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Oscillatory behavior of solutions of certain fractional difference equations

Advances in Difference Equations20182018:445

https://doi.org/10.1186/s13662-018-1905-3

  • Received: 2 October 2018
  • Accepted: 26 November 2018
  • Published:

Abstract

In this paper, we consider the oscillation behavior of solutions of the following fractional difference equation:
$$ \Delta \bigl( c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \bigr) +q ( t ) G ( t ) =0, $$
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.

Keywords

  • Oscillation
  • Oscillation criteria
  • Fractional difference operator
  • Riemann–Liouville
  • Fractional difference equations
  • Riccati technique
  • Hardy inequalities

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 [36], difference equations [79], dynamic equations on time scales [1014], and fractional differential (difference) equations [1531] obtaining some oscillation criteria. Recently, oscillation studies have become a very hot topic. That is why, we consider the following fractional difference equation:
$$ \Delta \bigl( c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \bigr) +q ( t ) G ( t ) =0, $$
(1)
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
$$ \Delta^{-v}f ( t ) =\frac{1}{\varGamma ( v ) }\sum_{s=a} ^{t-v} ( t-s-1 ) ^{v-1}f ( s ) , $$
(2)
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
$$ \Delta^{\mu }f ( t ) =\Delta^{m-v}f ( t ) =\Delta^{m} \Delta^{-v}f ( t ) , $$
(3)
where \(\lceil \mu \rceil \) is the ceiling function of μ.

Lemma 1

([33])

Assume that A and B are nonnegative real numbers. Then
$$ \lambda AB^{\lambda -1}-A^{\lambda }\leq ( \lambda -1 ) B ^{\lambda } $$
(4)
for all \(\lambda >1\).

2 Main results

Throughout this paper, we denote
$$ \phi ( t ) =\sum_{s=t_{1}}^{t-1} \frac{1}{c ( s ) };\quad\quad \vartheta ( t ) =\sum_{s=t_{2}}^{t-1} \frac{\phi ( s ) }{a ( s ) };\quad\quad \delta ( t ) =\sum_{s=t_{3}} ^{t-1}\frac{\vartheta ( s ) }{r ( s ) }. $$
For simplification, we consider
$$ \Delta \gamma_{+} ( s ) =\max \bigl\{ 0,\Delta \gamma ( s ) \bigr\} $$
and
$$ \Delta \beta_{+} ( s ) =\max \bigl\{ 0,\Delta \beta ( s ) \bigr\} . $$

Lemma 2

([28])

Let \(x ( t ) \) be a solution of Eq. (1), and let
$$ G ( t ) =\sum_{s=t_{0}}^{t-1+\alpha } ( t-s-1 ) ^{ ( -\alpha ) }x ( s ) , $$
(5)
then
$$ \Delta \bigl( G ( t ) \bigr) =\varGamma ( 1-\alpha ) \Delta^{\alpha }x ( t ) . $$
(6)

Lemma 3

Assume that \(x ( t ) \) is an eventually positive solution of Eq. (1). If
$$ \sum_{s=t_{0}}^{\infty }\frac{1}{c ( s ) }=\sum _{s=t_{0}} ^{\infty }\frac{1}{a ( s ) }=\sum _{s=t_{0}}^{\infty }\frac{1}{r ( s ) }=\infty , $$
(7)
then we have two possible cases for \(t\in [ t_{1},\infty ) \), \(t_{1}>t_{0}\) is sufficiently large:
  1. 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

