2018 2nd International Conference on Applied Mathematics, Modeling and Simulation (AMMS 2018) ISBN: 978-1-60595-580-3
Forced Oscillation of Nonlinear Fractional Delay Differential
Equations with Damping Term
Si-ying ZHU, Hui-juan LI
and An-ping LIU
*School of Mathematics and Physics, China University of Geosciences (Wuhan)
*Corresponding author
Keywords: Forced oscillation, Fractional differential equations, Damping term, Time delay.
Abstract. In this article, we study forced oscillatory properties of solutions to nonlinear fractional differential equations with a damping term and a time delay. Based on the properties of the Riemann-Liouville fractional derivative, we establish a sufficient condition for oscillation of all solutions.
Introduction
Fractional calculus is a theory of integrals and derivatives of any arbitrary real (or complex) order, it has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverseand widespread fields of science and engineering. It does indeed provides several potentially useful tools for solving differential and integral equations, and various other problems involving special functions of mathematical physics as well as their extensions and generalizations in one and more variables.It has a long history from 30 September 1695, when the derivative of order 1/2 was mentioned by Leibniz. The fractional differentiation and fractional integration go back to many great mathematicians such as Leibniz, Liouville, Grünwald, Letnikov, Riemann, Able, Riesz, and Weyl. The integrals and derivatives of non-integral order, and the fractional integro-differential equations have found many applications in recent studies in theoretical physics, mechanics and applied mathematics. Recently, there have been some books on the subject of fractional calculus and fractional differential equations, such as [1–6]. Many papers have investigated some aspects of fractional differential equations, some about the existence and uniqueness of solutions[7–9], Cauchy type problems[10, 11], the methods for explicit and numerical solutions[12–14] and so on; Others about oscillation properties[15–21], many of which has delays [22, 23].
In the paper[24], the authors discussed the oscillation of nonlinear fractional differential equations with damping term of this form:
) ( )) ( ( ) ( ) )( )( ( ) )(
(D01y t p t D0y t q t f y t g t (1) with initial condition I01- yb
, where b is a real number, and （0,1） is a constant, and D0 is the
Riemann-Liouville fractional derivatives of order α of y.
Since time-delays are inherent in many dynamic systems, The delay terms may degrade the achievable control performance or even cause instability. In recent years, considerable attention has been paid to fractional-order time-delay systems[7, 25–28], for the interdisciplinary nature of these contributions. For example, advances in stability analysis, computational techniques (symbolic and numerical), and results from many otherwise disconnected fields (automotive engineering, manufacturing, neuroscience, and control theory)[29, 30].
In this paper, we study forced oscillatory properties of solutions to nonlinear fractional differential equations with damping term and time delay:
) ( )) ( ( ) ( ) )( )( ( ) )(
y 0
1
0 t p t D y t q t f y t g t D
Where y(t)(t) when t[,0), and (t) is a given continuous function, limt0(t)0, b is a real number, where and （0,1） are constants, 0 y
D is the Riemann-Liouville fractional derivative of order α of y.
We will use the following conditions:
(A) p(t)C(R,R) , q(t)C(R,R) , f C(R,R) , and f(u)/u0 for all u0 , )
, ( ) (
g t C R R .
Definition 1.1. The solution y of problem (1.1) is called oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called non-oscillatory.
Preliminaries and Lemmas
In this section, we introduce the definitions of fractional integral and fractional derivative. There are several kinds of definitions of fractional integrals and fractional derivatives[1]. In this paper, we use Riemann-Liouville definition.
Definition 2.1. The Riemann-Liouville fractional integral Iay of order
R
is defined by:
t
a t v y v dv t a R t
y
I
, , ) ( ) ( ) (
1 ) )(
( a 1 (3)
Here () is the Gamma function defined by
0 1
)
( s e sds , for 0 , aR. This
integral is called the left-sided fractional integral.
Definition 2.2. The Riemann-Liouville fractional derivative Da y
of order R is defined by:
tv y v dv ta Rdt d n
t y
D t
a
n n
n
a
, , ) ( )
( )
( 1 )
)(
( 1 (4)
with n
1, where
means the integer part of . Lemma 2.1. [1] Let 0. Then for everyyL1[a,b]:) ( ) )(
(DaIay t y t (5) almost everywhere.
