Positive Green’s function and triple positive solutions of a second-order impulsive differential equation with integral boundary conditions and a delayed argument

Gaoli Lu,Meiqiang Feng

doi:10.1186/s13661-016-0595-6

Abstract

In this paper, we first establish the expression of positive Green’s function for a second-order impulsive differential equation with integral boundary conditions and a delayed argument. Furthermore, applying Legget-William’s fixed point theorem and Holder’s inequality, we obtain the existence results of at least three positive solutions under three cases: $p=1$ , $1< p<+\infty$ , and $p=+\infty$ . We discuss our problem with impulsive effects and a delayed argument. In this case, our results cover second-order boundary value problems without impulsive effects and delayed arguments and are compared with some recent results. Finally, we give an example to illustrate our main results.

Highlights

1 Introduction Functional differential equations with impulses are characterized by the fact that per sudden changing of their state the processes under consideration depend on their prehistory at each moment of time
They are used in many models of optimal control, physics, chemical technology, population dynamics, biology, biotechnology, industrial robotic, pharmacokinetics, etc. [ – ]
We investigate the existence of three positive solutions for a second-order boundary value problem with impulsive effects and a delayed argument of the form

Summary

Introduction

Functional differential equations with impulses are characterized by the fact that per sudden changing of their state the processes under consideration depend on their prehistory at each moment of time. To the best of our knowledge, there are almost no papers on the existence of three positive solutions for second-order impulsive differential equations with integral boundary conditions and a delayed argument, especially for Lp-integrable ω; for example, see [ , ] and the references therein. We investigate the existence of three positive solutions for a second-order boundary value problem with impulsive effects and a delayed argument of the form. The arguments are based upon a fixed point theorem due to Leggett and Williams, which deals with fixed points of a cone-preserving operator defined on an ordered Banach space Another contribution of this paper is to study the expression and properties of Green’s function associated with problem In Section , we use Leggett-Williams’ fixed point theorem to obtain the existence of three positive solutions for problem

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