The existence of solutions for Sturm\u2013Liouville differential equation with random impulses and boundary value problems

Zihan Li,Tengyuan Miao,Xiao-Bao Shu

doi:10.1186/s13661-021-01574-x

Abstract

In this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

Highlights

1 Introduction Impulsive dynamical systems are an emerging field drawing attention from both theoretical and applied disciplines. They are often typically described by ordinary differential equations with instantaneous state jumps [24, 29]
The impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states
Taking into account the stochastic effects, the models are better described as random impulsive differential equations (RIDEs) rather than impulsive differential equations or stochastic differential equations

Summary

Introduction

Impulsive dynamical systems are an emerging field drawing attention from both theoretical and applied disciplines. Hua, Cong, and Cheng [8] studied equation (1.2) in 2012, which is the existence and uniqueness of solutions for the periodic-integrable boundary value problem of second order differential equations. There are fewer people who studied the boundary value problem of Sturm–Liouville type differential equations with random impulses and the upper and lower solutions of this kind of equation. 3, we use Dhage’s fixed point theorem to study the existence of the solutions of equation (2.1), and the existence of solutions of general second order nonlinear random impulsive differential equations with boundary value problems is given. 4, we use the upper and lower solution method to give the monotonic iterative convergent sequence of the generalized Sturm–Liouville differential equation with random impulses.

Then we define the matrix Q as

And we can easily get