Lemma 2.2. [1] Let 0 , mN , and
dx
D d . If the fractional derivatives (Day)(x)
and(Da my)(x)
exist, then:
). )( (
) )(
(DmDay x Damy x
Lemma 2.3. Let 0 and n
1 . Assume that y is such that y(t)L1(a,b) , and]) , ([
a y AC a b
In m be the fractional integral of order n, then:
) ( lim
) (
) (
) ( ) )(
( 1
1
0
1
D I y zk a t
t y t
y D
I n k an
a z n
k
k
a
a (6)
Specifically, for01, we have:
). ( lim )
( ) (
) ( ) )(
( 1
1
I y z
a t t
y t
y D
I a
a z a
a (7)
Proof. We first note that the limits on the right-hand side exist because of our assumption on y that implies the continuity of D Ian y
n
1
. Moreover, because of this assumption, there exists some
1 L
such that:
. ) ( 1
1
1
a n a n n a n I a y I D y I D
This is a classical differential equation of order n1 forIany, its solution is easily seen to be of the form: ). ( ) ( lim ! ) ( ) ( 1 0
D I y z I tk a t t
y
I an
n a k a z n k k n
a (8)
Thus, by definition ofDay:
). ( ) ( lim ! ] ) [( ) ( ) ]( ) ( lim ! ) ( [ ) ( ) ( 1 0 1 0
t I z y I D k a t D I t I D I t I z y I D k a t D I t y I D I t y D I a n a k a z n k k n a a n a n a a n a n a k a z n k k n a a n a n a a a a (9)Because of D annihilates every summand in the sum. Next we apply the operator Dan to both
sides of the equation and find:
) ( ) ( lim ! ] ) [( ) ( 1 0
D I y z D I tk a t D t y n a n a n a k a z n k k n
a (10)
We now invoke the fractional derivatives of elementary functions to evaluate the terms in the sum and the semigroup property of fractional integration to manipulate the remaining term. This yields:
) ( ) ( lim ) 1 ( ) ( ) ( 1 0
D I y z I tn k a t t y a n a k a z n k n k (11)
Finally we substitute k in the sum by n-k-1, solve forIa(t) and combine the result with
) ( ) (
D y t I t
Ia a a to obtain:
) ( lim ) ( ) ( ) ( ) ( ) ( 1 1 0 1
D I y zk a t t y t I t y D I n a k n a z n k k a a a (12) as desired. Main Results
Theorem 3.1. Suppose that assumption (A) and the following conditions hold:
0 ) ) ( ) ( ( ) ( ) ( sup lim 0 0 1
V w M g sV s ds dw
0 ) ) ( ) ( ( ) ( ) ( sup lim 0 0 1
dw ds s V s g M w V w t w t t t (14)Here
st p v dv s V 0 ) ( exp )
( , M is an arbitrary constant. Then each solution of problem (1.2)
oscillates.
Proof. For the sake of contradiction, let y(t) be a non-oscillatory solution of. Without loss of generality, we can assume that there exists T 0, t0 T, such that y(t)0 for alltt0. According to equation (1.2) and assumption (A), the following inequality is satisfied:
). ( ) ( ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( ) )( ( ) ( ) )( ( )]' ( ) )( [( 0 1 0 0 t V t g t V t g t V t y f t q t V t p t y D t V t y D t V t y D
Integrating both sides of the above inequality from t0to t, we get:
t t t t ds s V s g M ds s V s g t V t y D t V t y D 0 0 . ) ( ) ( ) ( ) ( ) ( ) )( ( ) ( ) )(( 0 0 0 0
(15)
Where ( 0 )(0 ) (0 )
Dy t V t
M . From Lemma 2.3 and equation (3.3), we can obtain:
. ) ) ( ) ( ( ) ( ) ( ) ( 1 ) ( ) ( ] ) ( ) ( ) ( 1 ) ( [ ) ( ) ( ) )( ( ) ( 0 0 0 1 1 0 1 1 0 dw ds s V s g M w V w t t b ds s V s g t V t V M I t y I t y w t t t t
We substitute t in the sum by t to obtain:
. ) ) ( ) ( ( ) ( ) ( ) ( 1 ) ( ) ( 0 t 0 1 1 dw ds s V s g M w V w t t b t
y
t
w
Takingt, from the above inequality, we can obtain:
0 ) ) ( ) ( ( ) ( ) ( ) ( 1 inf lim ) ( inf lim ) ( inf lim 0 1 1 0
dw ds s V s g M w V w t t b t y t w t t t t Which contradicts the assumption that y(t)0.
