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

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 excellent 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 equations 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 [

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

∂ ∂ t [ a ( t ) g ( p ( t ) + q ( t ) D + ,t α u ( x , t ) ) ] + ∑ i = 1 m a i ( x , t ) f i ( ∫ 0 t ( t − s ) − α u ( x , s ) d s ) = b ( t ) h ( u ) Δ u ( x , t ) + ∑ i = 1 m b i ( t ) h i ( u ( x , t − τ i ) ) Δ u ( x , t − τ i ) , t ≥ t 0 > 0 (1.1)

where ( x , t ) ∈ Ω × R + = E , Δ is the Laplacian in R n , Ω is a bounded domain in R n with a piecewise smooth boundary ∂ Ω . 0 < α < 1 is a real number and D + α u ( x , t ) 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

∂ u ( x , t ) ∂ N + γ ( x , t ) u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × R + , (1.2)

and

u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × R + . (1.3)

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

The following conditions are assumed to hold:

(H1) f i , g ∈ C ( R ; R ) are convex in [ 0 , ∞ ) and g is a monotone increasing function with x f i ( x ) > 0 , x g ( x ) > 0 for x ≠ 0 , there exist positive constants k i , β such that f i ( x ) / x > k i , x / g ( x ) > β for x ≠ 0 . And τ i ≥ 0 , u ≠ 0 , h ( u ) > 0 , h i ( u ) > 0 , u h ' ( u ) > 0 , u h i ' ( u ) > 0 , h ( 0 ) > 0 , h i ( 0 ) > 0 .

(H2) a , a i , b , b i and q are positive continuous functions on t ∈ [ t 0 , ∞ ) for a certain t 0 > 0 , and p is a nonpositive continuous function on t ∈ [ t 0 , ∞ ) for a certain t 0 > 0 . There exists a constant M > 0 , q ( t ) ≤ M for t 0 > 0 . And

( − p ( t ) q ( t ) ) ′ ≠ 0 , t ∈ [ t 0 , ∞ ) , ∫ t 0 ∞ − p ( t ) q ( t ) d t < ∞ .

(H3) g - 1 ∈ C ( R ; R ) is continuous function with s g − 1 ( s ) > 0 for s ≠ 0 , there exists positive constant δ such that g − 1 ( u v ) ≤ δ g − 1 ( u ) g − 1 ( v ) for u v < 0 , and g − 1 ( u v ) ≥ δ g − 1 ( u ) g − 1 ( v ) for u v > 0 .

(H4) a i ∈ C ( E ¯ ; R + ) , and a i ( t ) = min x ∈ Ω ¯ a i ( x , t ) .

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

u ∈ C 1 + α ( E ¯ ; R + ) with ∫ 0 t u ( x , s ) ( t − s ) − α d s ∈ C 1 ( E ¯ ; R + ) ,

r ( t ) g ( p ( t ) + q ( t ) D + , t α u ( x , t ) ) ∈ C 1 ( 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 positive nor eventually negative, otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

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

Definition 2.1 [

( I 0 + α f ) ( t ) = 1 Γ ( α ) ∫ 0 t ( t - v ) α - 1 f ( v ) d v ,

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

Definition 2.2 [

( D + α u ) ( x , t ) = ∂ ∂ t 1 Γ ( 1 − α ) ∫ 0 t ( t − v ) − α u ( x , v ) d v .

Lemma 2.3 [

G ( t ) = ∫ 0 t ( t − v ) − α u ( v ) d v , α ∈ ( 0 , 1 ) , t > 0

Then

G ' ( t ) = Γ ( 1 − α ) ( D + α u ) ( t ) , α ∈ ( 0 , 1 ) , t > 0

Lemma 2.4 [

m X Y m − 1 − X m ≤ ( m − 1 ) Y m

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

For the sake of convenience, we set

U ( t ) = 1 | Ω | ∫ Ω u ( x , t ) d x , w h e r e | Ω | = ∫ Ω d x

z ( t ) = p ( t ) + q ( t ) ( D + α U ) ( t )

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

d d t [ a ( t ) g ( z ( t ) ) ] + ∑ i = 1 m k i a i ( t ) G ( t ) ≤ 0 (3.1)

has no eventually positive solution and the fractional differential inequality

d d t [ a ( t ) g ( z ( t ) ) ] + ∑ i = 1 m k i a i ( t ) G ( t ) ≥ 0 (3.2)

has no eventually negative solution, every solution of (1.1) and (1.2) is oscillatory 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 Ω × [ t 0 , ∞ ) . Integrating (1.1) with respect to x over Ω , we obtain

∫ Ω d d t [ a ( t ) g ( p ( t ) + q ( t ) D + ,t α u ( x , t ) ) ] d x + ∑ i = 1 m ∫ Ω a i ( x , t ) f i ( ∫ 0 t ( t − s ) − α u ( x , s ) d s ) d x = b ( t ) ∫ Ω h ( u ) Δ u ( x , t ) d x + ∑ i = 1 m b i ( t ) ∫ Ω h i ( u ( x , t − τ i ) ) Δ u ( x , t − τ i ) d x . (3.3)