     
  2. 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
$$ \Delta \bigl( c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \bigr) =-q ( t ) G ( t ) < 0. $$
(8)
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
$$ \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \leq -\frac{l_{1}}{c ( t ) }< 0. $$
Hence, there exist a constant \(l_{2}>0\) and sufficiently large \(t_{4}>t_{3}\) such that
$$ \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \leq - \frac{l_{2}}{a ( t ) }< 0. $$
(9)
Then there exist a constant \(l_{3}>0\) and sufficiently large \(t_{5}>t_{4}\) such that
$$ \Delta^{\alpha }x ( t ) \leq -\frac{l_{3}}{r ( t ) }, $$
that is,
$$ \Delta G ( t ) \leq -\frac{\varGamma ( 1-\alpha ) l _{3}}{r ( t ) }< 0. $$
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
$$ a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \geq c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }. $$
If there exists a positive sequence ϕ such that, for \(t\in [ t_{1},\infty ) \),
$$ \frac{\phi ( t ) }{c ( t ) \sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }}-\Delta \phi ( t ) \leq 0, $$
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
$$ r ( t ) \Delta^{\alpha }x ( t ) \geq \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \frac{a ( t ) }{\phi ( t ) }\sum _{s=t_{1}}^{t-1}\frac{\phi ( s ) }{a ( s ) }. $$
Furthermore, if there exists a positive sequence ϑ and \(t_{2}>t_{1}\) is sufficiently large such that, for \(t\in [ t_{2}, \infty ) \),
$$ \frac{\vartheta ( t ) }{\frac{a ( t ) }{\phi ( t ) }\sum_{s=t_{2}}^{t-1}\frac{\phi ( s ) }{a ( s ) }}-\Delta \vartheta ( t ) \leq 0, $$
then \(r ( t ) \Delta^{\alpha }x ( t ) /\vartheta ( t ) \) is a non-increasing sequence on \([ t_{2}, \infty ) \) and
$$ G ( t ) \geq \Delta G ( t ) \frac{r ( t ) }{ \vartheta ( t ) }\sum_{s=t_{2}}^{t-1} \frac{\vartheta ( s ) }{r ( s ) }. $$
Suppose also that there exists a positive sequence δ and \(t_{3}>t_{2}\) is sufficiently large such that, for \(t\in [ t_{3}, \infty ) \),
$$ \frac{\delta ( t ) }{\frac{r ( t ) }{\vartheta ( t ) }\sum_{s=t_{2}}^{t-1}\frac{\vartheta ( s ) }{r ( s ) }}-\Delta \delta ( t ) \leq 0. $$
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,
$$\begin{aligned} a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) =&a ( t_{0} ) \Delta \bigl( r ( t_{0} ) \Delta^{\alpha }x ( t_{0} ) \bigr) \\ &{}+\sum_{s=t_{0}}^{t-1}\frac{c ( s ) \Delta ( a ( s ) \Delta ( r ( s ) \Delta^{\alpha }x ( s ) ) ) }{c ( s ) } \\ \geq & c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \sum_{s=t_{0}}^{t-1} \frac{1}{c ( s ) }, \end{aligned}$$
and then
$$\begin{aligned}& \Delta \biggl( \frac{a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{\phi ( t ) } \biggr) \\& \quad = \frac{\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) \phi ( t ) -a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) \Delta \phi ( t ) }{\phi ( t ) \phi ( t+1 ) } \\& \quad \leq \frac{\Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{\phi ( t ) \phi ( t+1 ) } \biggl( \frac{\phi ( t ) }{c ( t ) \sum_{s=t_{1}}^{t-1}\frac{1}{c ( s ) }}-\Delta \phi ( t ) \biggr) \leq 0. \end{aligned}$$
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
$$\begin{aligned} r ( t ) \Delta^{\alpha }x ( t ) =&r ( t_{1} ) \Delta^{\alpha }x ( t_{1} ) +\sum_{s=t_{1}}^{t-1} \frac{a ( s ) \Delta ( r ( s ) \Delta^{\alpha }x ( s ) ) }{\phi ( s ) }\frac{\phi ( s ) }{a ( s ) } \\ \geq &\frac{a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{\phi ( t ) } \sum_{s=t_{1}}^{t-1} \frac{\phi ( s ) }{a ( s ) } \end{aligned}$$
and
