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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

超线性阻尼分数阶微分方程的振动性定理

孟凡伟

，邵晶

1,2

曲阜师范大学数学科学学院，曲阜　 273165

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

[D.X.Chen, 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： O175.1

Oscillation Theorems for Superlinear Damped

Fractional Diﬀerential Equations

MENG Fan-Wei

, SHAO Jing

1，2

School of Mathematical Sciences, Qufu Normal University, Qufu 273165

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 [D.X.Chen, 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

(r(t)D

x(t))

′

+ p(t)D

x(t) + q(t)f





(t − s)

−α

x(s)ds



= 0, (1)

where r, p, q : [0, ∞) → R = (−∞, +∞), f : R → R, are continuous functions, and r(t) > 0,

t ≥ 0, α ∈ (0, 1) is a constant. D

x is the Liouville fractional derivative of order α of x deﬁned

基金项目：国家自然科学基金 (No.s 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 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

by (D

x)(t) : =

Γ(1−α)





(t − s)

−α

x(s)ds



, here Γ(·) is the gamma function deﬁned by

Γ(t) :=



∞

t−1

−s

ds, for t ∈ R

= [0, ∞).

By a solution of (1) we mean a nontrivial function x ∈ C(R

, R) such that



(t −

−α

x(s)ds ∈ C

, R), r(t)D

x(t) ∈ C

, R), and satisfying (1) for t ≥ 0. Our attention

is restricted to those solution of (1) which exist on R

and satisfy sup{|x(t)| : t ≥ t

∗

} > 0 for

any t

∗

≥ 0.

A solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually

negative. Otherwise it is nonoscillatory. Equation (1) is said to be oscillatory if all its solutions

are oscillatory.

The theory of fractional derivative goes back to Leibniz’s note in his list to L’Hospital

[1], dated 30 September 1695, in which the meaning of the derivative of order

is discussed.

Recently, there have been severval books. On the subject of fractional derivatives and fractional

integrals, such as the books [2-6].

For three centuries the theory of fractional derivatives developed mainly as a pure theo-

retical ﬁeld of mathematics useful only for mathematicians. However, in the last few decades

many authors pointed out that fractional derivatives and fractional integrals are very suitable

for the desciption of properties of various real problems.

Many articles have investigated some aspects of fractional diﬀerential equations, such as

the existence and uniqueness of solutions to Cauchy type problems, the methods for explicit

and numerical solutions, and the stability of solutions, we refer to [7-14].

However, to the best of authors’ knowledge, very little is known regarding the oscillatory

behavior of fractional diﬀerential equations (excerpt for [15]). In particular, nothing is known

regarding the oscillation properties of superlinear fractional diﬀerential equation (1). To de-

velop the qualitative theory of fractional diﬀerential equations, it is of great interest to study

the oscillation of (1). In this paper, we establish several oscillation criteria for (1). It is no

doubt that the Riccati substitution and its generalized forms play a very important role in the

oscillatory theory of diﬀerential equations. In this article, we shall employ another method to

derive several oscillation criteria for (1) which are still new even in some particular cases . We

believe that our approach is simpler and more general.

We shall make use of the following conditions:

) yf (y) > 0, f

′

(y) ≥ 0, where y = 0;

)



∞

f(u)

< ∞,



−∞

f(u)

< ∞;

)



∞

√

′

(u)

f(u)

du < ∞,



−∞

√

′

(u)

f(u)

du < ∞;

) min



inf

u>0



′

(u)



∞

√

′

(s)

f(s)

ds, inf

u<0



′

(u)



−∞

√

′

(s)

f(s)



> 0.

- 2 -

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