AIMS Mathematics, 2020, 5(6): 6189-6210. doi: 10.3934/math.2020398.

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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

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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.
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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. AIMS Mathematics, 2020, 5(6): 6189-6210. doi: 10.3934/math.2020398

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