$$\begin{aligned} \Delta \biggl( \frac{r ( t ) \Delta^{\alpha }x ( t ) }{\vartheta ( t ) } \biggr) =&\frac{\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) \vartheta ( t ) -r ( t ) \Delta^{\alpha }x ( t ) \Delta \vartheta ( t ) }{\vartheta ( t ) \vartheta ( t+1 ) } \\ \leq &\frac{r ( t ) \Delta^{\alpha }x ( t ) }{ \vartheta ( t ) \vartheta ( t+1 ) } \biggl( \frac{ \vartheta ( t ) }{\frac{a ( t ) }{\phi ( t ) }\sum_{s=t_{1}}^{t-1}\frac{\phi ( s ) }{a ( s ) }}-\Delta \vartheta ( t ) \biggr) \leq 0. \end{aligned}$$
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
$$\begin{aligned} G ( t ) =&G ( t_{2} ) +\varGamma ( 1-\alpha ) \sum _{s=t_{2}}^{t-1}\frac{r ( s ) \Delta^{\alpha }x ( s ) }{\vartheta ( s ) }\frac{ \vartheta ( s ) }{r ( s ) } \\ \geq &\frac{r ( t ) \varGamma ( 1-\alpha ) \Delta^{\alpha }x ( t ) }{\vartheta ( t ) } \sum_{s=t_{2}}^{t-1} \frac{\vartheta ( s ) }{r ( s ) } \\ =&\Delta G ( t ) \frac{r ( t ) }{\vartheta ( t ) }\sum_{s=t_{2}}^{t-1} \frac{\vartheta ( s ) }{r ( s ) }, \end{aligned}$$
and then
$$\begin{aligned} \Delta \biggl( \frac{G ( t ) }{\delta ( t ) } \biggr) =&\frac{ ( \Delta G ( t ) ) \delta ( t ) -G ( t ) \Delta \delta ( t ) }{\delta ( t ) \delta ( t+1 ) } \\ \leq &\frac{G ( t ) }{\delta ( t ) \delta ( t+1 ) } \biggl( \frac{\delta ( t ) }{\frac{r ( t ) }{\vartheta ( t ) }\sum_{s=t_{2}}^{t-1}\frac{\vartheta ( s ) }{r ( s ) }}-\Delta \delta ( t ) \biggr) \leq 0. \end{aligned}$$
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,
$$ \lim_{t\rightarrow \infty }\sup \sum_{s=t_{3}}^{t-1} \Biggl( \frac{ \varGamma ( 1-\alpha ) \gamma ( s ) q ( s ) }{\vartheta ( s ) \phi ( s+1 ) }\sum_{u=t_{2}} ^{s-1} \frac{\vartheta ( u ) }{r ( u ) }\sum_{u=t _{1}}^{s-1} \frac{\phi ( u ) }{a ( u ) }-\frac{c ( s ) ( \Delta \gamma_{+} ( s ) ) ^{2}}{4 \gamma ( s ) } \Biggr) =\infty . $$
(10)
If there exist positive sequences β, λ such that, for all sufficiently large t,
$$ \frac{\lambda ( t ) }{r ( t ) \sum_{s=t_{1}}^{t-1}\frac{1}{r ( s ) }}-\Delta \lambda ( t ) \leq 0 $$
(11)
and
$$ \lim_{t\rightarrow \infty }\sup \sum_{\zeta =t_{2}}^{t-1} \Biggl( \frac{ \beta ( \zeta ) \lambda ( \zeta ) }{\lambda ( \zeta +1 ) a ( \zeta ) }\sum_{s=\zeta }^{\infty } \Biggl( \frac{1}{c ( s ) }\sum_{v=s}^{\infty }q ( v ) \Biggr) -\frac{r ( \zeta ) ( \Delta \beta_{+} ( \zeta ) ) ^{2}}{4\varGamma ( 1-\alpha ) \beta ( \zeta ) } \Biggr) =\infty . $$
(12)
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:
$$ \omega ( t ) =\gamma ( t ) \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }. $$
For \(t\in [ t_{0},\infty ) \), we have
$$\begin{aligned} \Delta \omega ( t ) =&\Delta \gamma ( t ) \frac{ \omega ( t+1 ) }{\gamma ( t+1 ) }+\gamma ( t ) \Delta \biggl( \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) } \biggr) \\ =&\Delta \gamma ( t ) \frac{\omega ( t+1 ) }{ \gamma ( t+1 ) }-\gamma ( t ) \frac{q ( t ) G ( t ) }{a ( t+1 ) \Delta ( r ( t+1 ) \Delta^{\alpha }x ( t+1 ) ) } \\ &{}-\gamma ( t ) \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) a ( t+1 ) \Delta ( r ( t+1 ) \Delta^{\alpha }x ( t+1 ) ) }. \end{aligned}$$
Since \(a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) /\phi ( t ) \) is a non-increasing sequence on \([ t_{1},\infty ) \), we have
$$ \frac{a ( t+1 ) \Delta ( r ( t+1 ) \Delta^{ \alpha }x ( t+1 ) ) }{\phi ( t+1 ) }\leq \frac{a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{\phi ( t ) }. $$
From Lemma 4, we obtain
$$\begin{aligned}& \frac{G ( t ) }{a ( t+1 ) \Delta ( r ( t+1 ) \Delta^{\alpha }x ( t+1 ) ) } \\& \quad = \frac{1}{a ( t+1 ) }\frac{G ( t ) }{\Delta G ( t ) }\frac{\Delta G ( t ) }{\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }\frac{\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{\Delta ( r ( t+1 ) \Delta^{\alpha }x ( t+1 ) ) } \\& \quad \geq \frac{1}{a ( t+1 ) } \Biggl( \frac{r ( t ) }{ \vartheta ( t ) }\sum _{s=t_{2}}^{t-1}\frac{\vartheta ( s ) }{r ( s ) } \Biggr) \Biggl( \frac{\varGamma ( 1- \alpha ) }{r ( t ) }\frac{a ( t ) }{\phi ( t ) }\sum_{s=t_{1}}^{t-1} \frac{\phi ( s ) }{a ( s ) } \Biggr) \frac{\phi ( t ) a ( t+1 ) }{\phi ( t+1 ) a ( t ) } \\& \quad = \frac{\varGamma ( 1-\alpha ) }{\vartheta ( t ) \phi ( t+1 ) }\sum_{s=t_{2}}^{t-1} \frac{\vartheta ( s ) }{r ( s ) } \Biggl( \sum_{s=t_{1}}^{t-1} \frac{\phi ( s ) }{a ( s ) } \Biggr) \end{aligned}$$
