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ISSN Online: 2327-4379 ISSN Print: 2327-4352

DOI: 10.4236/jamp.2019.77096 Jul. 10, 2019 1429 Journal of Applied Mathematics and Physics

Oscillation for a Class of Fractional Differential

Equation

Qian Feng, Anping Liu

School of Mathematics and Physics, China University of Geosciences, Wuhan, China

Abstract

We consider the oscillation of a class fractional differential equation with Robin and Dirichlet boundary conditions. By generalized Riccati transforma-tion technique and the differential inequality method, oscillatransforma-tion criteria for a class of nonlinear fractional differential equation are obtained.

Keywords

Oscillation, Fractional Derivative, Fractional Partial Differential Equation

1. Introduction

The fractional differential equations are used to describe mathematical models of numerous real processes and phenomena studied in many areas of science and engineering such as population dynamics, neural networks, industrial robotics, electric circuits, optimal control, biotechnology, economics and many other branches of science. Furthermore, the fractional calculus can also provide an ex-cellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a “memory” term in the model.

The oscillation theory as a part of the qualitative theory of differential equa-tions has been developed rapidly in the last decades, and there has been a great deal of works on the oscillatory behavior of integer order differential equations [1] [2] [3]. As a new cross-cutting area, recently some attention has been paid to oscillations of fractional differential equations [4] [5] [6] [7]. Some new devel-opments in the oscillatory behavior of solutions of fractional differential equa-tions with damping terms [8] [9] [10] [11] have been reported by authors.

In this paper, we consider the oscillatory behavior of solutions of the following fractional differential equation:

How to cite this paper: Feng, Q. and Liu, A.P. (2019) Oscillation for a Class of Frac-tional Differential Equation. Journal of Ap-plied Mathematics and Physics, 7, 1429- 1439.

https://doi.org/10.4236/jamp.2019.77096

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DOI: 10.4236/jamp.2019.77096 1430 Journal of Applied Mathematics and Physics

(

)

,t 0

1

0 1

( ) ( ( ) ( )D ( , )) ( , ) ( ) ( , )

( ) ( ) ( , ) ( ) ( ( , )) ( , ), 0

m t

i i i

m

i i i i

i

a t g p t q t u x t a x t f t s u x s ds

t

b t h u u x t b t h u x t u x t t t

α α

τ τ

− +

=

=

∂  + + −

= ∆ + − ∆ − ≥ >

(1.1)

where ( , )x t ∈ Ω×R+ =E, ∆ is the Laplacian in Rn, Ω is a bounded

do-main in Rn with a piecewise smooth boundary ∂Ω. 0< <α 1 is a real

num-ber and D u x tα ( , )

+ is the Riemann-Liouville left-sided fractional derivative of

order α ∈(0,1) of u for t R∈ +: (0, )= ∞ .

We shall consider Robin and Dirichlet boundary conditions

( , ) ( , ) ( , ) 0, ( , ) ,

u x t x t u x t x t R

N γ +

+ = ∈ ∂Ω×

∂ (1.2)

and

( , ) 0, ( , ) .

u x t = x t ∈ ∂Ω×R+ (1.3)

where γ∈ ∂ΩC[ , ]R+ is continuous function, N is the unit out normal vector to ∂Ω.

The following conditions are assumed to hold:

(H1) f g C R Ri, ∈ ( ; ) are convex in [0, )∞ and g is a monotone increasing

function with xf xi( ) 0, ( ) 0> xg x > for x≠0, there exist positive constants ki,

β

such that f x x k x g xi( ) > i, ( )>β for x≠0. And τ ≥i 0, u≠0,

( ) 0

h u > , h ui( ) 0> , uh u'( ) 0> , uh ui'( ) 0> , h(0) 0> , hi(0) 0> .

