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Oscillation results for certain forced fractional difference equations with damping term
Advances in Difference Equations volume 2016, Article number: 70 (2016)
Abstract
In this paper, we establish two sufficient conditions for the oscillation of forced fractional difference equations with damping term of the form
with initial condition \(\Delta^{\alpha-1}x(t)|_{t=0}=x_{0}\), where \(0<\alpha<1 \) is a constant, \(\Delta^{\alpha}x\) is the Riemann-Liouville fractional difference operator of order α of x, and \(\mathbb{N}_{0}=\{0,1,2,\ldots\}\).
1 Introduction
In the past few years, the theory of fractional differential equations and their applications have been investigated extensively. For example, see monographs [1–4]. In recent years, fractional difference equations, which are the discrete counterpart of the corresponding fractional differential equations, have comparably gained attention by some researchers. Many interesting results were established. For instance, see papers [5–20] and the references therein.
The oscillation theory as a part of the qualitative theory of differential equations and difference equations has been developed rapidly in the last decades, and there have been many results on the oscillatory behavior of integer-order differential equations and integer-order difference equations. In particular, we notice that the oscillation of fractional differential equations has been developed significantly in recent years. We refer the reader to [21–33] and the references therein. However, to the best of author’s knowledge, up to now, very little is known regarding the oscillatory behavior of fractional difference equations [18–20]. Unfortunately, the main results of paper [18] are incorrect. The main reason for the mistakes in [18] is an incorrect relation of \(t^{(\alpha-1)}\) and \(t^{(1-\alpha)}\). In fact, noting the definition of \(t^{(\alpha)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\alpha)}\), it is easy to observe that \(t^{(\alpha-1)} t^{(1-\alpha)}\neq1\).
In this paper we investigate the oscillation of forced fractional difference equations with damping term of the form
with initial condition \(\Delta^{\alpha-1}x(t)|_{t=0}=x_{0}\), where \(0<\alpha<1 \) is a constant, \(\Delta^{\alpha}x\) is the Riemann-Liouville difference operator of order α of x, and \(\mathbb{N}_{0}=\{0,1,2,\ldots\}\).
Throughout this paper, we assume that
-
(A)
\(p(t)\) and \(g(t)\) are real sequences, \(p(t)>-1\), \(f:\mathbb{N}_{0}\times\mathbb{R}\rightarrow\mathbb{R}\), and \(xf(t,x)>0\) for \(x\neq0\), \(t\in\mathbb{N}_{0}\).
A solution \(x(t)\) of the Eq. (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory.
2 Preliminaries
In this section, we present some preliminary results of discrete fractional calculus.
Definition 2.1
([7])
Let \(\nu>0\). The νth fractional sum f is defined by
where f is defined for \(s=a\ \operatorname{mod}(1)\), \(\Delta^{-\nu}f\) is defined for \(t=(a+\nu)\ \operatorname{mod}(1)\), and \(t^{(\nu)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\nu)}\). The fractional sum \(\Delta^{-\nu}f\) maps functions defined on \(\mathbb{N}_{a} =\{a,a+1,a+2,\ldots\}\) to functions defined on \(\mathbb{N}_{a+\nu}= \{a+\nu,a+\nu+1,a+\nu+2,\ldots\}\), where Γ is the gamma function.
Definition 2.2
([7])
Let \(\mu>0\) and \(m-1<\mu<m\), where m is a positive integer, \(m=\lceil\mu\rceil\). Set \(\nu=m-\mu\). The μth fractional difference is defined as
where \(\lceil\mu\rceil\) is the ceiling function of μ.
Lemma 2.3
([7])
Let f be a real-valued function defined on \(\mathbb{N}_{a}\), and let \(\mu,\nu>0\). Then the following equalities hold:
Lemma 2.4
Let \(x(t)\) be a solution of Eq. (1), and let
Then
Proof
Using Definition 2.1, it follows from (6) that
Therefore,
The proof of Lemma 2.4 is complete. □
Lemma 2.5
([6])
Let \(\mu\in\mathbb{R}\setminus\{\ldots ,-2,-1\}\). Then
3 Main results
In this section, we establish the oscillation results for Eq. (1).
Theorem 3.1
Suppose that, for \(t_{0}\in\mathbb{N}_{0}\),
and
where M is a constant, and
Then every solution \(x(t)\) of Eq. (1) is oscillatory.
Proof
Suppose to the contrary that there is a nonoscillatory solution \(x(t)\) of Eq. (1) which has no zero in \(\mathbb{N}_{t_{0}}=\{ t_{0},t_{0}+1,t_{0}+2,\ldots\}\). Then \(x(t)>0\) or \(x(t)<0\) for \(t\in\mathbb {N}_{t_{0}}\).