and
$$\begin{aligned} \Delta \omega ( t ) \leq &\Delta \gamma_{+} ( t ) \frac{\omega ( t+1 ) }{\gamma ( t+1 ) }-\gamma ( t ) q ( t ) \frac{\varGamma ( 1-\alpha ) }{\vartheta ( t ) \phi ( t+1 ) }\sum _{s=t_{2}} ^{t-1}\frac{\vartheta ( s ) }{r ( s ) } \Biggl( \sum _{s=t_{1}}^{t-1}\frac{\phi ( s ) }{a ( s ) } \Biggr) \\ &{}-\frac{\gamma ( t ) }{c ( t ) }\frac{\omega^{2} ( t+1 ) }{\gamma^{2} ( t+1 ) }. \end{aligned}$$
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
$$ \Delta \omega ( t ) \leq -\gamma ( t ) q ( t ) \frac{\varGamma ( 1-\alpha ) }{\vartheta ( t ) \phi ( t+1 ) }\sum _{s=t_{2}}^{t-1}\frac{\vartheta ( s ) }{r ( s ) } \Biggl( \sum _{s=t_{1}}^{t-1}\frac{ \phi ( s ) }{a ( s ) } \Biggr) + \frac{c ( t ) }{4\gamma ( t ) } \bigl( \Delta \gamma_{+} ( t ) \bigr) ^{2}. $$
Summing both sides of the above inequality from \(t_{3}\) to \(t-1\), we get
$$\begin{aligned}& \sum_{s=t_{3}}^{t-1} \Biggl( \frac{\varGamma ( 1-\alpha ) \gamma ( s ) q ( s ) }{\vartheta ( s ) \phi ( s+1 ) } \sum_{u=t_{2}}^{s-1} \frac{\vartheta ( u ) }{r ( u ) } \Biggl( \sum_{u=t_{1}}^{s-1} \frac{\phi ( u ) }{a ( u ) } \Biggr) - \frac{c ( s ) ( \Delta \gamma_{+} ( s ) ) ^{2}}{4\gamma ( s ) } \Biggr) \\& \quad \leq \omega ( t_{3} ) -\omega ( t ) \leq \omega ( t_{3} ) . \end{aligned}$$
This contradicts (10). Now we consider Case 2. Then we define the following function:
$$ \omega_{2} ( t ) =\beta ( t ) \frac{r ( t ) \Delta^{\alpha }x ( t ) }{G ( t ) }. $$
Then
$$\begin{aligned} \Delta \omega_{2} ( t ) =&\Delta \beta ( t ) \frac{ \omega ( t+1 ) }{\beta ( t+1 ) }+ \beta ( t ) \Delta \biggl( \frac{r ( t ) \Delta^{\alpha }x ( t ) }{G ( t ) } \biggr) \\ =&\Delta \beta ( t ) \frac{\omega ( t+1 ) }{ \beta ( t+1 ) }+\beta ( t ) \biggl( \frac{\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) G ( t ) -r ( t ) \Delta^{\alpha }x ( t ) \Delta G ( t ) }{G ( t ) G ( t+1 ) } \biggr) \\ =&\Delta \beta ( t ) \frac{\omega ( t+1 ) }{ \beta ( t+1 ) }+\beta ( t ) \frac{\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{G ( t+1 ) }-\beta ( t ) \frac{r ( t ) \Delta^{ \alpha }x ( t ) \Delta G ( t ) }{G ( t ) G ( t+1 ) }. \end{aligned}$$
Hence we have
$$\begin{aligned} G ( t ) =&G ( t_{1} ) +\varGamma ( 1-\alpha ) \sum _{s=t_{1}}^{t-1}\frac{r ( s ) \Delta^{\alpha }x ( s ) }{r ( s ) } \\ \geq &\varGamma ( 1-\alpha ) r ( t ) \Delta^{ \alpha }x ( t ) \sum _{s=t_{1}}^{t-1}\frac{1}{r ( s ) }. \end{aligned}$$
That is,
$$ \frac{G ( t ) }{r ( t ) \sum_{s=t_{1}}^{t-1}\frac{1}{r ( s ) }}\geq \varGamma ( 1-\alpha ) \Delta^{\alpha }x ( t ) = \Delta G ( t ) $$
and
$$\begin{aligned} \Delta \biggl( \frac{G ( t ) }{\lambda ( t ) } \biggr) =&\frac{\Delta G ( t ) \lambda ( t ) -G ( t ) \Delta \lambda ( t ) }{\lambda ( t ) \lambda ( t+1 ) } \\ \leq &\frac{G ( t ) }{\lambda ( t ) \lambda ( t+1 ) } \biggl( \frac{\lambda ( t ) }{r ( t ) \sum_{s=t_{1}}^{t-1}\frac{1}{r ( s ) }}-\Delta \lambda ( t ) \biggr) \leq 0. \end{aligned}$$
Thus we have \(G ( t ) /\lambda ( t ) \) is eventually non-increasing and
$$ \frac{G ( t ) }{G ( t+1 ) }\geq \frac{\lambda ( t ) }{\lambda ( t+1 ) }. $$
(13)
Using the fact that \(r ( t ) \Delta^{\alpha }x ( t ) \) is strictly decreasing, we have
$$ r ( t ) \Delta^{\alpha }x ( t ) \geq r ( t+1 ) \Delta^{\alpha }x ( t+1 ) $$
and \(\Delta G ( t ) >0\), then \(G ( t+1 ) >G ( t ) \), it follows that
