1 Introduction

In 1974, Diaz and Osler [1] presented a discrete fractional difference operator based on an infinite series. In 1988, Miller and Ross [2] introduced the definitions of noninteger-order differences and sums. Since then, the theory of fractional difference equations has been studied by several scholars. In recent years, some papers [3–21] on discrete fractional calculus were published, which helped to build up the theory of fractional difference equations. For example, Atici and Eloe [3] discussed the properties of the generalized falling function, the corresponding power rule for fractional delta operators, and the commutativity of fractional sums.

Very recently, the oscillation theory as a part of the qualitative theory of fractional differential equations and fractional difference equations has been developed. We refer the reader to [20–28] and the references therein. In particular, we notice that a few papers [24–28] studied the oscillation of fractional partial differential equations that involve the Riemann-Liouville fractional partial derivatives.

Motivated by the papers [24–29], we investigate the forced oscillation of the fractional partial difference equation of the form

$$ \Delta_{n}^{\alpha}u(m,n)=a(n)Lu(m,n)-q(m,n)u(m,n)+h(m,n),\quad (m,n)\in\Omega\times\mathbb{N}_{a}, $$
(1)

where \(m=(m_{1},m_{2},\ldots,m_{\ell})\), Ω is a convex connected solid net (for the definition of a convex connected solid net, we refer to [29]), and

$$ \Omega=\mathbb {N}(1,N_{1})\times \mathbb{N}(1,N_{2})\times \cdots\times\mathbb{N}(1,N_{\ell}), $$
(2)

\(\mathbb{N}(1,N_{i})=\{1,2,\ldots N_{i}\}\), \(i=1,2,\ldots,\ell\), L is the discrete Laplacian on Ω defined as

$$ Lu(m,n)=\sum_{i=1}^{\ell}\Delta_{m_{i}}^{2}u \bigl((m_{1},\ldots,m_{i-1},m_{i}-1,m_{i+1}, \ldots ,m_{\ell }),n\bigr), $$
(3)

\(\Delta_{n}^{\alpha}u(m,n)\) is the Riemann-Liouville fractional difference operator of order α of u with respect to n, \(\alpha\in(0,1)\) is a constant, \(\mathbb{N}_{a}=\{a,a+1,a+2,\ldots\} \), and \(a\geq0\) is a real number.

Throughout this paper, we always assume that

  1. (A)

    \(a(n)\geq0, n\in\mathbb{N}_{a}\); \(q(m,n)\geq0\), \(q(n)=\min_{m\in \Omega}q(m,n)\), \((m,n)\in\Omega\times\mathbb{N}_{a}\); and \(h:\Omega\times\mathbb {N}_{a}\rightarrow\mathbb{R}\).

Consider one of the two following boundary conditions:

$$(\mathrm{B}1)\qquad\Delta_{N}u(m-1,n)+g(m,n)u(m,n)=0,\quad (m,n)\in \partial\Omega\times\mathbb{N}_{a}, $$

or

$$(\mathrm{B}2)\qquad \Delta_{N}u(m-1,n)=\phi(m,n), \quad (m,n)\in \partial\Omega\times\mathbb {N}_{a}, $$

where

$$\begin{aligned} \partial\Omega={}& \bigcup_{i=1}^{\ell} \bigl\{ (m_{1},\ldots,m_{i-1},0,m_{i+1}, \ldots,m_{\ell}), (m_{1},\ldots,m_{i-1}, \\ &{}N_{i}+1,m_{i+1},\ldots,m_{\ell}) \bigr\} , \quad m_{i}\in\mathbb{N}(1,N_{i}), 1\leq i\leq\ell, \end{aligned}$$
(4)

\(\Delta_{N}u(m-1,n)\) is the normal difference at \((m,n)\in\partial\Omega\times\mathbb{N}_{a}\) defined by

$$\Delta_{N}u(m-1,n)=\sum_{\mathrm{all} \ m\pm1\notin\Omega} \bigl( \Delta_{m}\bigl(u(m,n)\bigr)-\Delta_{m}u(m-1,n) \bigr)=\sum _{\mathrm{all}\ m\pm1\notin\Omega}\Delta_{m}^{2}u(m,n), $$

N is the unit exterior normal vector to ∂Ω, \(m+1:=\{m_{1}+1,m_{2},\ldots,m_{\ell}\}\cup\cdots\cup\{m_{1},\ldots , m_{\ell -1},m_{\ell}+1\}\), \(m-1:=\{m_{1}-1,m_{2},\ldots,m_{\ell}\}\cup\cdots\cup\{m_{1},\ldots, m_{\ell -1},m_{\ell}-1\}\), and \(g(m,n)\geq0, \phi(m,n)\geq0, (m,n)\in\partial\Omega\times\mathbb{N}_{a}\). For the details on ∂Ω and \(\Delta_{N}u(m-1,n)\), we refer to the monograph [30] and paper [29], respectively.

