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Oscillation criteria of fractional differential equations

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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.

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Publié le 01 janvier 2012
Nombre de lectures 4
Langue English

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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(vt)y(v)dvfor= 0t>0 t α whereDis 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 authors 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    ηαα r(t)D y(t)q(t)f(vt)y(v)dvfor= 0t>0(1:1) t α whereaÎ(0, 1) is a constant,h> 0 is a quotient of odd positive integers,Dis the Liouville rightsided fractional derivative of orderaofydefined by α1dα (D y)(t) :=(vt)y(v)dfortÎ+:= (0,), hereΓis the gamma (1α)dt t1v function defined by(t) :=v edfortÎ+, and the following conditions are assumed to hold:
(A)randqare positive continuous functions on [t0,) for a certaint0> 0 andf: h ®is a continuous function such thatf(u)/(u)Kfor a certain constantK> 0 and for allu0.
By a solution of (1.1) we mean a nontrivial functionyÎC(+,) 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.