(H2) a,ai,b,bi and q are positive continuous functions on t t∈[ , )0 ∞ for

a certain t0>0, and p is a nonpositive continuous function on t t∈[ , )0 ∞

for a certain t0>0. There exists a constant M>0, q t( )≤M for t0>0. And

( ) 0

( )

p t q t

− 

 

  , t t∈[ , )0 ∞ , 0 ( ) ( )

t q tp t dt

∞−

< ∞

.

(H3) g-1C R R( ; ) is continuous function with sg s−1( ) 0> for s0,

there exists positive constant δ such that g uv−1( )

δ

g u g v−1( ) −1( ) for uv<0,

and g uv−1( )

δ

g u g v−1( ) −1( ) for uv>0.

(H4) a C E Ri∈ ( ; )+ , and a ti( ) min= x∈Ωa x ti( , ).

By a solution of (1.1), (1.2) and (1.3), it mean a nontrivial function

1 ( ; )

u CE R

+

∈ with 1

0 ( , )( ) ( ; )

t

u x s t s ds C E R−α

+

− ∈

,

1 ,

( ) ( ( ) ( ) t ( , )) ( ; )

r t g p t q t D u x tα C E R

+ +

+ ∈ satisfies (1.1) for t>0 on E and the boundary conditions (1.2) and (1.3).

A solution u of (1.1) is said to be oscillatory if it is neither eventually posi-tive nor eventually negaposi-tive, otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

2. Preliminaries

In this section, there are several kinds of definitions of fractional derivatives and integrals and some lemmas which are useful throughout this paper.

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DOI: 10.4236/jamp.2019.77096 1431 Journal of Applied Mathematics and Physics

of a function f R: + →R on the half-axis R+ is given by

-1

0 1 0

( )( ) ( - ) ( ) ,

( )

t

I f tα t vα f v dv

α

+ =Γ

provided that the left side is pointwise defined on R+, where Γ is the gamma

function.

Definition 2.2 [12] The Riemann-Liouville fractional partial derivative of or-der 0< <α 1 with respect to t of a function u x t( , ) is given by

0

1

( )( , ) ( ) ( , ) .

(1 )

t

D u x t t v u x v dv

t

α α

α −

+

= −

∂ Γ −

Lemma 2.3 [12] Let

0

( ) t( ) ( ) , (0,1), 0

G t = t v u v dv −α α t>

Then

'( ) (1 )( )( ), (0,1), 0

G t

α

D u tα

α

t

+

= Γ − ∈ >

Lemma 2.4 [4] If X and Y are nonnegative, then

1 ( 1)

m m m

mXYX m Y

where the equality holds if and only if X Y= .

3. Oscillation of (1.1) and (1.2)

For the sake of convenience, we set

1

( ) ( , ) ,

U t = u x t dx whereΩ = dx

( ) ( ) ( )( )( )

z t p t q t D U tα

+

= +

Theorem 3.1 Suppose that (H1)–(H5) hold and if the fractional differential inequality

1

[ ( ) ( ( ))] m i i( ) ( ) 0

i

d a t g z t k a t G t

dt =

+

≤ (3.1) has no eventually positive solution and the fractional differential inequality

1

[ ( ) ( ( ))] m i i( ) ( ) 0

i

d a t g z t k a t G t

dt =

+

≥ (3.2) has no eventually negative solution, every solution of (1.1) and (1.2) is oscillato-ry in E.

Proof Suppose that u x t( , ) is a nonoscillatory solution of (1.1) and (1.2), it is either eventually positive or eventually negative. Without loss of generality, we may assume that u x t( , ) is an eventually positive solution of (1.1) and (1.2) in

0

[ , )t

Ω× ∞ . Integrating (1.1) with respect to x over Ω, we obtain

(

)

,t 0

1

1

( ) ( ( ) ( )D ( , )) ( , ) ( ) ( , )

( ) ( ) ( , ) ( ) ( ( , )) ( , ) .

m t

i i i

m

i i i i

i

d a t g p t q t u x t dx a x t f t s u x s ds dx

dt

b t h u u x t dx b t h u x t u x t dx

α α

τ τ

− +

=

=

 +  + −

 