Using Green’s formula and boundary condition (1.2), it is obvious that

∫ Ω h ( u ) Δ u ( x , t ) d x = ∫ ∂ Ω h ( u ) ∂ u ( x , t ) ∂ N d s − ∫ Ω h ′ ( u ) | g r a d u | 2 d x = − ∫ ∂ Ω γ ( x , t ) u h ( u ) d s − ∫ Ω h ′ ( u ) | g r a d u | 2 d x ≤ 0 , t ≥ t 1 (3.4)

∫ Ω h i ( u ( x , t − τ i ) ) Δ u ( x , t − τ i ) d x ≤ 0 , t ≥ t 1 , (3.5)

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

∫ Ω a i ( x , t ) f i ( ∫ 0 t ( t − s ) − α u ( x , s ) d s ) d x ≥ a i ( t ) ∫ Ω d x f i [ ∫ 0 t ( t − s ) − α ( ∫ Ω u ( x , s ) d x ( ∫ Ω d x ) − 1 ) d s ] ≥ k i a i ( t ) ∫ Ω d x G ( t ) (3.6)

Combining (3.3)?(3.6), we obtain

d d t [ a ( t ) g ( z ( t ) ) ] + ∑ i = 1 m k i a i ( t ) G ( t ) ≤ 0 (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

U ( t ) = 1 | Ω | ∫ Ω u ( x , t ) d x is an eventually negative solution of the Equation (3.2).

This completes the proof.

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

∫ t 0 ∞ g − 1 ( 1 a ( t ) ) d t = ∞ (3.8)

hold. if there exists a positive function r ∈ C 1 [ t 0 , ∞ ) such that

lim sup t → ∞ ∫ t 0 t [ r ( t ) ∑ i = 1 m k i a i ( s ) − M a ( s ) [ r ′ ( s ) ] 2 4 β Γ ( 1 − α ) r ( s ) ] d s = ∞ (3.9)

where k i , 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

[ a ( t ) g ( z ( t ) ) ] ′ ≤ − ∑ i = 1 m k i a i ( t ) G ( t ) ≤ 0 , t ∈ [ t 1 , ∞ ) . (3.10)

Thus, a ( t ) g ( z ( t ) ) is strictly decreasing on a ( t ) > 0 . Since a ( t ) > 0 for t ∈ [ t 1 , ∞ ) and (H1), we see that z ( t ) is eventually of one sign. We claim that

z ( t ) > 0 , t ∈ [ t 1 , ∞ ) . (3.11)

If not, there exists t 2 ≥ t 1 such that z ( t 2 ) < 0 . Since a ( t ) g ( z ( t ) ) is strictly decreasing on [ t 1 , ∞ ) and it is clear that a ( t ) g ( z ( t ) ) ≤ a ( t 2 ) g ( z ( t 2 ) ) = c < 0 , where c is a constant for t ∈ [ t 2 , ∞ ) . Therefore, we have

z ( t ) ≤ g − 1 ( c a ( t ) ) . (3.12)

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

z ( t ) q ( t ) = p ( t ) q ( t ) + D + α U ( t ) < g − 1 ( c a ( t ) ) q ( t ) ≤ δ g − 1 ( c ) g − 1 ( 1 a ( t ) ) M , (3.13)

Integrating the above inequality from t 2 to t , from Lemma 2.3, we have

∫ t 2 t ( p ( t ) q ( t ) + G ′ ( s ) Γ ( 1 − α ) ) d s < ∫ t 2 t δ g − 1 ( c ) g − 1 ( 1 a ( t ) ) M d s , (3.14)

which yields

G ( t ) ≤ G ( t 2 ) + Γ ( 1 − α ) [ ∫ t 2 t − p ( t ) q ( t ) d s + δ g − 1 ( c ) M ∫ t 2 t g − 1 ( 1 a ( t ) ) d s ] . (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

z ( t ) = p ( t ) + q ( t ) ( D + α U ) ( t ) = p ( t ) + q ( t ) G ′ ( t ) Γ ( 1 − α ) , (3.16)

therefore,

G ′ ( t ) = Γ ( 1 − α ) z ( t ) - p ( t ) q ( t ) ≥ Γ ( 1 − α ) z ( t ) q ( t ) ≥ Γ ( 1 − α ) z ( t ) M (3.17)

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

w ( t ) = r ( t ) a ( t ) g ( z ( t ) ) G ( t ) , t ∈ [ t 1 , ∞ ) . (3.18)

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

w ′ ( t ) = ( r ( t ) G ( t ) ) ′ [ a ( t ) g ( z ( t ) ) ] + r ( t ) G ( t ) [ a ( t ) g ( z ( t ) ) ] ′ ≤ r ′ ( t ) r ( t ) w ( t ) − r ( t ) ∑ i = 1 m k i a i ( t ) − Γ ( 1 − α ) z ( t ) M w ( t ) G ( t ) ≤ r ′ ( t ) r ( t ) w ( t ) − r ( t ) ∑ i = 1 m k i a i ( t ) − z ( t ) g ( z ( t ) ) Γ ( 1 − α ) w 2 ( t ) M r ( t ) a ( t ) ≤ r ′ ( t ) r ( t ) w ( t ) − r ( t ) ∑ i = 1 m k i a i ( t ) − β Γ ( 1 − α ) M r ( t ) a ( t ) w 2 ( t ) . (3.19)

