Abstract
In this paper, we consider the oscillation behavior of solutions of the following fractional difference equation:
where \(t\in \mathbf{N}_{t_{0}+1-\alpha }\), \(G ( t ) = \sum_{s=t_{0}}^{t-1+\alpha } ( t-s-1 ) ^{-\alpha }x ( s ) \), and \(\Delta^{\alpha }\) denotes a Riemann–Liouville fractional difference operator of order \(0<\alpha \leq 1\). By using the generalized Riccati transformation technique, we obtain some oscillation criteria. Finally we give an example.
Similar content being viewed by others
1 Introduction and preliminaries
Fractional differential (or difference) equations are a more general form of differential equations with integer order. And there is an increasing interest in the study of them due to some important contributions [1, 2].
Many authors have been focused on various equations like ordinary and partial differential equations [3,4,5,6], difference equations [7,8,9], dynamic equations on time scales [10,11,12,13,14], and fractional differential (difference) equations [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] obtaining some oscillation criteria. Recently, oscillation studies have become a very hot topic. That is why, we consider the following fractional difference equation:
where \(t\in \mathbf{N} _{t_{0}+1-\alpha }\), \(G ( t ) = \sum_{s=t_{0}}^{t-1+\alpha } ( t-s-1 ) ^{ ( -\alpha ) }x ( s ) \), \(c ( t ) \), \(a ( t ) \), \(r ( t ) \), and \(q ( t ) \) are positive sequences, and \(\Delta^{\alpha }\) denotes the Riemann–Liouville fractional difference operator of order \(0<\alpha \leq 1\).
By a solution of Eq. (1), we mean a real-valued sequence \(x ( t ) \) satisfying Eq. (1) for \(t\in \mathbf{N} _{t_{0}}\). A solution \(x ( t ) \) of Eq. (1) is called oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called non-oscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.
Definition 1
([32])
Let \(v>0\). The vth fractional sum f is defined by
where f is defined for \(s\equiv a \mathbf{mod} ( 1 ) \), \(\Delta^{-v}f\) is defined for \(t\equiv ( a+v ) \mathbf{mod} ( 1 ) \), and \(t^{ ( v ) }=\frac{\varGamma ( t+1 ) }{\varGamma ( t-v+1 ) }\). The fractional sum \(\Delta^{-v}f\) maps functions defined on \(\mathbf{N} _{a}\) to functions defined on \(\mathbf{N} _{a+v}\), where \(\mathbf{N} _{t}= \{ t,t+1,t+2,\ldots \} \).
Definition 2
([32])
Let \(v>0\) and \(m-1<\mu <m\), where m denotes a positive integer, \(m= \lceil \mu \rceil \). Set \(v=m-\mu \). The μth fractional difference is defined as
where \(\lceil \mu \rceil \) is the ceiling function of μ.
Lemma 1
([33])
Assume that A and B are nonnegative real numbers. Then
for all \(\lambda >1\).
2 Main results
Throughout this paper, we denote
For simplification, we consider
and
Lemma 2
([28])
Let \(x ( t ) \) be a solution of Eq. (1), and let
then
Lemma 3
Assume that \(x ( t ) \) is an eventually positive solution of Eq. (1). If
then we have two possible cases for \(t\in [ t_{1},\infty ) \), \(t_{1}>t_{0}\) is sufficiently large:
-
Case 1
\(\Delta^{\alpha }x ( t ) >0\), \(\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) >0\), \(\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) >0\) or
-
Case 2
\(\Delta^{\alpha }x ( t ) >0\), \(\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) <0\), \(\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) >0\).