= ∆ + − ∆ −

(3.3)

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DOI: 10.4236/jamp.2019.77096 1432 Journal of Applied Mathematics and Physics

2

2

1

( , )

( ) ( , ) ( ) ( )

( , ) ( ) ( ) 0,

u x t

h u u x t dx h u ds h u grad u dx

N

x t uh u ds h u grad u dx t t

γ

Ω ∂Ω Ω

∂Ω Ω

∆ = −

= − − ≤ ≥

(3.4)

1

( ( , )) ( , ) 0, ,

i i i

h u x t τ u x t τ dx t t

Ω − ∆ − ≤ ≥

(3.5)

By using Jensen’s inequality and (H1), (H4), we get

(

)

(

)

0

1 0

( , ) ( ) ( , )

( ) ( ) ( , ) ( ) ( ) ( )

t i i

t

i i i i

a x t f t s u x s ds dx

a t dx f t s u x s dx dx ds k a t dxG t

α

α − Ω

− −

Ω Ω Ω Ω

 

 

(3.6)

Combining (3.3)–(3.6), we obtain

1

[ ( ) ( ( ))] m i i( ) ( ) 0

i

d a t g z t k a t G t

dt =

+

(3.7)

Therefore, U t( ). is an eventually positive solution of (3.1), this contradicts the hypothesis.

Secondly, if u x t( , ) is an eventually negative solution of the problem (1.1) and (1.2), then using above procedure, we can easily show that

1

( ) ( , )

U t = u x t dx

is an eventually negative solution of the Equation (3.2). This completes the proof.

Theorem 3.2 Suppose that (H1)–(H4) and

0

1( 1 )

( )

t g a t dt

= ∞

(3.8)

hold. if there exists a positive function 1 0

[ , )

r C t∈ ∞ such that

0

2

1

( )[ ( )]

limsup [ ( ) ( ) ]

4 (1 ) ( )

m t

i i t

t i

Ma s r s

r t k a s ds

r s

β α

→∞ =

− = ∞

Γ −

(3.9)

where ki,

β

are defined as in (H1), then every solution of (3.1) and (3.2) is

oscillatory.

Proof Suppose that U t( ) is a nonoscillatory solution of (3.1). Without loss of generality, we may assume that U t( ) is an eventually positive solution of (3.1). Then there exists G t( ) 0,> t t∈[ , )1 ∞ , where G t( ) is defined as in Lemma

2.3.

It follows from (3.7) that

1 1

[ ( ) ( ( ))] m i i( ) ( ) 0, [ , ).

i

a t g z t k a t G t t t

=

′ ≤ −

≤ ∈ ∞ (3.10)

Thus, a t g z t( ) ( ( )) is strictly decreasing on a t( ) 0> . Since a t( ) 0> for

1

[ , )

t t∈ ∞ and (H1), we see that z t( ) is eventually of one sign. We claim that

1

( ) 0, [ , ).

z t > t t∈ ∞ (3.11)

If not, there exists t2≥t1 such that z t( ) 02 < . Since a t g z t( ) ( ( )) is strictly

decreasing on [ , )t1 ∞ and it is clear that a t g z t( ) ( ( ))≤a t g z t( ) ( ( ))2 2 = <c 0,

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DOI: 10.4236/jamp.2019.77096 1433 Journal of Applied Mathematics and Physics

1

( ) ( ).

( )

c

z t g

a t

(3.12)

Due to q t( ) 0> and g c−1( ) 0< , we get

1 1( ) 1 1

( ) ( )

( ) ( ) ( ) ,

( ) ( ) ( )

c

g g c g

a t a t

z t p t D U t

q t q t α q t M

δ

− − −

+

   

   

   

= + < ≤ (3.13)

Integrating the above inequality from t2 to t, from Lemma 2.3, we have

2 2

1( ) 1 1

( )

( ) ( ) ,

( ) (1 )

t t

t t

g c g a t

p t G s ds ds

q t M

δ

α

− −  

 

 

+ <

Γ −

 

(3.14)

which yields

2 2

1

1

2 ( ) ( ) 1

( ) ( ) (1 ) .

