Forced oscillation of certain fractional differential equations

Da-Xue Chen,Yong-Hong Lan,Pei-Xin Qu

doi:10.1186/1687-1847-2013-125

Abstract

The paper deals with the forced oscillation of the fractional differential equation (D qx)(t )+ f1(t, x(t)) = v(t )+ f2(t, x(t)) for t > a ≥ 0 with the initial conditions (D q–k a x)(a )= bk (k =1 , 2, ... , m –1 ) and limt→a+(I m–q a x)(t )= bm, where D qx is the Riemann-Liouville fractional derivative of order q of x, m –1< q ≤ m, m ≥ 1 is an integer, I m–q a x is the Riemann-Liouville fractional integral of order m – q of x ,a ndbk (k =1 , 2, ... , m) are/is constants/constant. We obtain some oscillation theorems for the equation by reducing the fractional differential equation to the equivalent Volterra fractional integral equation and by applying Young’s inequality. We also establish some new oscillation criteria for the equation when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator. The results obtained here improve and extend some existing results. An example is given to illustrate our theoretical results. MSC: 34A08; 34C10

Highlights

The results are stated when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator
We get some new oscillatory properties of ( . ) when the Riemann-Liouville fractional operator is replaced by the Caputo fractional operator
4 Results with the Caputo fractional derivative The Riemann-Liouville fractional derivatives played an important role in the development of the theory of fractional derivatives and integrals and for their applications in pure mathematics

Summary

Introduction

1 Introduction The aim of the paper is to establish several oscillation theorems for forced fractional differential equation with initial conditions of the form (Dqax)(t) + f (t, x(t)) = v(t) + f (t, x(t)), t > a ≥ , (Dqa–kx)(a) = bk There has been a significant development in ordinary and partial differential equations involving both Riemann-Liouville and Caputo fractional derivatives in recent years. Chen [ ] established some oscillation criteria for the fractional differential equation r(t) Dq–x η(t) – p(t)f (s – t)–qx(s) ds = for t > , t where q ∈ ( , ) is a constant, η > is a quotient of odd positive integers, Dq–x is the Liouville right-sided fractional derivative of order q of x defined by

Objectives

Results

Conclusion