Proof
From the hypothesis, there exists \(t_{1}\) such that \(x ( t ) >0\) on \([ t_{1},\infty ) \), so that \(G ( t ) >0\) on \([ t_{1},\infty ) \), and from Eq. (1), we have
Then \(c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) \) is an eventually non-increasing sequence on \([ t_{1},\infty ) \). We know that \(\Delta^{\alpha }x ( t ) \), \(\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) \), and \(\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) \) are eventually of one sign. For \(t_{2}>t_{1}\) is sufficiently large, we claim that \(\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) >0\) on \([ t_{2}, \infty ) \). Otherwise, assume that there exists sufficiently large \(t_{3}>t_{2}\) such that \(\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) <0\) on \([ t_{3},\infty ) \). For \([ t_{3},\infty ) \) and there exists a constant \(l_{1}>0\), we have
Hence, there exist a constant \(l_{2}>0\) and sufficiently large \(t_{4}>t_{3}\) such that
Then there exist a constant \(l_{3}>0\) and sufficiently large \(t_{5}>t_{4}\) such that
that is,
By (7), we obtain \(\lim_{t\rightarrow \infty }G ( t ) =-\infty \). This is a contradiction. If \(\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) <0\), then \(\Delta^{\alpha }x ( t ) >0\) due to \(\sum_{s=t_{0}}^{\infty }\frac{1}{r ( s ) }=\infty \). If \(\Delta ( r ( t ) \Delta^{ \alpha }x ( t ) ) >0\), then \(\Delta^{\alpha }x ( t ) >0\) due to \(\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) >0\). So, the proof is complete. □
Lemma 4
Assume that \(x ( t ) \) is an eventually positive solution of Eq. (1), which satisfies Case 1 of Lemma 3. Then
If there exists a positive sequence ϕ such that, for \(t\in [ t_{1},\infty ) \),
where \(t_{1}\) is sufficiently large, then \(a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) / \phi ( t ) \) is a non-increasing sequence on \([ t_{1}, \infty ) \) and
Furthermore, if there exists a positive sequence ϑ and \(t_{2}>t_{1}\) is sufficiently large such that, for \(t\in [ t_{2}, \infty ) \),
then \(r ( t ) \Delta^{\alpha }x ( t ) /\vartheta ( t ) \) is a non-increasing sequence on \([ t_{2}, \infty ) \) and
Suppose also that there exists a positive sequence δ and \(t_{3}>t_{2}\) is sufficiently large such that, for \(t\in [ t_{3}, \infty ) \),
Then \(G ( t ) /\delta ( t ) \) is a non-increasing sequence on \([ t_{3},\infty ) \).
Proof
Assume that x is an eventually positive solution of Eq. (1). Then we have that \(\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) >0\) and \(\Delta ( c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) ) <0\) on \([ t_{0},\infty ) \). So,
and then
Hence, \(a ( t ) \Delta ( r ( t ) \Delta^{ \alpha }x ( t ) ) /\phi ( t ) \) is a non-increasing sequence on \([ t_{1},\infty ) \) where \(t_{1}>t_{0}\) is sufficiently large. Then we have
and
So \(r ( t ) \Delta^{\alpha }x ( t ) /\vartheta ( t ) \) is a non-increasing sequence on \([ t_{2}, \infty ) \) where \(t_{2}>t_{1}\) is sufficiently large. Then we have
and then
Then \(G ( t ) /\delta ( t ) \) is a non-increasing sequence on \([ t_{3},\infty ) \) where \(t_{3}>t_{2}\) is sufficiently large. So the proof is complete. □
Theorem 1
Assume that (7) holds and there exists a positive sequence γ such that, for all sufficiently large t,
If there exist positive sequences β, λ such that, for all sufficiently large t,
and
Then every solution of Eq. (1) is oscillatory.
Proof
Suppose to the contrary that \(x(t)\) is a non-oscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there is a solution \(x ( t ) \) of Eq. (1) such that \(x ( t ) >0\) on \([ t_{0},\infty ) \), where \(t_{0}\) is sufficiently large. From Lemma 3, \(x ( t ) \) satisfies Case 1 or Case 2. Firstly, let Case 1 hold. Then we define the following function:
For \(t\in [ t_{0},\infty ) \), we have
Since \(a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) /\phi ( t ) \) is a non-increasing sequence on \([ t_{1},\infty ) \), we have
From Lemma 4, we obtain
and
Setting \(\lambda =2\), \(A= ( \frac{\gamma ( t ) }{c ( t ) } ) ^{1/2}\frac{\omega ( t+1 ) }{\phi ( t+1 ) }\), and \(B=\frac{1}{2} ( \frac{c ( t ) }{ \gamma ( t ) } ) ^{1/2}\Delta \gamma_{+} ( t ) \) using Lemma 1, we obtain
Summing both sides of the above inequality from \(t_{3}\) to \(t-1\), we get
This contradicts (10). Now we consider Case 2. Then we define the following function:
Then
Hence we have
That is,
and
Thus we have \(G ( t ) /\lambda ( t ) \) is eventually non-increasing and
Using the fact that \(r ( t ) \Delta^{\alpha }x ( t ) \) is strictly decreasing, we have
and \(\Delta G ( t ) >0\), then \(G ( t+1 ) >G ( t ) \), it follows that
From 8, we have
for \(\Delta G ( t ) >0\), and letting \(u\rightarrow \infty \), we get
or
And so
Letting \(u\rightarrow \infty \), we have
due to \(\lim_{u\rightarrow \infty }a ( u ) \Delta ( r ( u ) \Delta^{\alpha }x ( u ) ) =k<0\). Then, by (13), we obtain
So,
Setting \(\lambda =2\), \(A= ( \frac{\varGamma ( 1-\alpha ) \beta ( t ) }{r ( t ) } ) ^{1/2}\frac{\omega_{2} ( t+1 ) }{\beta ( t+1 ) }\), and \(B=\frac{1}{2} ( \frac{r ( t ) }{\varGamma ( 1-\alpha ) \beta ( t ) } ) ^{1/2}\Delta \beta_{+} ( t ) \) using Lemma 1, we obtain
Summing both sides of the above inequality from \(t_{2}\) to \(t-1\), we have
which contradicts (12). So, the proof is complete. □
Theorem 2
Let (7) hold. Assume that there exists a positive sequence γ such that, for all sufficiently large t,
If there exist positive sequences β, λ such that (11) and (12) hold, then Eq. (1) is oscillatory.