$$\begin{aligned} \Delta \omega_{2} ( t ) \leq &\Delta \beta_{+} ( t ) \frac{\omega ( t+1 ) }{\beta ( t+1 ) }+\beta ( t ) \frac{\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{G ( t+1 ) } \\ &{}-\frac{\varGamma ( 1-\alpha ) \beta ( t ) }{r ( t ) }\frac{\omega_{2}^{2} ( t+1 ) }{\beta^{2} ( t+1 ) }. \end{aligned}$$
From 8, we have
$$\begin{aligned}& c ( u ) \Delta \bigl( a ( u ) \Delta \bigl( r ( u ) \Delta^{\alpha }x ( u ) \bigr) \bigr) -c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \\ & \quad =-\sum_{s=t}^{u-1}q ( s ) G ( s ) \end{aligned}$$
for \(\Delta G ( t ) >0\), and letting \(u\rightarrow \infty \), we get
$$ -c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \leq -G ( t ) \sum_{s=t}^{\infty }q ( s ) $$
or
$$ \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) \geq \frac{G ( t ) }{c ( t ) }\sum_{s=t}^{\infty }q ( s ) . $$
And so
$$ a ( u ) \Delta \bigl( r ( u ) \Delta^{\alpha }x ( u ) \bigr) -a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \geq G ( t ) \sum _{s=t} ^{u-1} \Biggl( \frac{1}{c ( s ) }\sum _{v=s}^{\infty }q ( v ) \Biggr) . $$
Letting \(u\rightarrow \infty \), we have
$$ \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \leq -G ( t ) \frac{1}{a ( t ) }\sum_{s=t}^{\infty } \Biggl( \frac{1}{c ( s ) }\sum_{v=s}^{\infty }q ( v ) \Biggr) $$
due to \(\lim_{u\rightarrow \infty }a ( u ) \Delta ( r ( u ) \Delta^{\alpha }x ( u ) ) =k<0\). Then, by (13), we obtain
$$\begin{aligned} \frac{\Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{G ( t+1 ) } \leq& -\frac{G ( t ) }{G ( t+1 ) }\frac{1}{a ( t ) }\sum _{s=t}^{\infty } \Biggl( \frac{1}{c ( s ) }\sum _{v=s}^{\infty }q ( v ) \Biggr) \\ \leq &-\frac{\lambda ( t ) }{\lambda ( t+1 ) }\frac{1}{a ( t ) }\sum _{s=t}^{\infty } \Biggl( \frac{1}{c ( s ) }\sum _{v=s}^{\infty }q ( v ) \Biggr) . \end{aligned}$$
So,
$$\begin{aligned} \begin{aligned} \Delta \omega_{2} ( t ) &\leq \Delta \beta_{+} ( t ) \frac{\omega_{2} ( t+1 ) }{\beta ( t+1 ) }-\beta ( t ) \frac{\lambda ( t ) }{\lambda ( t+1 ) } \frac{1}{a ( t ) }\sum_{s=t}^{\infty } \Biggl( \frac{1}{c ( s ) }\sum_{v=s}^{\infty }q ( v ) \Biggr) \\ &{}-\frac{\varGamma ( 1-\alpha ) \beta ( t ) }{r ( t ) }\frac{\omega_{2}^{2} ( t+1 ) }{\beta^{2} ( t+1 ) }.\end{aligned} \end{aligned}$$
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
$$ \Delta \omega_{2} ( t ) \leq -\beta ( t ) \frac{ \lambda ( t ) }{\lambda ( t+1 ) } \frac{1}{a ( t ) }\sum_{s=t}^{\infty } \Biggl( \frac{1}{c ( s ) }\sum_{v=s} ^{\infty }q ( v ) \Biggr) +\frac{r ( t ) ( \Delta \beta_{+} ( t ) ) ^{2}}{4\varGamma ( 1-\alpha ) \beta ( t ) }. $$
Summing both sides of the above inequality from \(t_{2}\) to \(t-1\), we have
$$\begin{aligned}& \sum_{\zeta =t_{2}}^{t-1} \Biggl( \beta ( \zeta ) \frac{ \lambda ( \zeta ) }{\lambda ( \zeta +1 ) }\frac{1}{a ( \zeta ) }\sum_{s=\zeta }^{\infty } \Biggl( \frac{1}{c ( s ) }\sum_{v=s}^{\infty }q ( v ) \Biggr) -\frac{r ( \zeta ) ( \Delta \beta_{+} ( \zeta ) ) ^{2}}{4\varGamma ( 1-\alpha ) \beta ( \zeta ) } \Biggr) \\& \quad \leq \omega_{2} ( t_{2} ) -\omega_{2} ( t ) \leq \omega_{2} ( t_{2} ) < \infty , \end{aligned}$$
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,
$$ \lim_{t\rightarrow \infty }\sup \sum_{s=t_{3}}^{t-1} \Biggl( \gamma ( s ) q ( s ) \frac{\varGamma ( 1-\alpha ) }{\vartheta ( s+1 ) }\sum _{u=t_{2}}^{s-1}\frac{\vartheta ( u ) }{r ( u ) }-\frac{a ( s ) \vartheta ( s+1 ) ( \Delta \gamma_{+} ( s ) ) ^{2}}{4\gamma ( s ) \vartheta ( s ) \sum_{u=t_{0}} ^{s-1}\frac{1}{c ( u ) }} \Biggr) =\infty . $$
(14)
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:
$$ \pi ( t ) =\gamma ( t ) \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{r ( t ) \Delta^{\alpha }x ( t ) }. $$
For \(t\in [ t_{0},\infty ) \), we have