Taking

m = 2 , X = β Γ ( 1 − α ) M r ( t ) a ( t ) w ( t ) , Y = 1 2 M r ( t ) a ( t ) β Γ ( 1 − α ) r ′ ( t ) r ( t ) (3.20)

from Lemma 2.4 and (3.19), we get

w ′ ( t ) ≤ − r ( t ) ∑ i = 1 m k i a i ( t ) + M a ( t ) [ r ′ ( t ) ] 2 4 β Γ ( 1 − α ) r ( t ) , (3.21)

Integrating both sides of the inequality (3.21) from t 0 to t , and taking the limit supremum of both sides of the above inequality as t → ∞ , we get

lim sup t → ∞ ∫ t 0 t [ r ( t ) ∑ i = 1 m k i a i ( s ) − M a ( s ) [ r ′ ( s ) ] 2 4 β Γ ( 1 − α ) r ( s ) ] d s < w ( t 0 ) < ∞ . (3.22)

Which contradicts (3.9). The proof is complete.

Secondly, if U ( t ) is an eventually negative solution of the fractional differential inequality (3.2) and there exists G ( t ) < 0 , t ∈ [ t 1 , ∞ ) . When (3.2) is oscillatory 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 r ∈ C 1 [ t 0 , ∞ ) and a function H ∈ C ( D , R ) , where D : = { ( s , t ) : s ≥ t ≥ t 0 } , such that

H ( t , t ) = 0 , f o r t ≥ t 0

H ( s , t ) = 0 , f o r ( s , t ) ∈ D 0

where D 0 : = { ( s , t ) : s ≥ t ≥ t 0 } and H has a nonpositive continuous partial derivative H ′ t ( s , t ) = ∂ H ( s , t ) / ∂ t with respect to the second variable and satisfies

lim sup s → ∞ 1 H ( s , t 0 ) ∫ t 0 s H ( s , t ) [ r ( t ) ∑ i = 1 m k i a i ( t ) − M a ( t s ) [ r ′ ( t ) ] 2 4 β Γ ( 1 − α ) r ( t ) ] d t = ∞ (3.23)

where k i , 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 t 0 to s , for s ∈ [ t 0 , ∞ ) , we derive

∫ t 0 s H ( s , t ) [ r ( t ) ∑ i = 1 m k i a i ( t ) − M a ( t s ) [ r ′ ( t ) ] 2 4 β Γ ( 1 − α ) r ( t ) ] d t ≤ − [ H ( s , t ) w ( t ) ] | t 0 s + ∫ t 0 s H ′ t ( s , t ) w ( t ) d t ≤ H ( s , t 0 ) w ( t 0 ) , (3.24)

Therefore,

1 H ( s , t 0 ) ∫ t 0 s H ( s , t ) [ r ( t ) ∑ i = 1 m k i a i ( t ) − M a ( t s ) [ r ′ ( t ) ] 2 4 β Γ ( 1 − α ) r ( t ) ] d t ≤ w ( t 0 ) < ∞ , (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 differential 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

∫ t 0 ∞ g − 1 ( 1 a ( t ) ) d t < ∞ (3.26)

which is different from (3.8). In this case, we have the following results.

Theorem 3.4 Assume that (H1)?(H4) and (3.26) hold, and that there exist a positive function r ∈ C 1 ( [ t 0 , ∞ ) ; R ) such that (3.9) holds. If for every constant T = max { t 2 , t 3 } , such that

∫ T ∞ ( 1 a ( t ) ∑ i = 1 m ∫ T t a i ( s ) d s ) d t = ∞ (3.27)

Then every solutions U ( t ) of (3.1) and (3.2) are oscillatory or satisfies lim t → ∞ G ( t ) = 0 or lim t → ∞ G ′ ( t ) = 0 , where G ( t ) is defined as Lemma 2.3.

Proof Suppose that U ( t ) is a nonoscillatory solution of (3.1). We may assume 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

Since

Letting

If

Integrating both sides of (3.31) from

where

Integrating both sides of (3.33) from

using

Secondly, if

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

Proof Suppose that

When

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

is positive and the corresponding eigenfunction

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

Proof Suppose that

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

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

Set

Combining (4.2)-(4.5), we obtain

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

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

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.

Example 1 Consider the fractional differential equation

Here,

Taking

Then, we get

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

which satisfies condition (3.10). For every constant

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

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

The authors declare no conflicts of interest regarding the publication of this paper.

Feng, Q. and Liu, A.P. (2019) Oscillation for a Class of Fractional Differential Equation. Journal of Applied Mathematics and Physics, 7, 1429- 1439. https://doi.org/10.4236/jamp.2019.77096