OSCILLATION PROPERTIES OF SOLUTIONS
OF FRACTIONAL DIFFERENCE EQUATIONS
by
Mustafa BAYRAM a* and Aydin SECER b
a Department of Computer Engineering, Istanbul Gelisim University, Istanbul, Turkey b Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey
Original scientific paper https://doi.org/10.2298/TSCI181017342B
In this article, studied the properties of the oscillation of fractional difference equa-tions, and we obtain some results. The results we obtained are an expansion and further development of highly known results. Then we showed them with examples.
Key words: fractional difference equation, oscillatory solutions,
oscillation theory Introduction and preliminaries
In the investigations of qualitative properties for differential equations, research on time scales of the dynamic equations, oscillation of differential (or difference) equations and fractional differential equations have been a very important issue in the science and engineer-ing. We refer to [1-25] and the references therein.
We first investigated following fractional difference equations:
{
1}
2(
)
0 1 1 1 ( ) ( ) ( ) ( ) ( ) 0 1 i n t i s t i a t t x t q t x s t s η δ α δ α α ψ + − = = ∆ ∆ ∆ + = − − ∑
∑
(1)We can rewrite eq. (1):
{
1}
2 1 ( ) ( ) ( ) n ( ) i( ) 0 i i a t ψ t αx t δ δ q t G tη = ∆ ∆ ∆ + = ∑
(2) where 0 0 1 1 1 1 2 N , ( ) ( ), , ( 1) t t s t t G t x s t s α α α δ δ + − + − = ∈ = − −∑
and ηi are the division of two odd positive integers. The ψ( ), ( )t a t , and q ti( ) are positive coef-ficient sequences, and ∆α demonstrate that the Riemann-Liouville fractional difference
opera-tor of order α where 0< ≤α 1. Therefore, in our results we use the following conditions: C1. 1 0 1/ 1 ( ) s t ψ δ s ∞ = = ∞
∑
and 2 0 1/ 1 ( ) s t a δ s ∞ = = ∞∑
(3)C2. 1 0 1/1( ) s t ψ δ s ∞ = < ∞
∑
and 2 0 1/1( ) s t a δ s ∞ = < ∞∑
. (4)By a solution of eq. (2), we mean a real-valued sequence x t
( )
satisfying eq. (2) for 0t
t ∈ . A solution x t( ) of eq. (2) is called oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called non-oscillatory. Equation (2) is called oscillatory if all its solutions are oscillaory.
Definition 1. [26]. We define vth fractional sum f as:
[
]
1 1 ( ) ( ) t v( 1) ( ), 0 v v s a f t v − − t s f s v − − = ∆ = Γ∑
− − > (5)where we define f for s a≡ mod(1), ∆−vf for t≡(a v+ )mod(1) and t( )v = Γ + Γ + −(1 )/ (1t t v). The fractional sum ∆−vf maps functions defined on
a
to functions defined on a v+ , where
{ , 1, 2,...} t = t t+ t+
.
Definition 2. [26] Let m− < <1 µ m and v >0, where m denotes a positive integer,
m= µ . Set v m= −µ. Then we define that µth fractional difference:
( ) m v ( ) m v ( )
f t f t f t
µ − −
∆ = ∆ = ∆ ∆ (6)
Oscillation properties of equation (2)
In this section, we work the oscillation properties of equation (2).
Lemma 1. [22]. Suppose that x t( ) be a solution of eq. (2) and let:
( )
(
)
( )
0 1 ( ) 1 t s t G t α+ − t s −α x s = =∑
− − (7) then[
G t( )]
(1 α) αx t( ) ∆ = Γ − ∆ (8)Theorem 1. Assume C1 holds and furthermore, for all suficiently large t:
( )
( )
( )
1 2 3 1/ 1/ 1 1 n 1 i s t s i q s a δ δ τ ζ τ ζ ψ τ ∞ ∞ ∞ = = = = = ∞ ∑
∑ ∑
∑
(9) and( )
(
1( )
)
3 2 1 1/ 1 1 i n s i i s t t q s η δ τ α ψ τ ∞ − = = = Γ − = ∞ ∑∑
∑
(10)Then every solution of eq. (2) is either oscillatory or limt→∞G t( ) 0= .
