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

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 solutio...

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Main Authors:	Zihan Li, Xiao-Bao Shu, Fei Xu
Format:	Article
Language:	English
Published:	AIMS Press 2020-08-01
Series:	AIMS Mathematics
Subjects:	random impulse differential equation upper and lower solutions monotonic iterative sequences boundary value problem
Online Access:	https://www.aimspress.com/article/10.3934/math.2020398/fulltext.html

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spelling	doaj-f3216c7a0b1a4a64a74410ff4b323fbe2020-11-25T03:10:01ZengAIMS PressAIMS Mathematics2473-69882020-08-01566189621010.3934/math.2020398The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problemZihan Li0Xiao-Bao Shu1Fei Xu21 College of Mathematics, Hunan University, Changsha, Hunan 410082, PR China1 College of Mathematics, Hunan University, Changsha, Hunan 410082, PR China2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, CanadaIn 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.https://www.aimspress.com/article/10.3934/math.2020398/fulltext.htmlrandom impulse differential equationupper and lower solutionsmonotonic iterative sequencesboundary value problem
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Zihan Li Xiao-Bao Shu Fei Xu
spellingShingle	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 random impulse differential equation upper and lower solutions monotonic iterative sequences boundary value problem
author_facet	Zihan Li Xiao-Bao Shu Fei Xu
author_sort	Zihan Li
title	The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem
title_short	The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem
title_full	The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem
title_fullStr	The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem
title_full_unstemmed	The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem
title_sort	existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem
publisher	AIMS Press
series	AIMS Mathematics
issn	2473-6988
publishDate	2020-08-01
description	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.
topic	random impulse differential equation upper and lower solutions monotonic iterative sequences boundary value problem
url	https://www.aimspress.com/article/10.3934/math.2020398/fulltext.html
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The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem

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