( ) ( )

t t

t t

p t g c

G t G t ds g ds

q t M a t

δ

α  − − −   

≤ + Γ − +

 

 (3.15)

By (H2) and (3.8), letting t→∞, we get lim ( )

t→∞G t = −∞. This contradicts the

fact that G t( ) 0> . Hence, (3.11) holds.

From Lemma 2.3

( )

( ) ( ) ( )( )( ) ( ) ( ) ,

(1 )

G t

z t p t q t D U tα p t q t

α +

= + = +

Γ − (3.16)

therefore,

( )- ( ) ( ) ( )

( ) (1 ) (1 ) (1 )

( ) ( )

z t p t z t z t

G t

q t q t M

α α α

′ = Γ − ≥ Γ − ≥ Γ − (3.17)

Define the function w t( ) by the generalized Riccati substitution

1

( ) ( ( ))

( ) ( ) , [ , ).

( )

a t g z t

w t r t t t

G t

= ∈ ∞ (3.18)

Then we have w t( ) 0> for t t∈[ , )0, and from (3.18), it follows that

[

]

[

]

1

2

1

1

( ) ( )

( ) ( ) ( ( )) ( ) ( ( ))

( ) ( )

( ) ( ) ( ) ( ) (1 ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( ) (1 ) ( )

( ) ( ( )) ( ) ( )

( ) ( ) ( ) ( ) ( )

m i i i

m i i i

m i i i

r t r t

w t a t g z t a t g z t

G t G t

r t w t r t k a t z t w t

r t M G t

r t w t r t k a t z t w t

r t g z t Mr t a t

r t w t r t k a t r t

α

α

β =

=

= ′

  ′

′ = +

 

′ Γ −

≤ − −

′ Γ −

≤ − −

≤ − −

(1 ) 2( ).

( ) ( )w t

Mr t a t

α Γ −

(3.19)

Taking

(1 ) 1 ( ) ( ) ( )

2, ( ),

( ) ( ) 2 (1 ) ( )

Mr t a t r t

m X w t Y

Mr t a t r t

β α

β α

′ Γ −

= = =

Γ − (3.20)

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DOI: 10.4236/jamp.2019.77096 1434 Journal of Applied Mathematics and Physics

2

1

( )[ ( )]

( ) ( ) ( ) ,

4 (1 ) ( )

m i i i

Ma t r t

w t r t k a t

r t

β α

=

′ ≤ − +

Γ −

(3.21)

Integrating both sides of the inequality (3.21) from t0 to t, and taking the

limit supremum of both sides of the above inequality as t→ ∞, we get

0

2

0 1

( )[ ( )]

limsup [ ( ) ( ) ] ( ) .

4 (1 ) ( )

m t

i i t

t i

Ma s r s

r t k a s ds w t

r s

β α

→∞ =

− < < ∞

Γ −

(3.22)

Which contradicts (3.9). The proof is complete.

Secondly, if U t( ) is an eventually negative solution of the fractional diffe-rential inequality (3.2) and there exists G t( ) 0,< t t∈[ , )1 ∞ . When (3.2) is

oscil-latory is similar to that of above procedure, and hence is omitted.

Theorem 3.3 Assume that (H1) - (H4) and (3.8) hold. Furthermore, suppose that there exist a positive function 1

0

[ , )

r C t∈ ∞ and a function H C D R∈ ( , ), where D: {( , ) := s t s t t≥ ≥ 0}, such that

0

( , ) 0,

H t t = for t t

0

( , ) 0, ( , )

H s t = for s tD

where D0: {( , ) := s t s t t≥ ≥ 0} and Hhas a nonpositive continuous partial

de-rivative Ht′(s,t)=∂H(s,t) ∂t with respect to the second variable and satisfies

0

2

1 0

1 ( )[ ( )]

limsup ( , )[ ( ) ( ) ]

( , ) 4 (1 ) ( )

m s

i i t

s i

Ma ts r t

H s t r t k a t dt

H s t β α r t

→∞ =

− = ∞

Γ −

(3.23)

where ki,

β

and r t( ) are defined as in Theorem 3.2. Then all solutions of

(3.1) and (3.2) are oscillatory.