On the other hand, we can assume that there exists T 0,t0 T , such that y(t)0, for all tt0. According to equation (1.2) and assumption (A), the following inequality is satisfied:
Integrating both sides of the above inequality from t0tot , we get:
t
t
t
t
ds s V s g M
ds s V s g t
V t y D t
V t y D
0
0
) ( ) (
) ( ) ( ) ( ) )( ( ) ( ) )(
( 0 0 0 0
(16)
Where ( 0 )(0) (0 )
Dy t V t
M . From Lemma 2.3 and equation (3.4), we can obtain:
dw ds s V s g M w
V w t
t b
ds s V s g t
V t
V M I
t y
I t y
w
t t
t
t
) ) ( ) ( (
) (
) (
) (
1 )
( ) (
] ) ( ) ( ) (
1 )
( [ )
( ) (
) )( (
) (
0 0
0
1 1
0 1 1
0
We substitute t in the sum byt to obtain:
0 ) ) ( ) ( (
) (
) ( ) (
1 )
( ) (
0
0
1
1
dw ds s V s g M w
V w t t
b t
y w
t t
Takingt, from the above inequality, we can obtain:
0
) ) ( ) ( (
) (
) ( ) (
1 sup lim )
( sup lim
) ( sup lim
0
0
1 1
dw ds s V s g M w
V w t t
b t y
w
t t
t t
t
which contradicts the assumption that y(t)0. This proof is complete.
Example
Consider the fractional delay differential equation:
0 , sin )
1 )( ( ) 1 )(
( 2 2 2
1 0 2
3
0y t Dy t t ye e t t
D y t (17)
where , ( ) 1, ( ) , ( ) , ( ) , ( ) sin , 1
2
1 2 0 2
p t q t t f u ueu V s et s g t e t t . Then:
. sin sin
) ( ) (
0 0 0
0 0
2
sds e
ds se e
ds s V s
g w
t s t s
t w
t
w
t s
Set 4
0
dw ds w e Me w t dw sds e M e w t dw sds e M e w t dw ds s V s g M w V w t w w t w s w t w s t w t w t t ) 4 sin( 2 2 ( ) ( ) sin ( ) ( ) sin ( ) ( ) ) ( ) ( ( ) ( ) ( 2 4 1 0 2 1 4 1 4 1 4 0 2 1 4 1 4 0 1 2 1 0 1 0 0
Set 2
s w
t , then the above integral can be written as the following form:
0 2 2
2 2 0 2 2 0 4 1 2 ) ( 2 4 1 0 t sin ) 4 cos( 2 cos ) 4 sin( 2 2 ) 2 )( ) 4 sin( 2 2 ( 1 2 2 2 2 2 t s t t s t t s t s t s t ds s e t e ds s e t e ds e Me ds s ds s t e Me s
Let t , because e2s2coss2 e2s2 and
4 2 lim
0
2 2
t s
t e ds , we know that
ds s e t s t
0 2 cos 2
lim 2 is convergent as well.
Set: B ds s e A ds s e t s t t s
t
0 2 2
0 2 2
sin lim
, cos
lim 2 2 .
Select the sequence etk
t
k },lim
A B -arctan -2k 4 2 3 { }
{t , then we calculate:
]) sin ) 4 cos( 2 cos ) 4 sin( 2 2 [ ( lim
0 2 2
2 0 2 0 4 1 2 2 2 ds s e t e ds s e t e ds e Me e t s t t s t t s t k k k k k k
(18)Firstly, we consider the following limit:
. ) 2 3 sin( ) arctan arctan 2 3 sin( ) arctan 2 3 cos( ) arctan 2 3 sin( ) arctan 2 2 3 cos( lim ) arctan 2 2 3 sin( lim ) sin ) 4 cos( cos ) 4 (sin( lim 2 2 2 2 2 2
0 2 2 2 0 2 2 2
B A B A A B A B B A A B B A B A A B k B A B k A ds s e t ds s e t k k t t s k s k
k k k
Secondly, we know that tk
t e
lim and 1 4 0 1 4 1 4
2 2
2
lim 2
Me
e ds Me Mek
t s
t .
Hence, for equation (4.2), we have:
)] )(
( )[
(
)]} sin
) 4 cos( 2
cos )
4 (sin( 2
2 [ { lim
2 2 4
1
0
2 2
0
2 2
0 4
1 2 2 2
B A Me
ds s e
t e ds s e
t e ds e Me
e
k k
k k
k k
t s t
t s t
t s t
k
Then we obtain
. 0
)} sin )
4 cos( 2
cos )
4 sin( 2
2 { lim
) ) ( ) ( (
) (
) ( inf lim
0
2 2 2
0
2 2
2 0
4 1 0
1
2 2
2 0
ds s e
t e ds s e
t e ds e Me
dw ds s V s g M w
V w t
k k
k k
k t t s t t s
t s
K t k
w t t
t
Similarly, selecting the sequence 2 arctan }
4 2 { } {
A B j
tj , we can obtain:
0 )
) ( ) ( (
) (
) ( sup lim
0
0
1
V w M g sV s dw
w
t w
t t
t
Therefore, by Theorem 3.1 all solutions of equation (4.1) are oscillatory.
Acknowledgement
This research was financially supported by the National Science Foundation of China (No.41630643 and No.11701533).
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