Positive solutions for impulsive fractional differential equations with generalizedperiodic boundary value conditions

Kaihong Zhao,Ping Gong

doi:10.1186/1687-1847-2014-255

Abstract

By constructing Green’s function, we give the natural formulae of solutions forthe following nonlinear impulsive fractional differential equation with generalizedperiodic boundary value conditions: where is a real number, is the standard Caputo differentiation. We present theproperties of Green’s function. Some sufficient conditions for the existence ofsingle or multiple positive solutions of the above nonlinear fractional differentialequation are established. Our analysis relies on a nonlinear alternative of theSchauder and Guo-Krasnosel’skii fixed point theorem on cones. As applications,some interesting examples are provided to illustrate the main results. MSC: 34B10, 34B15, 34B37.

Highlights

In recent years, the fractional order differential equation has aroused great attention due to both the further development of fractional order calculus theory and the important applications for the theory of fractional order calculus in the fields of science and engineering such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth
In order to describe the dynamics of populations subject to abrupt changes as well as other phenomena such as harvesting, diseases, and so on, some authors have used an impulsive differential system to describe these kinds of phenomena since the last century
For the basic theory on impulsive differential equations, the reader can refer to the monographs of Bainov and Simeonov [ ], Lakshmikantham et al [ ] and Benchohra et al [ ]

Summary

Introduction

The fractional order differential equation has aroused great attention due to both the further development of fractional order calculus theory and the important applications for the theory of fractional order calculus in the fields of science and engineering such as physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth. We consider the following nonlinear impulsive fractional differential equation with generalized periodic boundary value conditions In Section , we present some useful definitions, lemmas and the properties of Green’s function. In Section , we give some sufficient conditions for the existence of a single positive solution for BVPs (see [ , ]) The Caputo fractional derivative of order α > of a continuous function f : ( , +∞) → R is given by cDα +f (t) =. Let < b < a < +∞, Green’s functions G (t, s), G (t, ti) and G (t, ti) defined by

It is obvious that

Let a m am

Take c