Proof Suppose that U t( ) is a nonoscillatory solution of (3.1). Without loss of generality, we may assume that U t( ) is an eventually positive solution of (3.1). We proceed as in the proof of Theorem 3.2 to get (3.21), Multiplying (3.21) by H s t( , ) and integrating from t0 to s, for s t∈[ , )0 ∞ , we derive

0

0 0

2

1

0 0

( )[ ( )]

( , )[ ( ) ( ) ] [ ( , ) ( )] ( , ) ( )

4 (1 ) ( )

( , ) ( ),

m

s s s

i i t t

t t

i

Ma ts r t

H s t r t k a t dt H s t w t H s t w t dt

r t

H s t w t

β α

=

− ≤ − +

Γ −

(3.24) Therefore,

0

2

0 1

0

1 ( , )[ ( ) ( ) ( )[ ( )] ] ( ) ,

( , ) 4 (1 ) ( )

m s

i i

t i

Ma ts r t

H s t r t k a t dt w t

H s t = β α r t

− ≤ < ∞

Γ −

(3.25)

which is a contradiction to (3.23). The proof is complete.

Secondly, if U t( ) is an eventually negative solution of the fractional diffe-rential inequality (3.2). The proof when (3.2) is oscillatory is similar to that of above procedure, and hence is omitted.

Next, we consider the case

0

1( 1 )

( )

t g a t dt

< ∞

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DOI: 10.4236/jamp.2019.77096 1435 Journal of Applied Mathematics and Physics

Theorem 3.4 Assume that (H1)–(H4) and (3.26) hold, and that there exist a positive function 1

(

)

0

[ , );

r C t∈ ∞ R such that (3.9) holds. If for every constant 2 3

max{ , }

T = t t , such that

1

1 ( )

( )

m t i

T T

i a s ds dt

a t

=

 

= ∞

 

(3.27)

Then every solutions U t( )of (3.1) and (3.2) are oscillatory or satisfies

lim ( ) 0

t→∞G t = or lim ( ) 0t→∞G t′ = , where G t( ) is defined as Lemma 2.3.

Proof Suppose that U t( ) is a nonoscillatory solution of (3.1). We may as-sume that U t( ) is an eventually positive solution of (3.1), proceeding as in the proof of Theorem 3.2 to get (3.10). Then there are two cases for the sign of z t( ). When z t( ) is eventually positive is similar to that of Theorem 3.2, we get that every solution U t( ) of (3.1) is oscillatory.

If z t( ) is eventually negative, there exists t1≥t0, such that z t( ) 0< for 1

t t. From (3.16), therefore,

( )

( ) (1 ) .

( )

p t G t

q t

α −

′ < Γ − (3.28)

Since ( ) 0 ( )

p t q t

− 

 

  and 0 ( ) ( )

t q tp t dt

∞−

< ∞

holds, then we obtain

( )

lim 0.

( )

t

p t q t

→∞ −

= (3.29)

Letting t→ ∞ in (3.28), we have

lim ( ) 0.

t→∞G t′ ≤ (3.30)

If lim ( ) 0

t→∞G t′ < , then there exists t2≥t1 such that G t′( ) 0< for t t≥ 2.

Thus, we get lim ( ) 0

t→∞G t = ≥A and G t( )≥A, t t≥ 2. Now we claim that A=0.