$$\begin{aligned} \Delta \pi ( t ) =&\Delta \gamma ( t ) \frac{ \pi ( t+1 ) }{\gamma ( t+1 ) }+\gamma ( t ) \Delta \biggl( \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{r ( t ) \Delta^{\alpha }x ( t ) } \biggr) \\ =&\Delta \gamma ( t ) \frac{\pi ( t+1 ) }{ \gamma ( t+1 ) }-\gamma ( t ) \frac{q ( t ) G ( t ) }{r ( t+1 ) \Delta^{\alpha }x ( t+1 ) } \\ &{}-\gamma ( t ) \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) }{r ( t ) \Delta^{\alpha }x ( t ) r ( t+1 ) \Delta^{\alpha }x ( t+1 ) }. \end{aligned}$$
From Lemma 4, we obtain
$$\begin{aligned}& \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \geq \frac{\sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }}{a ( t ) }c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) , \\& 1\leq \frac{r ( t+1 ) \Delta^{\alpha }x ( t+1 ) }{r ( t ) \Delta^{\alpha }x ( t ) }\leq \frac{\vartheta ( t+1 ) }{\vartheta ( t ) }, \\& \frac{\vartheta ( t ) }{\vartheta ( t+1 ) }\leq \frac{r ( t+1 ) \Delta^{\alpha }x ( t+1 ) }{r ( t ) \Delta^{\alpha }x ( t ) } \end{aligned}$$
or
$$ \frac{r ( t+1 ) \vartheta ( t ) }{r ( t ) \vartheta ( t+1 ) }\leq \frac{\Delta G ( t ) }{ \Delta G ( t+1 ) } $$
and
$$\begin{aligned} \frac{G ( t ) }{r ( t+1 ) \Delta^{\alpha }x ( t+1 ) } =&\frac{\varGamma ( 1-\alpha ) }{r ( t+1 ) }\frac{G ( t ) }{\Delta G ( t ) } \frac{ \Delta G ( t ) }{\Delta G ( t+1 ) } \\ \geq &\frac{\varGamma ( 1-\alpha ) }{r ( t+1 ) } \Biggl( \frac{r ( t ) }{\vartheta ( t ) }\sum _{s=t _{2}}^{t-1}\frac{\vartheta ( s ) }{r ( s ) } \Biggr) \frac{r ( t+1 ) \vartheta ( t ) }{r ( t ) \vartheta ( t+1 ) } \\ =&\frac{\varGamma ( 1-\alpha ) }{\vartheta ( t+1 ) }\sum_{s=t_{2}}^{t-1} \frac{\vartheta ( s ) }{r ( s ) }. \end{aligned}$$
Hence,
$$\begin{aligned} \Delta \pi ( t ) \leq &\Delta \gamma_{+} ( t ) \frac{ \pi ( t+1 ) }{\gamma ( t+1 ) }- \gamma ( t ) q ( t ) \frac{\varGamma ( 1-\alpha ) }{ \vartheta ( t+1 ) }\sum_{s=t_{2}}^{t-1} \frac{\vartheta ( s ) }{r ( s ) } \\ &{}-\frac{\gamma ( t ) \vartheta ( t ) }{\vartheta ( t+1 ) }\frac{\sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }}{a ( t ) }\frac{\pi^{2} ( t+1 ) }{\gamma^{2} ( t+1 ) }. \end{aligned}$$
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
$$ \Delta \pi ( t ) \leq -\gamma ( t ) q ( t ) \frac{\varGamma ( 1-\alpha ) }{\vartheta ( t+1 ) } \sum _{s=t_{2}}^{t-1}\frac{\vartheta ( s ) }{r ( s ) }+\frac{a ( t ) \vartheta ( t+1 ) ( \Delta \gamma_{+} ( t ) ) ^{2}}{4\gamma ( t ) \vartheta ( t ) \sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }}. $$
Summing both sides of the above inequality from \(t_{3}\) to \(t-1\), we have
$$\begin{aligned}& \sum_{s=t_{3}}^{t-1} \Biggl( \gamma ( s ) q ( s ) \frac{ \varGamma ( 1-\alpha ) }{\vartheta ( s+1 ) } \sum_{u=t_{2}}^{s-1} \frac{\vartheta ( u ) }{r ( u ) }-\frac{a ( s ) \vartheta ( s+1 ) ( \Delta \gamma_{+} ( s ) ) ^{2}}{4\gamma ( s ) \vartheta ( s ) \sum_{u=t_{0}}^{s-1}\frac{1}{c ( u ) }} \Biggr) \\& \quad \leq \pi ( t_{1} ) -\pi ( t ) \\& \quad \leq \pi ( t_{2} ) < \infty , \end{aligned}$$
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,
$$ \lim_{t\rightarrow \infty }\sup \sum_{s=t_{2}}^{t-1} \biggl( \gamma ( s ) q ( s ) \frac{\delta ( s ) }{ \delta ( s+1 ) }-\frac{r ( s ) \phi ( s ) ( \Delta \gamma_{+} ( s ) ) ^{2}}{4\gamma ( s ) \sum_{s=t_{1}}^{u-1}\frac{\phi ( u ) }{a ( u ) }\sum_{u=t_{0}}^{s-1}\frac{1}{c ( u ) }} \biggr) = \infty . $$
(15)
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:
$$ \nu ( t ) =\gamma ( t ) \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{G ( t ) }. $$
For \(t\in [ t_{0},\infty ) \), we get
