The method of upper and lower solutions to impulsive differential equations with integral boundary conditions

Huihui Pang,Meng Lu,Chen Cai

doi:10.1186/1687-1847-2014-183

Abstract

This paper considers a second-order impulsive differential equation with integral boundary conditions. Some sufficient conditions for the existence of solutions are proposed by using the method of upper and lower solutions and Leray-Schauder degree theory.

Highlights

1 Introduction The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments
The theory of impulsive differential equations has become an important area of investigation in recent years, and it is much richer than the corresponding theory of differential equations
The theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics

Summary

Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. Some well-known works, such as Hao et al [ ], Zhang et al [ ] and Ding and Wang [ ], deal with impulsive differential equations with integral boundary conditions. Most of these results are obtained by using the fixed point theorem in cones. In [ ], Shen and Wang applied the method of upper and lower solutions to solve impulsive differential equations with nonlinear boundary conditions as follows:.

Preliminaries

Existence results