Oscillation criteria of fractional differential equations

Chen

Description

In this article, we are concerned with the oscillation of the fractional differential equation r ( t ) D - α y η ( t ) ′ - q ( t ) f ∫ t ∞ ( v - t ) - α y ( v ) d v = 0 for t > 0 , where D - α y is the Liouville right-sided fractional derivative of order α ∈ (0,1) of y and η > 0 is a quotient of odd positive integers. We establish some oscillation criteria for the equation by using a generalized Riccati transformation technique and an inequality. Examples are shown to illustrate our main results. To the best of author's knowledge, nothing is known regarding the oscillatory behavior of the equation, so this article initiates the study. MSC (2010) : 34A08; 34C10.

Sujets

• Oscillation
• Fractional calculus

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 4
• Langue: English

Extrait

ChenAdvances in Difference Equations2012,2012:33 http://www.advancesindifferenceequations.com/content/2012/1/33

R E S E A R C HOpen Access Oscillation criteria of fractional differential equations DaXue Chen

Correspondence: cdx2003@163. com College of Science, Hunan Institute of Engineering, 88 East Fuxing Road, Xiangtan, Hunan 411104, P. R. China

Abstract In this article, we are concerned with the oscillation of the fractional differential equation   ∞    η   − α α r(t)D y(t)−q(t)f(v−t)y(v)dvfor= 0t>0 − t α where D is the Liouville rightsided fractional derivative of orderaÎ(0,1) ofyand − h>0 is a quotient of odd positive integers. We establish some oscillation criteria for the equation by using a generalized Riccati transformation technique and an inequality. Examples are shown to illustrate our main results. To the best of author’s knowledge, nothing is known regarding the oscillatory behavior of the equation, so this article initiates the study. MSC (2010): 34A08; 34C10. Keywords:oscillation, fractional derivative, fractional differential equation

1 Introduction The goal of this article is to obtain several oscillation theorems for the fractional differ ential equation   ∞    η  α −α r(t)D y(t)−q(t)f(v−t)y(v)dvfor= 0t>0 (1:1) − t α whereaÎ(0, 1) is a constant,h> 0 is a quotient of odd positive integers, D is − the Liouville rightsided fractional derivative of orderaofydefined by  ∞ α 1d −α (D y)(t) :=(v−t)y(v)dfortÎℝ + := (0,∞), hereΓis the gamma − (1−α)dt  ∞ t−1−v function defined by(t) :=v edfortÎℝ + , and the following conditions are assumed to hold:

(A)randqare positive continuous functions on [t 0 ,∞) for a certaint 0 > 0 andf: h ℝ®ℝis a continuous function such thatf(u)/(u)≥Kfor a certain constantK> 0 and for allu≠0.

By a solution of (1.1) we mean a nontrivial functionyÎC(ℝ + ,ℝ) such that

© 2012 Chen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.