$$\begin{aligned} \Delta \nu ( t ) =&\Delta \gamma ( t ) \frac{ \nu ( t+1 ) }{\gamma ( t+1 ) }+\gamma ( t ) \Delta \biggl( \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) }{G ( t ) } \biggr) \\ =&\Delta \gamma ( t ) \frac{\nu ( t+1 ) }{ \alpha ( t+1 ) }-\gamma ( t ) \frac{q ( t ) G ( t ) }{G ( t+1 ) } \\ &{}-\gamma ( t ) \frac{c ( t ) \Delta ( a ( t ) \Delta ( r ( t ) \Delta^{\alpha }x ( t ) ) ) \Delta G ( t ) }{G ( t ) G ( t+1 ) }. \end{aligned}$$
From Lemma 4, we have
$$ \Delta G ( t ) \geq \frac{1}{r ( t ) } \Biggl( \frac{a ( t ) }{\phi ( t ) }\sum _{s=t_{1}}^{t-1}\frac{ \phi ( s ) }{a ( s ) } \Biggr) \frac{\sum_{s=t_{0}} ^{t-1}\frac{1}{c ( s ) }}{a ( t ) }c ( t ) \Delta \bigl( a ( t ) \Delta \bigl( r ( t ) \Delta^{\alpha }x ( t ) \bigr) \bigr) $$
and
$$ \frac{G ( t ) }{G ( t+1 ) }\geq \frac{\delta ( t ) }{\delta ( t+1 ) }. $$
Thus we obtain
$$\begin{aligned} \Delta \nu ( t ) \leq &\Delta \gamma_{+} ( t ) \frac{ \nu ( t+1 ) }{\gamma ( t+1 ) }- \gamma ( t ) p ( t ) \frac{\delta ( t ) }{\delta ( t+1 ) } \\ &{}-\frac{\gamma ( t ) }{r ( t ) \phi ( t ) }\sum_{s=t_{1}}^{t-1} \frac{\phi ( s ) }{a ( s ) } \sum_{s=t_{0}}^{t-1} \frac{1}{c ( s ) }\frac{\nu^{2} ( t+1 ) }{\gamma^{2} ( t+1 ) }. \end{aligned}$$
Then, setting \(\lambda =2\),
$$\begin{aligned}& A= \Biggl( \frac{\gamma ( t ) }{r ( t ) \phi ( t ) }\sum_{s=t_{1}}^{t-1} \frac{ \phi ( s ) }{a ( s ) }\sum_{s=t_{0}}^{t-1} \frac{1}{c ( s ) } \Biggr) ^{1/2}\frac{\nu ( t+1 ) }{\gamma ( t+1 ) },\quad \text{and} \\& B=\frac{1}{2} \biggl( \frac{r ( t ) \phi ( t ) }{\gamma ( t ) \sum_{s=t_{1}}^{t-1}\frac{ \phi ( s ) }{a ( s ) }\sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }} \biggr) ^{1/2}\Delta \gamma_{+} ( t ) \end{aligned}$$
using Lemma 1, we obtain
$$ \Delta \nu ( t ) \leq -\gamma ( t ) q ( t ) \frac{\delta ( t ) }{\delta ( t+1 ) }+ \frac{r ( t ) \phi ( t ) ( \Delta \gamma_{+} ( t ) ) ^{2}}{4\gamma ( t ) \sum_{s=t_{1}}^{t-1}\frac{ \phi ( s ) }{a ( s ) }\sum_{s=t_{0}}^{t-1}\frac{1}{c ( s ) }}. $$
Summing both sides of the above inequality from \(t_{2}\) to \(t-1\), we have
$$\begin{aligned} \begin{aligned} \sum_{s=t_{2}}^{t-1} \biggl( \gamma ( s ) q ( s ) \frac{ \delta ( s ) }{\delta ( s+1 ) }-\frac{r ( s ) \phi ( s ) ( \Delta \gamma_{+} ( s ) ) ^{2}}{4\gamma ( s ) \sum_{s=t_{1}}^{u-1}\frac{ \phi ( u ) }{a ( u ) }\sum_{u=t_{0}}^{s-1}\frac{1}{c ( u ) }} \biggr) & \leq \nu ( t_{2} ) -\nu ( t ) \\ & \leq \nu ( t_{2} ) < \infty ,\end{aligned} \end{aligned}$$
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 \(t\geq 2\):
$$ \Delta^{3+\alpha }x ( t ) +t^{-2} \Biggl( \sum _{s=t_{0}}^{t-1+ \alpha } ( t-s-1 ) ^{ ( -\alpha ) }x ( s ) \Biggr) =0. $$
(16)
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,
$$\begin{aligned}& \lim_{t\rightarrow \infty }\sup \sum_{s=t_{3}}^{t-1} \Biggl( \frac{ \varGamma ( 1-\alpha ) \gamma ( s ) q ( s ) }{\vartheta ( s ) \phi ( s+1 ) }\sum_{u=t_{2}} ^{s-1} \frac{\vartheta ( u ) }{r ( u ) }\sum_{u=t _{1}}^{s-1} \frac{\phi ( u ) }{a ( u ) }-\frac{c ( s ) ( \Delta \gamma_{+} ( s ) ) ^{2}}{4 \gamma ( s ) } \Biggr) \\& \quad \geq \lim_{t\rightarrow \infty }\sup \sum_{s=t_{3}}^{t-1} \biggl( \frac{ \varGamma ( 1-\alpha ) k^{5}s^{2}}{6 ( s+1 ) }- \frac{1}{4s} \biggr) =\infty \end{aligned}$$
and
$$\begin{aligned}& \lim_{t\rightarrow \infty }\sup \sum_{\zeta =t_{2}}^{t-1} \Biggl( \frac{ \beta ( \zeta ) \lambda ( \zeta ) }{\lambda ( \zeta +1 ) a ( \zeta ) }\sum_{s=\zeta }^{\infty } \Biggl( \frac{1}{c ( s ) }\sum_{v=s}^{\infty }q ( v ) \Biggr) -\frac{r ( \zeta ) ( \Delta \beta_{+} ( \zeta ) ) ^{2}}{4\varGamma ( 1-\alpha ) \beta ( \zeta ) } \Biggr) \\& \quad \geq \lim_{t\rightarrow \infty }\sup \sum_{\zeta =t_{2}}^{t-1} \Biggl( \frac{\zeta^{2}}{ ( \zeta +1 ) }\sum_{s=\zeta }^{ \infty } \Biggl( \sum_{v=s}^{\infty }v^{-2} \Biggr) -\frac{1}{4\varGamma ( 1-\alpha ) \zeta } \Biggr) \\& \quad =\infty . \end{aligned}$$
Thus, (16) is oscillatory from Theorem 1.

