Multi-parameter fourth order impulsive integral boundary value problems with one-dimensional m-Laplacian and deviating arguments

Meiqiang Feng,Jingliang Qiu

doi:10.1186/s13660-015-0587-6

Abstract

Using inequality techniques and fixed point theories, several new and more general existence and multiplicity results are derived in terms of different values of \(\lambda>0\) and \(\mu>0\) for a fourth order impulsive integral boundary value problem with one-dimensional m-Laplacian and deviating arguments. We discuss our problems under two cases when the deviating arguments are delayed and advanced. Moreover, the nonexistence of a positive solution is also studied. In this paper, our results cover fourth order boundary value problems without deviating arguments and impulsive effect and are compared with some recent results by Jankowski.

Highlights

1 Introduction Functional differential equations with impulse effect occur in many applications, such as population dynamics, biology, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc., and can be expressed by functional differential equations with impulses, see [ – ]
Functional differential equations with impulses are characterized by sudden changing of their state and by the fact that the processes under consideration depend on their prehistory at each moment of time
We investigate a fourth order impulsive integral boundary value problem with one-dimensional m-Laplacian and deviating arguments

Summary

Introduction

Functional differential equations with impulse effect occur in many applications, such as population dynamics, biology, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc., and can be expressed by functional differential equations with impulses, see [ – ]. In [ ], Zhang and Liu studied the following fourth order four-point boundary value problem without impulsive effect: By using the upper and lower solution method, fixed point theorems and the properties of Green’s function G(t, s) and H(t, s), the authors give sufficient conditions for the existence of one positive solution. To the best of our knowledge, no paper has considered the existence, multiplicity and nonexistence of positive solutions for fourth order impulsive differential equations with one-dimensional m-Laplacian, multiple parameters and deviating arguments till ; for example, see [ – ] and the references therein.

Results

Conclusion