Proof. Assume that the contrary that x t( ) is non-oscillatory solution of eq. (2). Then without loss of generality, we may assume that there is a solution x t( ) of eq. (2) such that
( ) 0
x t > on [ , )t ∞1 , where t1 is sufficiently large, so that G t >( ) 0 on [ , )t ∞1 . And all of q ti( )’s are not identically zero on [ , )t ∞1 for i=1,2,...,n. From eq. (2), we have:
{
}
2 1 1 ( ) ( ) ( ) ( ) i( ) 0 n i i a t ψ t αx t δ δ q t G tη = ∆ ∆ ∆ = − < ∑
(11)In that case
{
1}
2 ( ) ( ) ( ) a t ∆ψ t ∆αx t δ δ is an eventually non-increasing sequence on [ , )t ∞1 . So, we understand that ∆{ ( )[ψ t ∆αx t( )] }δ1
and ∆αx t( ) are ultimately of one sign. For
2 1
t >t is big enough, ∆{ ( )[ψ t ∆αx t( )] }γ1 and ∆αx t( ) have a fixed sign on [ , )t ∞2 . We then consider the following conditions:
– Case 1. ∆αx t( ) 0< and ∆{ ( )[ψ t ∆αx t( )] } 0δ1 < ;
– Case 2. ∆{ ( )[ψ t ∆αx t( )] } 0δ1 < and 0< ∆αx t( ); – Case 3. ∆{ ( )[ψ t ∆αx t( )] } 0δ1 > and 0> ∆αx t( ); – Case 4. ∆{ ( )[ψ t ∆αx t( )] } 0δ1 > and 0< ∆αx t
( )
.For the Case 1, we have:
( )
(
)
(
( )
)
{
}
(
( )
)
( )
( )
{
}
1 1 1 2 1 1 1 2 1/ 1 2 2 1/ 1/ 1 2 2 1/ ( ) ( ) 1 1 ( ) 1 1 ( ) t s t t s t s x s G t G t G t s t x t s δ δ α δ δ δ α δ ψ α α ψ α ψ ψ − = − = ∆ = + ≤ + Γ − Γ − Γ − + ∆ ∑
∑
Then, by C1, we obtain limt→∞G t( )= −∞ which contradicts with 0<G t( ). For the Case 2, we have from eq. (9):
( )
( )
( )
( )
( )
{
( )
( )
}
( )
( )
( )
( )
{
( )
( )
}
( )
2 2 1 1 1 2 2 2 2 1 1 2 2 1/ 1 2 2 1/ 1/ 1 2 2 2 2 2 1/1 t s t t s t a s s x s t x t t x t a s t x t a t t x t a s δ δ δ α δ δ α α δ δ δ δ δ α α δ ψ ψ ψ ψ ψ − = − = ∆ ∆ ∆ = ∆ + ≤ ≤ ∆ + ∆ ∆ ∑
∑
Then, by C1, we obtain lim ( )[ ( )]1
t→∞ψ t ∆αx t γ = −∞ which contradicts with 0< ∆αx t( ).