Declarations

Acknowledgements

The author is grateful to the scholars who provided the literature sources.

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Funding

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Authors’ contributions

HA contributed to the work totally, and he read and approved the final version of the manuscript.

Competing interests

The author declares that they have no competing interests.

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.

Authors’ Affiliations

(1)
Department of Mechatronics Engineering, Istanbul Gelisim University, Istanbul, Turkey

References

  1. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) View ArticleGoogle Scholar
  2. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
  3. Öğrekçi, S.: Interval oscillation criteria for second-order functional differential equations. SIGMA 36(2), 351–359 (2018) Google Scholar
  4. 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) MathSciNetView ArticleGoogle Scholar
  5. Öğrekçi, S.: New interval oscillation criteria for second-order functional differential equations with nonlinear damping. Open Math. 13(1), 239–246 (2015) MathSciNetView ArticleGoogle Scholar
  6. 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) Google Scholar
  7. 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) MathSciNetView ArticleGoogle Scholar
  8. Hasil, P., Veselý, M.: Oscillation constants for half-linear difference equations with coefficients having mean values. Adv. Differ. Equ. 2015, 210 (2015) MathSciNetView ArticleGoogle Scholar
  9. 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) MathSciNetView ArticleGoogle Scholar
  10. Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of fourth-order delay dynamic equations. Sci. China Math. 58(1), 143–160 (2015) MathSciNetView ArticleGoogle Scholar
  11. 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) MathSciNetMATHGoogle Scholar
  12. Grace, S.R., Bohner, M., Sun, S.: Oscillation of fourth-order dynamic equations. Hacet. J. Math. Stat. 39, 545–553 (2010) MathSciNetMATHGoogle Scholar
  13. 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) MathSciNetMATHGoogle Scholar
  14. 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) MathSciNetMATHGoogle Scholar
  15. Liu, T., Zheng, B., Meng, F.: Oscillation on a class of differential equations of fractional order. Math. Probl. Eng. 2013, Article ID 830836 (2013) MathSciNetMATHGoogle Scholar
  16. Qin, H., Zheng, B.: Oscillation of a class of fractional differential equations with damping term. Sci. World J. 2013, Article ID 685621 (2013) Google Scholar
  17. Ogrekci, S.: Interval oscillation criteria for functional differential equations of fractional order. Adv. Differ. Equ. 2015, 3 (2015) MathSciNetView ArticleGoogle Scholar
  18. 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) Google Scholar
  19. Chen, D.-X.: Oscillation criteria of fractional differential equations. Adv. Differ. Equ. 2012, 33 (2012) MathSciNetView ArticleGoogle Scholar
  20. Zheng, B.: Oscillation for a class of nonlinear fractional differential equations with damping term. J. Adv. Math. Stud. 6(1), 107–115 (2013) MathSciNetMATHGoogle Scholar
  21. Xu, R.: Oscillation criteria for nonlinear fractional differential equations. J. Appl. Math. 2013, Article ID 971357 (2013) MathSciNetMATHGoogle Scholar
  22. Secer, A., Adiguzel, H.: Oscillation of solutions for a class of nonlinear fractional difference equations. J. Nonlinear Sci. Appl. 9(11), 5862–5869 (2016) MathSciNetView ArticleGoogle Scholar
  23. Abdalla, B.: On the oscillation of q-fractional difference equations. Adv. Differ. Equ. 2017, 254 (2017) MathSciNetView ArticleGoogle Scholar
  24. 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) MathSciNetView ArticleGoogle Scholar
  25. Alzabut, J.O., Abdeljawad, T.: Sufficient conditions for the oscillation of nonlinear fractional difference equations. J. Fract. Calc. Appl. 5(1), 177–187 (2014) MathSciNetGoogle Scholar
  26. 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) MathSciNetView ArticleGoogle Scholar
  27. 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) Google Scholar
  28. Sagayaraj, M.R., Selvam, A.G.M., Loganathan, M.P.: On the oscillation of nonlinear fractional difference equations. Math. Æterna 4, 220–224 (2014) MathSciNetMATHGoogle Scholar
  29. 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) Google Scholar
  30. Li, W.N.: Oscillation results for certain forced fractional difference equations with damping term. Adv. Differ. Equ. 2016, 70 (2016) MathSciNetView ArticleGoogle Scholar
  31. 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) MATHGoogle Scholar
  32. Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2008) MathSciNetView ArticleGoogle Scholar
  33. Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1988) MATHGoogle Scholar

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