If not, that is A>0, then from (3.10), we derive

2

1 1

[ ( ) ( ( ))] m i i( ) ( ) m i i( ) , [ , ).

i i

a t g z t k a t G t k a t A t t

= =

′ ≤ −

≤ −

∈ ∞ (3.31)

Integrating both sides of (3.31) from t2 to t, we have

2 2

2 2 2

1 1

( ) ( ( )) ( ) ( ( )) m i tt i( ) m i tt i( ) [ , )

i i

a t g z t a t g z t A k a s ds A k a s ds t t

= =

≤ −

≤ −

, ∈ ∞,

(3.32) where a t g z t( ) ( ( )) 02 2 > , Hence, from (H2) and (3.32), we get

2

1 1 ( )

( ) ( ) ( ) 1 .

( ) ( ) (1 ) ( )

m t i t i i

A k a s ds

z t p t G t g

q t q t α M a t

− =

 

= + ≤

Γ −  

 

 

∑ ∫

(3.33)

Integrating both sides of (3.33) from t3 to t, we obtain

2

2 2

1

1 1

3

( )

( ) ( )

( ) ( ) (1 ) ,

( ) ( )

m t i t i

t t i

t t

k a s ds

p t g A

G t G t ds g du

q t M a t

δ

α − − =

   

  

   

≤ + Γ − +

   

 

 

∑ ∫

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DOI: 10.4236/jamp.2019.77096 1436 Journal of Applied Mathematics and Physics

using g−1( ) 0A <

and (3.27), as t→ ∞, lim ( )t G t

→∞ = −∞. which contradicts

the fact that G t( ) 0> . Therefore, we have A=0, that is lim ( ) 0

t→∞G t = . The

proof is complete.

Secondly, if U t( ) is an eventually negative solution of the fractional diffe-rential inequality (3.2). The proof when (3.2) is oscillatory is similar to that of above procedure, and hence is omitted.

Theorem 3.5 Assume that (H1) - (H4) and (3.26) hold, Let r t( ) and

( , )

H s t be defined as in Theorem 3.3 such that (3.23) holds. Furthermore, as-sume that (3.27) holds for every T =max{ , }t t2 3 . Then every solutions U t( ) of

(3.1) and (3.2) are oscillatory or satisfies lim ( ) 0

t→∞G t = or lim ( ) 0t→∞G t′ = , where ( )

G t is defined as Lemma 2.3.

Proof Suppose that U t( ) is a nonoscillatory solution of (3.1). Without loss of generality, assume that U t( ) is an eventually positive solution of (3.1), and proceeding as in the proof of Theorem 3.2 to get (3.11), there are two cases for the sign of z t( ).

When z t( ) is eventually positive, the proof is similar to that of Theorem 3.3.

( )

z t is eventually negative, the proof is similar to that of Theorem 3.4. Here we omitted it.

4. Oscillation of (1.1) and (1.3)

In the next we establish sufficient conditions for the oscillation of all solutions of (1.1), (1.3). For this we need the following:

The smallest eigenvalue λ0 of the Dirichlet problem

( ) ( ) 0,

( ) 0,

w x w x in

w x on

λ

∆ + = Ω

= ∂Ω (4.1)

is positive and the corresponding eigenfunction

φ

is positive in Ω.

Theorem 4.1 Let all the conditions of Theorem 3.2 and 3.3 be hold. Then every solution of (1.1) and (1.3) oscillates in E.

Proof Suppose that u x t( , ) is a nonoscillatory solution of (1.1) and (1.3). Without loss of generality, we may assume that u x t( , ) is an eventually positive solution of (1.1) and (1.3) in Ω×[ , )t0 ∞ for t0>0. Multiplying both sides of

the Equation (1.1) by φ( ) 0x > and then integrating with respect to x over

, we obtain for t t> 1,

[

]

. ) ( ) , ( )) , ( ( ) ( ) ( ) , ( ) ( ) ( ) ( ) , ( ) ( ) , ( ) ( )) , ( D ) ( ) ( ( ) ( 1 1 0 t, dx x t x u t x u h t b dx x t x u u h t b dx x ds s x u s t f t x a dx x t x u t q t p g t a dt d m i i i i i m i t i i φ τ τ φ φ φ α α