For the Case 3, we have limt→∞G t( )=k1≥0 and limt→∞ψ( )[t ∆αx t( )]δ1 =k2≤0. If
we suppose that k >1 0, then G t( )>k1 for t t≤ ≤3 t2. Therefore, if we sum both sides of eq. (2)
from t to ∞, we obtain:
( )
{
1}
2( )
( )
( )
1 1 1 ( ) ( ) i i n n i i s t i i s t a t ψ t αx t δ δ ∞ q s G sη kη ∞q s = = = = ∆ ∆ < − ≤ − ∑∑
∑ ∑
that is:{
1}
( )
( )
2 1/ 1 1 ( ) ( ) n i i i s t k t x t q s a t δ η δ α ψ ∞ = = ∆ ∆ ≤ − ∑
∑
(12)If we sum both sides of the eq. (12) from t to ∞, we have: 2 1 1/ 1 2 1 ( ) ( ) ( ) ( ) i n i s t i s k t x t k q a s δ η δ α τ ψ ∞ ∞ τ = = = ∆ − ≤ −
∑ ∑
∑
which means for k ≤2 0: 1 2 1/ 1/ 1 1 1 1 ( ) ( ) n i ( ) ( ) i s t i s x t t k a s q δ δ η α τ ψ − ∞ − ∞ τ = = = ∆ ≤ −
∑ ∑
∑
(13)If we sum both sides of the eq. (13) from t3 to t −1, we obtain:
( )
(
)
(
( )
)
( )
( )
( )
1 2 3 1/ 1/ 1 3 1 1 1 1 1 i t n i s t s i G t G t k q s a δ δ η τ ζ τ ζ α α ψ τ − ∞ ∞ = = = = ≤ − Γ − Γ − ∑
∑ ∑
∑
Therefore, by eq. (9), we obtain limt→∞G t( )= −∞ with contradicts with G t >( ) 0. For the Case 4, we have:
( )
(
)
(
( )
)
( )
( )
{
}
( )
{
( )
( )
}
( )
1 1 1 1 1 1 2 2 1/ 1/ 1 1 2 2 2 1/ 1/ 1 0 1 1 t t s t s t s x s G t G t t x t s s δ δ α γ δ α δ δ ψ ψ α α ψ ψ − − = = ∆ = + > ∆ > Γ − Γ −∑
∑
That is:( )
( )
{
}
(
)
( )
( )
1 1 1 2 1/ 1 2 2 1/ 1 i i t s t t x t G t s η δ δ η α δ α ψ ψ − = Γ − ∆ < ∑
Then from eq. (2):
( )
{
( )
( )
}
(
)
( )
( )
{
( )
( )
}
2 1 1 1 1 2 1/ 1 2 2 1/ 1 1 i n t i i s t q t t x t a t t x t s η δ δ δ δ α α δ α ψ ψ ψ − = = Γ − ∆ < −∆ ∆ ∆ ∑
∑
(14)If we sum both sides of the eq. (14) from t3 to t −1, we obtain:
( )
( )
{
1}
1( )
(
( )
)
( )
{
( )
( )
1}
2 1 3 2 1/ 1 1 2 2 1/ 3 3 3 1 1 i i n t s i i s t t t x t q s a t t x t η η δ δ δ δ α α δ τ α ψ ψ ψ τ − − = = = Γ − ∆ < ∆ ∆ ∑
∑
∑
If we take t → ∞, we get a contradiction with eq. (10). Therefore, the proof of the
Theorem 1 is complete
Theorem 2. Suppose that C2, eqs. (9) and (10) hold. Furthermore, for all sufficiently
large t:
(
)
( )
( )
( )
(
( )
)
1 2 1 4 3 2 1/ 1/ 1 1 1/ 1 1 s n 1 1 i i s t t i t q s a δ δ η τ δ τ ζ ξ ζ α α ζ ψ τ ψ ξ ∞ − − ∞ = = = = = Γ − Γ − = ∞ ∑
∑ ∑
∑
∑
(15) and( )
( )
( )
(
( )
)
2 1 2 1 4 3 2 1/ 1/ 1 1 1/ 1/ 1 1 1 s n 1 i i s t t i t q a s a δ η δ τ δ δ τ ζ τ ζ α τ ζ ψ ζ ∞ − ∞ − = = = = = Γ − = ∞ ∑
∑ ∑
∑
∑
(16)Therefore, each solution of eq. (2) is either limt→∞G t( ) 0= or oscillatory.
