超线性阻尼分数阶微分方程的振动性定理.doc

超线性阻尼分数阶微分方程的振动性定理
孟凡伟1,邵晶1,2
1
曲阜师范大学数学科学学院,曲阜 273165
2
济宁学院数学系,曲阜 273155
摘要:本文讨论了含α-阶 Riemann-Liouville 分数阶导数的超线性分数阶阻尼微分方程的振动
性问题。在比较一般的条件下,得到了一些新的振动性定理,所采用的方法有别于以前文献
[, oscillation criteria of fractional differential equations, Advances in Difference
equations, 33(2012),1-10] 给出的方法。并给出实例说明结果的重要性。
关键词:分数阶微分方程,超线性,振动性
中图分类号:
Oscillation Theorems for Superlinear Damped
Fractional Differential Equations
MENG Fan-Wei1, SHAO Jing1,2
1
School of Mathematical Sciences, Qufu Normal University, Qufu 273165
2
Department of Mathematics, Jining University, Qufu, 273155
Abstract: In this paper, the oscillation of the superlinear damped fractional differential
equation with Riemann-Liouville fractional derivative of order α∈(0, 1) is considered. Several
new oscillation criteria are established under quite general assumptions. Our methodology is
somewhat different from that of previous author [, oscillation criteria of fractional
differential equations, Advances in Difference equations, 33(2012),1-10]. Example is also given
to illustrate the results.
Key words: Fractional differential equation, superlinear, Oscillation.
0 Introduction
We consider the oscillatory behavior of the superlinear fractional differential equation
(∫ t )
α′α−α

(1)
0
where r, p, q : [0, ∞) → R = (−∞, +∞), f : R → R, are continuous functions, and r(t) > 0,
t ≥ 0, α∈(0, 1) is a constant. Dαx is the Liouville fractional derivative of order α of x defined
基金项目: 国家自然科学基金( 11171178, 11271225), 教育部博士点基金(No. 20103705110003), 山东省高等学校科技
发展计划项目(J12LI52).
作者简介: Meng Fanwei(1963-),male,professor,major research direction:qualitative properties of differential equa-
tions. Correspondence author:Shao Jing(1983-),female,major research direction:qualitative properties of differential
equations.
-1-(t − s)
(r(t)D x(t)) + p(t)D x(t) + q(t)f
x(s)ds
= 0,
by (Dαx)(t) : =

1 d
Γ(1−α) dt

(∫

t
0

)
(t − s)−αx(s)ds , here Γ(·) is the gamma function de