∑ ∫

∑∫

= Ω Ω = Ω − Ω + − ∆ − + ∆ =       + + (4.2)

Using Green's formula and boundary condition (1.3), it is obvious that

. , 0 ) ( ) , ( ) ( ) , ( ) ( ) ,

(x t xdx u x t xdx 0 u x t xdx t t1

u = ∆ =− ≤ ≥

Ω φ Ω φ λ Ω φ

(4.3) . , 0 ) ( ) , ( ) ( ) ,

(x t x dx 0 u x t x dx t t1

ui =− − i ≤ ≥

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DOI: 10.4236/jamp.2019.77096 1437 Journal of Applied Mathematics and Physics

By using Jensen's inequality and (H1) and (H4), we get

). ( ) ( ) ( ) ) ( ( ) ( ) , ( ) ( ) ( ) ( ) ( ) , ( ) ( ) , ( 1 0 0 t G dx x t a k ds dx x dx x s x u s t f dx x t a dx x ds s x u s t f t x a i i t i i t i i

Ω Ω − Ω − Ω Ω − ≥          − ≥       φ φ φ φ φ α α (4.5) Set

Ω − Ω

Ω  Ω=

 

 

= u xt xdx xdx where x dx

t

U() ( , )φ( ) φ( ) 1, φ( )

Combining (4.2)-(4.5), we obtain

. 0 ) ( ) ( ))] ( ( ) ( [ 1 ≤ +

= t G t a k t z g t a dt d i m i i (4.6)

The rest of the proof is similar to that of Theorems 3.2 and 3.3, and hence the details are omitted.

Theorem 4.1 Let the conditions of Theorem 3.4 hold. Then every solution

( )

U t of (1.1) and (1.3) is oscillatory or satisfies lim ( ) 0

t→∞G t = or lim ( ) 0t→∞G t′ = ,

where U t( ) is defined as Lemma 2.3.

Theorem 4.2 Let the conditions of Theorem 3.5 hold; Then every solution

( )

U t of (4.6) is oscillatory or satisfies lim ( ) 0

t→∞G t = or lim ( ) 0t→∞G t′ = , where ( )

G t is defined as Lemma 2.3.

The proofs of Theorem 4.1 and 4.2 are similar to that of Theorems 3.2-3.5 and hence the details are omitted.

5. Applications

Example 1 Consider the fractional differential equation

2 1 3 5

3 2 2 3

,t 0

1 8

(- D ( , )) ( ) ( ) ( , ) ( , )

3

( , ), 0

t t

t t u x t x t t s u x s ds e u x t

t t

u x t t

α α π − +   ∂ + + + =   ∂  

+∆ − >

(5.5)

Here, a t( )=t23, p t( ) 1

t

= − , q t( )=t−12,

2 3

( )

b t =t , 32 53

1( , ) 83

a x t =x + x and

( ) ( )

f x =g x =x.

Taking τi =π , Ω =1 (0, )π , m=1 , t0=1, k1=1 ,

β

=4 , M =9 , 2

1( ) min ( , )x 1

a t a x t t

∈Ω

= = .

Then, we get

0 1 2 1 3 1 1 ( ) , ( )

t g a t dt dt

t

= < ∞

( ) 1 32 0,

( ) 2

p t t

q t

− ′

−  = −

 

  0 1 1

2

( ) 1 .