Proof. Let’s the contrary that x t( ) is non-oscillatory solution of eq. (2). Then with-out loss of generality, we assume that there is a solution x t( ) of eq. (2) such that 0<x t( ) on
1
[ , )t ∞ , where t1 is sufficiently large, so that G t >( ) 0 on [ , )t ∞1 . It appears that all of q ti( )’s are not identically zero on [ , )t ∞1 for i=1,2,...,n. From eq. (11), we obtained that
1 2 ( )( { ( )[ ( )] })
a t ∆ψ t ∆αx t δ δ is an eventually non-increasing sequence on
[
)
1,t ∞ . For the Case 1, we have:
( )
(
)
{
( )
( )
}
( )
{
( )
( )
}
( )
( )
( )
{
}
( )
( )
1 1 1 1 1 1 1 1 1 1 1/ 1 1/ 1 1/ 1/ 1/ 1 1 1/ 1/ 1 1 1 1 s t s t s t s t G t s x s s t x t s t x t s K s δ δ δ δ α δ α δ δ δ α δ δ ψ ψ ψ ψ α ψ ψ ψ ∞ − ∞ − = = ∞ − ∞ − = = −Γ − < ∆ < ∆ < < ∆ = ∑
∑
∑
∑
Then from the last inequality and eq. (2), we obtain:
( )
{
( )
( )
1}
2( ) (
)
( )
1 1 1/ 1 1 1 i n i i s t a t t x t q t K s η δ δ α δ ψ α ψ ∞ = = ∆ ∆ ∆ < Γ − ∑
∑
(17)If we sum both sides of the eq. (17) from t2 to t −1:
( )
{
( )
( )
1}
2(
(
)
)
( )
( )
1 2 1 1 1/ 1 1 1 i i n t i i s t s a t t x t K q s η δ δ η α δ τ ψ α ψ τ − ∞ = = = ∆ ∆ < Γ − ∑
∑
∑
that is:( )
( )
{
}
(
)
( )
( )
( )
2 1 1 3 1/ 1 1 1/ 1 1 i 1 i n t i i s t s K t x t q s a t δ η η δ α δ τ α ψ ψ τ − ∞ = = = Γ − ∆ ∆ < ∑
∑
∑
(18)If we sum both sides of the eq. (18) from t3 to t −1:
( )
( )
(
( )
)
( )
( )
2 1 1 3 2 1/ 1 1 1 1/ 1 1 i 1 i t n s i s t i t K t x t q a s δ η η δ α δ τ ζ τ α ψ τ ψ ζ − − ∞ = = = = Γ − ∆ < ∑ ∑
∑
∑
then we get:( )
(
( )
)
(
( )
)
( )
( )
1 2 1 3 2 1/ 1/ 1 1 1 1/ 1 1 1 t n i s 1 i i s t i t K G t q t a s δ δ η η δ τ ζ τ α α τ ψ ψ ζ − − ∞ = = = = Γ − Γ − ∆ < ∑ ∑
∑
∑
(19)If we sum both sides of the the eq. (19) from t4 to t −1, we have:
( )
( )
(
( )
)
(
( )
)
( )
( )
1 2 1 4 3 2 1/ 1/ 1 1 1 1 4 1/ 1 1 1 i 1 i t s n i s t t i t K G t G t q s a δ δ η η τ δ τ ζ ξ ζ α α ζ ψ τ ψ ξ − − − ∞ = = = = = Γ − Γ − − < ∑
∑ ∑
∑
∑
By eq. (14), we obtain limt→∞G t( )= −∞ due to K <1 0, which conradicts with
0<G t( ).
For the Case 2:
( )
(
) ( )
{
( )
}
( )
(
) ( )
{
( )
}
( )
1 1 1 1 1 1 2 2 1/ 1/ 1 1 1/ 1/ 1 1 1 t t s t s t s x s G t t x t s s δ δ α δ δ α δ δ α ψ α ψ ψ ψ − − = = Γ − ∆ >∑
> Γ − ∆ ∑
and( )
( )
( )
{
( )
( )
}
( )
( )
{
( )
( )
}
( )
( )
{
( )
( )
}
( )
( )
2 2 1 1 2 2 2 1 2 2 2 1 2 2 1/ 1/ 1/ 1/ 1/ 2 2 2 1/ 2 1/ 1 1 1 s t s t s t s t a s s x s t x t a s a t t x t a s a t t x t K a s a s δ δ δ α δ α δ δ δ δ α δ δ δ δ α δ δ ψ ψ ψ ψ ∞ = ∞ = ∞ ∞ = = ∆ ∆ − ∆ ≤ < < ∆ ∆ < < ∆ ∆ = ∑
∑
∑
∑
Thereore, we have:( )
(
)
( )
( )
1 2 1 2 1/ 1 2 1/1 1/1 1 t s t s t G t K a s s δ δ δ α ψ ∞ − = = > −Γ − ∑
∑
Thus, from eq. (2), we obtain:
( )
{
( )
( )
}
( ) (
)
( )
( )
1 2 1 2 1 2 1/ 1 2 1/ 1/ 1 1 1 1 i n t i i s t s t a t t x t q t K a s s η δ δ δ α δ δ ψ α ψ ∞ − = = = ∆ ∆ ∆ = Γ − ∑
∑
∑
(20)If we sum two sides of the eq. (20) from t3 to t −1 , we have:
( )
{
( )
( )
1}
2( ) (
)
( )
1( )
2 1 3 2 1/ 1 1 2 1/ 1/ 1 1 1 1 i t n s i s t i s t a t t x t q s K a η δ δ δ α δ δ τ τ ψ α τ ψ τ − ∞ − = = = = ∆ ∆ = Γ − ∑ ∑
∑
∑
Then:( )
( )
( )
( ) (
)
( )
( )
2 1 1 2 1 4 3 2 1/ 1/ 1 1 1 4 4 2 1/ 1/ 1 1 1 1 1 i t s n i s t t i t t x t q K a s a δ η δ τ δ α δ δ τ ζ τ ζ ψ τ α ζ ψ ζ − − ∞ − = = = = = ∆ = Γ − ∑
∑ ∑
∑
∑
letting t → ∞, we obtain:( )
( ) (
)
( )
( )
2 1 2 1 4 3 2 1/ 1/ 1 1 2 1/ 1/ 1 1 s n 1 1 1 i i s t t i t q K a s a δ η δ τ δ δ τ ζ τ ζ τ α ζ ψ ζ ∞ − ∞ − = = = = = Γ − < ∞ ∑
∑ ∑
∑
∑
which contradicts with eq. (16). The rest of the proof is made similar to the proof of the
Theo-rem 1. Thus the proof of the theoTheo-rem is completed.
Application
Let as consider the following fractional difference equation as an example:
( )
{
}
(
)
( )( )
0 3 1 7 2 1 t 1 0, 2 s t t x t s t x s t t α α α − + − = ∆ ∆ ∆ + ∑
− + − = ≤ (21) This corresponds to eq. (2) with t =0 2, δ1=7, δ2=1/2, α∈(0,1], a t( )=t1/2, ψ( ) 1,t = 2 ( ) 1/ , 1 q t = t n= , and η1=3. However,( )
1 0 0 1/ 1 1 s t ψ γ s s t ∞ ∞ = = = = ∞∑
∑
and( )
2 0 2 1/ 1 1 s t a δ s s t s ∞ ∞ = = = = ∞∑
∑
Then C1 holds. So, we have:
( )
( )
( )
1 2 3 3 1/ 1/7 1/ 2 2 1/2 1 1 n 1 1 i s t s i s t s q s a δ δ τ ζ τ τ ζ τ ζ ζ ψ τ τ ∞ ∞ ∞ ∞ ∞ ∞ − = = = = = = = = = ∞ ∑
∑ ∑
∑
∑ ∑
∑
and( )
(
1( )
)
(
)
(
)(
)
3 2 3 3 3 1 1 3 1/ 2 2 1 2 1 i 1 1 1 1 2 n s s i i s t t s t s t q s s s s η δ τ τ α α α ψ τ ∞ − ∞ − ∞ = = = = = = Γ − = Γ − = Γ − − = ∞ ∑∑
∑
∑ ∑
∑
Therefore, eqs. (9) and (10) holds, and then we say that eq. (21) is limt→∞G t( ) 0= or oscillatory by Theorem 1.
Conclusion
In this work, we studied the qualitative behavior of solutions of non-linear fractional difference equations (FDE) with fractional Riemann-Liouville difference operator. Because there was a gap for the oscillatory solutions of FDE under the condition (C2) in the literature, we con-sidered the equation with the conditions (C1) and (C2). By using some techniques, we obtained some oscillation results. The obtained results improved the many criteria in the literature. References
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Paper submitted: October 17, 2018 Paper revised: October 30, 2018 Paper accepted: November 26, 2018
© 2019 Society of Thermal Engineers of Serbia Published by the Vinča Institute of Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4.0 terms and conditions