Proof
Suppose to the contrary that \(x(t)\) is a non-oscillatory solution of (1). Then, without loss of generality, we may assume that there is a solution \(x ( t ) \) of Eq. (1) such that \(x ( t ) >0\) on \([ t_{0},\infty ) \) where \(t_{0}\) is sufficiently large. From Lemma 3, \(x ( t ) \) satisfies Case 1 or Case 2. Firstly, let Case 1 hold. Then we define the following function:
For \(t\in [ t_{0},\infty ) \), we have
From Lemma 4, we obtain
or
and
Hence,
In Lemma 1, choosing \(\lambda =2\), \(A= ( \frac{\gamma ( t ) \vartheta ( t ) }{\vartheta ( t+1 ) }\frac{ \sum_{s=t_{1}}^{t-1}\frac{1}{c ( s ) }}{a ( t ) } ) ^{1/2}\frac{\pi ( t+1 ) }{\gamma ( t+1 ) }\), and \(B=\frac{1}{2} ( \frac{a ( t ) \vartheta ( t+1 ) }{\gamma ( t ) \vartheta ( t ) \sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }} ) ^{1/2} \Delta \gamma_{+} ( t ) \), we obtain
Summing both sides of the above inequality from \(t_{3}\) to \(t-1\), we have
which contradicts (14). And the proof of Case 2 is the same as that of Theorem 1 and hence is omitted. This completes the proof. □
Theorem 3
Let (7) hold. Assume that there exists a positive sequence γ such that, for all sufficiently large t,
If there exist positive sequences β, λ such that (11) and (12) hold, then Eq. (1) is oscillatory.
Proof
Suppose to the contrary that \(x(t)\) is a non-oscillatory solution of (1). Then, without loss of generality, we may assume that there is a solution \(x ( t ) \) of Eq. (1) such that \(x ( t ) >0\) on \([ t_{0},\infty ) \), where \(t_{0}\) is sufficiently large. From Lemma 3, \(x ( t ) \) satisfies Case 1 or Case 2. Firstly, let Case 1 hold. Then we define the following function:
For \(t\in [ t_{0},\infty ) \), we get
From Lemma 4, we have
and
Thus we obtain
Then, setting \(\lambda =2\),
using Lemma 1, we obtain
Summing both sides of the above inequality from \(t_{2}\) to \(t-1\), we have
which contradicts (15). The proof of Case 2 is the same as that of Theorem 1 and hence is omitted. This completes the proof. □
3 Applications
Example 1
Consider the following fractional difference equation for \(t\geq 2\):
This corresponds to Eq. (1) with \(\alpha \in ( 0,1 ] \), \(t_{0}=2\), \(c ( t ) =a ( t ) =r ( t ) =1\), and \(q ( t ) =t^{-2}\). Then \(\phi ( t ) =\lambda ( t ) =t-t_{1}\), \(\vartheta ( t ) =\sum_{s=t_{2}} ^{t-1} ( s-t_{1} ) \), \(\gamma ( t ) =\beta ( t ) =t\). For \(k\in ( 0,1 ) \), it can be written \(kt \leq \phi ( t ) \leq t\), \(k^{2}t^{2}/2\leq \vartheta ( t ) \leq t^{2}/2\), \(k^{3}t^{3}/3\leq \sum_{s=t_{3}}^{t-1}k^{2}s ^{2}\leq t^{3}/3\). So,
and
References
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)
Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Öğrekçi, S.: Interval oscillation criteria for second-order functional differential equations. SIGMA 36(2), 351–359 (2018)
Öğrekçi, S., Misir, A., Tiryaki, A.: On the oscillation of a second-order nonlinear differential equations with damping. Miskolc Math. Notes 18(1), 365–378 (2017)
Öğrekçi, S.: New interval oscillation criteria for second-order functional differential equations with nonlinear damping. Open Math. 13(1), 239–246 (2015)
Sadhasivam, V., Kavitha, J., Nagajothi, N.: Oscillation of neutral fractional order partial differential equations with damping term. Int. J. Pure Appl. Math. 115(9), 47–64 (2017)
Hasil, P., Veselý, M.: Oscillation and non-oscillation criteria for linear and half-linear difference equations. J. Math. Anal. Appl. 452(1), 401–428 (2017)
Hasil, P., Veselý, M.: Oscillation constants for half-linear difference equations with coefficients having mean values. Adv. Differ. Equ. 2015, 210 (2015)
Sugie, J., Tanaka, M.: Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation. Proc. Am. Math. Soc. 145(5), 2059–2073 (2017)
Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of fourth-order delay dynamic equations. Sci. China Math. 58(1), 143–160 (2015)