The function \(u(m,n)\) is said to be a solution of problem (1)-(B1) (or (1)-(B2)) if it satisfies (1) for \((m,n)\in\Omega\times\mathbb{N}_{a}\) and satisfies (B1) (or (B2)) for \((m,n)\in\partial\Omega\times\mathbb{N}_{a}\).

The solution \(u(m,n)\) of problem (1)-(B1) (or (1)-(B2)) is said to be oscillatory in \(\Omega\times\mathbb{N}_{a}\) 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 and partial differences.

Definition 2.1

[3]

Let \(0<\nu<1\). The νth fractional sum of f is defined by

$$ \Delta_{a}^{-\nu}f(t)=\frac{1}{\Gamma(\nu)}\sum _{s=a}^{t-\nu}(t-s-1)^{(\nu-1)}f(s), $$
(5)

where f is defined for \(s\in\mathbb{N}_{a}\), \(\Delta_{a}^{-\nu}f\) is defined for \(s\in\mathbb{N}_{a+\nu}= \{a+\nu,a+\nu+1,a+\nu+2,\ldots\}\), Γ is the gamma function, and

$$t^{(\nu)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\nu)}. $$

Definition 2.2

Let \(0<\nu<1\). The νth fractional sum with respect to n of \(u(m,n)\) is defined by

$$ \Delta_{n}^{-\nu}u(m,n)=\frac{1}{\Gamma(\nu)}\sum _{s=a}^{n-\nu}(n-s-1)^{(\nu-1)}u(m,s). $$
(6)

Definition 2.3

[3]

Let \(\mu>0\) and \(k-1<\mu<k\), where k denotes a positive integer, \(k=\lceil\mu\rceil\). Set \(\nu=k-\mu\). The μth fractional difference is defined as

$$ \Delta^{\mu}f(t)=\Delta^{k-\nu}f(t)=\Delta^{k} \Delta^{-\nu}f(t), $$
(7)

where \(\lceil\mu\rceil\) is the ceiling function of μ.

Definition 2.4

Let \(0<\mu<1\) and \(\nu=1-\mu\). The μth fractional partial difference with respect to n of a function \(u(m,n)\) is defined as

$$ \Delta_{n}^{\mu}u(m,n)=\Delta_{n}^{1-\nu}u(m,n)= \Delta_{n}\Delta _{n}^{-\nu}u(m,n). $$
(8)

Lemma 2.5

[3]

Let f be a real-valued function defined on \(\mathbb{N}_{a}\), and let \(\mu,\nu>0\). Then the following equalities hold:

$$\begin{aligned}& \Delta^{-\nu}\bigl[\Delta^{-\mu}f(t)\bigr]=\Delta^{-(\mu+\nu)}f(t)= \Delta ^{-\mu }\bigl[\Delta^{-\nu}f(t)\bigr]; \end{aligned}$$
(9)
$$\begin{aligned}& \Delta^{-\nu}\Delta f(t)=\Delta\Delta^{-\nu}f(t)- \frac{(t-a)^{(\nu-1)}}{\Gamma(\nu)}f(a). \end{aligned}$$
(10)

Lemma 2.6

For \(n_{0}\in\mathbb{N}_{a}\), let

$$ E(n)=\sum_{s=n_{0}}^{n-1+\alpha}(n-s-1)^{(-\alpha)}x(n), \quad n\in \mathbb{N}_{a}, \alpha\in(0,1). $$
(11)

Then

$$ \Delta E(n)=\Gamma(1-\alpha)\Delta^{\alpha}x(n). $$
(12)

Proof

By Definition 2.1, from (11) we have

$$\begin{aligned} E(n) =& \sum_{s=n_{0}}^{n-1+\alpha}(n-s-1)^{(-\alpha )}x(s)= \sum_{s=n_{0}}^{n-(1-\alpha)}(n-s-1)^{((1-\alpha)-1)}x(s) \\ =&\Gamma(1-\alpha)\Delta^{-(1-\alpha)}x(n). \end{aligned}$$
(13)

Using Definition 2.3, from (13) it follows that

$$\Delta E(n)=\Gamma(1-\alpha)\Delta\Delta^{-(1-\alpha)}x(n)=\Gamma (1-\alpha ) \Delta^{\alpha}x(n). $$

The proof of Lemma 2.6 is complete. □

Lemma 2.7

Discrete Gaussian formula [29]

Let Ω be a convex connected solid net. Then

$$ \sum_{m\in\Omega}Ly(m,n)=\sum_{m\in\partial\Omega} \Delta_{N}y(m-1,n). $$
(14)

Lemma 2.8

[31]