( )

t q tp t dt dt

t

∞− ∞

= < ∞

It is clear that conditions (H1) - (H4) and (3.1) hold. Furthermore, taking

4 3

( )

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DOI: 10.4236/jamp.2019.77096 1438 Journal of Applied Mathematics and Physics 0

2 2

2

3 3

2

3

4 1

1 3

4

9 ( )

( )[ ( )] 8 3

limsup [ ( ) ( ) ] limsup [ ]

4 (1 ) ( ) 3

16 (1 )

m

t t

i i t

t i t

s s

Ma s r s

r s k a s ds s ds

r s

s

β α

α

→∞ = →∞

− = − = ∞

Γ −

Γ −

which satisfies condition (3.10). For every constant T t≥ 0, t∈[2 , )T ∞ , we

ob-tain

5 8

3 3

2 2

1 3 3

1 ( ) 1 8 1 , ,

( ) 3

m t t

i

T T T T T

i a s ds dt s ds dt t dt t

a t

t t

∞ ∞ ∞

=

   

 

= ≥ → ∞ → ∞

 

Which shows that (3.27) holds. Therefore, by Theorem 3.4 every solution of (5.5) is oscillatory or satisfies lim ( ) 0

t→∞G t = or lim ( ) 0.t→∞G t′ = .

Acknowledgements

This research was partially supported by grants from the National Basic Re-search Program of China, No. 41630643 and by the Science Foundation for The Excellent Youth Scholars of Ministry of Education of China, No. 11801530.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this pa-per.

References

[1] Yang J., Liu, A. and Liu, G. (2013) Oscillation of Solutions to Neutral Nonlinear Impulsive Hyperbolic Equations with Several Delays. Electronic Journal of Differen-tial Equations, 2013, 207-211.

[2] Liu, A., Ma, Q. and He, M. (2010) Oscillation of Nonlinear Impulsive Parabolic Eq-uations of Neutral Type. Rocky Mountain Journal of Mathematics, 36, 1011-1026. https://doi.org/10.1216/rmjm/1181069442

[3] Liu, A., Xiao, L., Liu, T. and Zou, M. (2007) Oscillation of Nonlinear Impulsive Hyperbolic Equation with Several Delays. Rocky Mountain Journal of Mathematics, 37, 1669-1684. https://doi.org/10.1216/rmjm/1194275940

[4] Xiang, S., Han, Z., Zhao, P. and Sun, Y. (2014) Oscillation Behavior for a Class of Differential Equation with Fractional-Order Derivatives. Abstract and Applied Analy-sis, 2014, Article ID: 419597. https://doi.org/10.1155/2014/419597

[5] Grace, S.R., Agarwal, R.P., Wong, P.J.Y. and Zafer, A. (2012) On the Oscillation of Fractional Differential Equations. Fractional Calculus and Applied Analysis, 15, 222-231. https://doi.org/10.2478/s13540-012-0016-1

[6] Han, Z., Zhao, Y., Sun, Y. and Zhang, C. (2013) Oscillation Theorem for a Kind of Fractional Differential Equations. Discrete Dynamics in Nature and Society, 2013, 216-219. https://doi.org/10.1155/2013/390282

[7] Wang, Y., Han, Z., Zhao, P. and Sun, S. (2014) On the Oscillation and Asymptotic Behavior for a Kind of Fractional Differential Equations. Advances in Difference Equations,2014, 50. https://doi.org/10.1186/1687-1847-2014-50

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DOI: 10.4236/jamp.2019.77096 1439 Journal of Applied Mathematics and Physics [9] Qi, C. and Huang, S. (2013) Interval Oscillation Criteria for a Class of Fractional Differential Equations with Damping Term. Mathematical Problems in Engineering, 2013, Article ID: 301085. https://doi.org/10.1155/2013/301085

[10]Zheng, B. (2013) Oscillation for a Class of Nonlinear Fractional Differential Equa-tions with Damping Term. Journal of Advanced Mathematical Studies, 6, 107-115. https://doi.org/10.1155/2013/912072

[11]Prakash, P., Harikrishnan, S. and Benchohra, M. (2015) Oscillation of Certain Non-linear Fractional Partial Differential Equation with Damping Term. Applied Ma-thematics Letters, 43, 72-79. https://doi.org/10.1016/j.aml.2014.11.018

References

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