Grace, S.R., Agarwal, R.P., Sae-Jie, W.: Monotone and oscillatory behavior of certain fourth order nonlinear dynamic equations. Dyn. Syst. Appl. 19(1), 25–32 (2010)
Grace, S.R., Bohner, M., Sun, S.: Oscillation of fourth-order dynamic equations. Hacet. J. Math. Stat. 39, 545–553 (2010)
Grace, S.R., Argawal, R.P., Pinelas, S.: On the oscillation of fourth order superlinear dynamic equations on time scales. Dyn. Syst. Appl. 20, 45–54 (2011)
Li, T., Thandapani, E., Tang, S.: Oscillation theorems for fourth-order delay dynamic equations on time scales. Bull. Math. Anal. Appl. 3, 190–199 (2011)
Liu, T., Zheng, B., Meng, F.: Oscillation on a class of differential equations of fractional order. Math. Probl. Eng. 2013, Article ID 830836 (2013)
Qin, H., Zheng, B.: Oscillation of a class of fractional differential equations with damping term. Sci. World J. 2013, Article ID 685621 (2013)
Ogrekci, S.: Interval oscillation criteria for functional differential equations of fractional order. Adv. Differ. Equ. 2015, 3 (2015)
Muthulakshmi, V., Pavithra, S.: Interval oscillation criteria for forced fractional differential equations with mixed nonlinearities. Glob. J. Pure Appl. Math. 13(9), 6343–6353 (2017)
Chen, D.-X.: Oscillation criteria of fractional differential equations. Adv. Differ. Equ. 2012, 33 (2012)
Zheng, B.: Oscillation for a class of nonlinear fractional differential equations with damping term. J. Adv. Math. Stud. 6(1), 107–115 (2013)
Xu, R.: Oscillation criteria for nonlinear fractional differential equations. J. Appl. Math. 2013, Article ID 971357 (2013)
Secer, A., Adiguzel, H.: Oscillation of solutions for a class of nonlinear fractional difference equations. J. Nonlinear Sci. Appl. 9(11), 5862–5869 (2016)
Abdalla, B.: On the oscillation of q-fractional difference equations. Adv. Differ. Equ. 2017, 254 (2017)
Abdalla, B., Abodayeh, K., Abdeljawad, T. Alzabut, J.: New oscillation criteria for forced nonlinear fractional difference equations. Vietnam J. Math. 45(4), 609–618 (2017)
Alzabut, J.O., Abdeljawad, T.: Sufficient conditions for the oscillation of nonlinear fractional difference equations. J. Fract. Calc. Appl. 5(1), 177–187 (2014)
Bai, Z., Xu, R.: The asymptotic behavior of solutions for a class of nonlinear fractional difference equations with damping term. Discrete Dyn. Nat. Soc. 2018, Article ID 5232147 (2018)
Chatzarakis, G.E., Gokulraj, P., Kalaimani, T., Sadhasivam, V.: Oscillatory solutions of nonlinear fractional difference equations. Int. J. Differ. Equ. 13(1), 19–31 (2018)
Sagayaraj, M.R., Selvam, A.G.M., Loganathan, M.P.: On the oscillation of nonlinear fractional difference equations. Math. Æterna 4, 220–224 (2014)
Selvam, A.G.M., Sagayaraj, M.R., Loganathan, M.P.: Oscillatory behavior of a class of fractional difference equations with damping. Int. J. Appl. Math. Res. 3(3), 220–224 (2014)
Li, W.N.: Oscillation results for certain forced fractional difference equations with damping term. Adv. Differ. Equ. 2016, 70 (2016)
Sagayaraj, M.R., Selvam, A.G.M., Loganathan, M.P.: Oscillation criteria for a class of discrete nonlinear fractional equations. Bull. Soc. Math. Serv. Stand. 3(1), 27–35 (2014)
Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2008)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1988)
Acknowledgements
The author is grateful to the scholars who provided the literature sources.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
HA contributed to the work totally, and he read and approved the final version of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Adiguzel, H. Oscillatory behavior of solutions of certain fractional difference equations. Adv Differ Equ 2018, 445 (2018). https://doi.org/10.1186/s13662-018-1905-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-018-1905-3