Case 1. \(x(t)>0\), \(t\in\mathbb{N}_{t_{0}}\). Noting assumption (A), from Eq. (1) we have
Therefore, using the fundamental property of Δ and noting the definition of \(V(t)\), we get
Summing both sides of (13) from \(t_{0}\) to \(t-1\), we obtain
where \(M=(\Delta^{\alpha}x(t_{0}))V(t_{0})\), that is,
Applying the \(\Delta^{-\alpha}\) operator to inequality (14), we have
On the one hand, applying Lemma 2.3 to the left-hand side of (15), we obtain
On the other hand, using Definition 2.1, it follows from the right-hand side of (15) that
It follows from (18) that
By using the Stirling formula [20]
we obtain
From (20), taking then limit as \(t\rightarrow\infty\) in (19), we have
which contradicts with \(x(t)>0\).
Case 2. \(x(t)<0\), \(t\in\mathbb{N}_{t_{0}}\). By assumption (A), from Eq. (1) we have
Therefore,
Summing both sides of (22) from \(t_{0}\) to \(t-1\), we obtain
where \(M=(\Delta^{\alpha}x(t_{0}))V(t_{0})\), that is,
Using the procedure of the proof of Case 1, we conclude that
By (20), taking the limit as \(t\rightarrow\infty\) in (24), we have
which contradicts with \(x(t)<0\). The proof of Theorem 3.1 is complete. □
Theorem 3.2
Suppose that, for \(t_{0}\in\mathbb{N}_{0}\),
and
where M is a constant, and \(V(t)\) is defined by (11). Then every solution \(x(t)\) of Eq. (1) is oscillatory.
Proof
Suppose to the contrary that there is a nonoscillatory solution \(x(t)\) of Eq. (1) that has no zero in \(\mathbb{N}_{t_{0}}\). Then \(x(t)>0\) or \(x(t)<0\) for \(t\in\mathbb{N}_{t_{0}}\).
Case 1. \(x(t)>0\), \(t\in\mathbb{N}_{t_{0}}\). As in the proof of Case 1 in Theorem 3.1, we obtain (14). By Lemma 2.4 it follows from (14) that
Summing both sides of (27) from \(t_{0}\) to \(t-1\), we have
Letting \(t\rightarrow\infty\) in (28), we obtain a contradiction with \(E(t)>0\).
Case 2. \(x(t)<0\), \(t\in\mathbb{N}_{t_{0}}\). As in the proof of Case 2 in Theorem 3.1, we obtain (23). By Lemma 2.4 it follows from (23) that
Summing both sides of (29) from \(t_{0}\) to \(t-1\), we have
Letting \(t\rightarrow\infty\) in (30), we obtain a contradiction with \(E(t)<0\). This completes the proof of Theorem 3.2. □
4 Examples
In this section, we conclude from the following two examples that the assumptions of Theorem 3.1 and Theorem 3.2 cannot be dropped.
Example 4.1
Consider the following fractional difference equation:
with the initial condition \(\Delta^{-\frac{1}{3}}x(t)|_{t=0}=0\).
Here \(\alpha=\frac{2}{3}\), \(p(t)=-\frac{1}{3}\), \(f(t,x(t))=\frac{\Gamma(t+\frac{1}{3})}{ 3t\Gamma(t)}x(t)\), \(g(t)=\frac{3-2\Gamma(\frac{2}{3})}{9}\). It is easy to see that
Therefore, we have
which shows that condition (9) of Theorem 3.1 does not hold. It is not difficult to see that \(x(t)=t^{(\frac{2}{3})}\) is a nonoscillatory solution of Eq. (31).
Indeed, on the one hand, using Lemma 2.5, we obtain
and
On the other hand, we have
Combining (33)-(35), we conclude that \(x(t)=t^{(\frac{2}{3})}\) is a solution of Eq. (31).
Example 4.2
Consider the following fractional difference equation:
with the initial condition \(\Delta^{-\frac{2}{3}}x(t)|_{t=0}=0\).
Here \(\alpha=\frac{1}{3}\), \(p(t)=-\frac{1}{2}\), \(f(t,x(t))=\frac{\Gamma(t+\frac{2}{3})}{ 2t\Gamma(t)}x(t)\), \(g(t)=\frac{3-\Gamma(\frac{1}{3})}{6}\). Obviously,
Therefore, we have
Thus, condition (25) of Theorem 3.2 does not hold. In fact, we can easily verify that \(x(t)=t^{(\frac{1}{3})}\) is a nonoscillatory solution of Eq. (36).
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Acknowledgements
This work is supported by the National Natural Science Foundation of China (10971018). The author thanks the referees very much for their valuable comments and suggestions on this paper.
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Li, W.N. Oscillation results for certain forced fractional difference equations with damping term. Adv Differ Equ 2016, 70 (2016). https://doi.org/10.1186/s13662-016-0798-2
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DOI: https://doi.org/10.1186/s13662-016-0798-2
MSC
- 26A33
- 39A12
- 39A21
Keywords
- oscillation
- forced fractional difference equation
- damping term