 06 May 2020 Accepted: 06 July 2020 Published: 31 July 2020
MSC : 34A37, 34B, 34F05

In this article, we consider the existence of upper and lower solutions to a second-order random impulsive differential equation. We first study the solution form of the corresponding linear impulsive system of the second-order random impulsive differential equation. Based on the form of the solution, we define the resolvent operator. Then, we prove that the fixed point of this operator is the solution to the equation. Finally, we construct the sum of two monotonic iterative sequences and prove that they are convergent. Thus, we conclude that the system has upper and lower solutions.
- random impulse differential equation,
- upper and lower solutions,
- monotonic iterative sequences,
- boundary value problem
Citation: Zihan Li, Xiao-Bao Shu, Fei Xu. The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem[J]. AIMS Mathematics, 2020, 5(6): 6189-6210. doi: 10.3934/math.2020398

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Abstract

In this article, we consider the existence of upper and lower solutions to a second-order random impulsive differential equation. We first study the solution form of the corresponding linear impulsive system of the second-order random impulsive differential equation. Based on the form of the solution, we define the resolvent operator. Then, we prove that the fixed point of this operator is the solution to the equation. Finally, we construct the sum of two monotonic iterative sequences and prove that they are convergent. Thus, we conclude that the system has upper and lower solutions.

References

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