For \(\varepsilon>0\),

$$ \lim_{t\rightarrow\infty}\frac{\Gamma (t)t^{\varepsilon}}{\Gamma (t+\varepsilon)}=1. $$
(15)

For convenience, we introduce the following notations:

$$ U(n)=\sum_{m\in\Omega}u(m,n), \qquad H(n)= \sum_{m\in \Omega}h(m,n), \qquad \Phi (n)=\sum _{m\in\partial\Omega}\phi(m,n). $$
(16)

3 Oscillation of problem (1)-(B1)

Theorem 3.1

For \(n_{0}\in \mathbb{N}_{a}\), if

$$ \liminf_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1}H(s)=- \infty, $$
(17)

and

$$ \limsup_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1} H(s)=+\infty, $$
(18)

where \(H(n)\) is defined as in (16), then every solution \(u(m,n)\) of problem (1)-(B1) is oscillatory in \(\Omega\times\mathbb{N}_{a}\).

Proof

Suppose to the contrary that there is a nonoscillatory solution \(u(m,n)\) of problem (1)-(B1) that has no zero in \(\Omega\times \mathbb{N}_{a}\) for some \(n^{*}\geq a\). Then \(u(m,n)>0\) or \(u(m,n)<0\) for \(n\geq n^{*}\).

Case 1. \(u(m,n)>0, n\geq n^{*}\). Summing equation (1) over Ω, we have

$$\begin{aligned} \sum_{m\in\Omega}\Delta_{n}^{\alpha }u(m,n) =& a(n)\sum_{m\in\Omega}Lu(m,n) -\sum _{m\in\Omega}q(m,n)u(m,n) \\ &{} + \sum_{m\in\Omega}h(m,n),\quad n\in \mathbb{N}_{a}. \end{aligned}$$
(19)

The discrete Gaussian formula and (B1) yield

$$ \sum_{m\in\Omega}Lu(m,n)=\sum_{m\in\partial\Omega} \Delta _{N}u(m-1,n)=\sum_{m\in\partial\Omega}-g(m,n)u(m,n) \leq 0, \quad n\in\mathbb{N}_{a}. $$
(20)

From assumption (A) we have

$$ \sum_{m\in\Omega}q(m,n)u(m,n)\geq q(n)\sum _{m\in\Omega}u(m,n), \quad n\in\mathbb{N}_{a}. $$
(21)

Combining (19)-(21), we obtain

$$ \Delta^{\alpha}U(n)+q(n)U(n)\leq H(n), \quad n\in \mathbb{N}_{a}, $$
(22)

where \(U(n)\) is defined as in (16). It follows from (22) that

$$ \Delta^{\alpha}U(n)\leq H(n), \quad n\in\mathbb{N}_{a}. $$
(23)

Using Lemma 2.6, from (23) we have

$$ \Delta G(n)\leq\Gamma(1-\alpha)H(n), $$
(24)

where

$$G(n)=\sum_{s=n^{*}}^{n-1+\alpha}(n-s-1)^{(-\alpha)}U(n), \quad n\in\mathbb{N}_{a}. $$

Summing both sides of (24) from \(n^{*}\) to \(n-1\), we obtain

$$ G(n)\leq G\bigl(n^{*}\bigr)+\Gamma(1-\alpha)\sum _{s=n^{*}}^{n-1}H(s). $$
(25)

Taking \(n\rightarrow\infty\) in (25), we have

$$\liminf_{n\rightarrow\infty}G(n)=-\infty, $$

which contradicts with \(G(n)>0\).

Case 2. \(u(m,n)<0, n\geq n^{*}\). As in the proof of Case 1, we obtain (19). The discrete Gaussian formula and (B1) yield

$$ \sum_{m\in\Omega}Lu(m,n)=\sum _{m\in\partial\Omega}\Delta _{N}u(m-1,n)=\sum _{m\in\partial\Omega}-g(m,n)u(m,n)\geq 0,\quad n\in\mathbb{N}_{a}. $$
(26)

From assumption (A) we have

$$ \sum_{m\in\Omega}q(m,n)u(m,n)\leq q(n)\sum _{m\in\Omega}u(m,n), \quad n\in\mathbb{N}_{a}. $$
(27)

Combining (19), (26), and (27), we obtain

$$ \Delta^{\alpha}U(n)+q(n)U(n)\geq H(n), \quad n\in \mathbb{N}_{a}. $$
(28)

Then we have

$$ \Delta^{\alpha}U(n)\geq H(n),\quad n\in\mathbb{N}_{a}. $$
(29)

Using the above-mentioned method in Case 1, we easily obtain a contradiction. This completes the proof of Theorem 3.1. □

Theorem 3.2

If

$$ \liminf_{n\rightarrow\infty}(n-a)^{1-\alpha} \Biggl\{ \sum _{s=a}^{n-\alpha }(n-s-1)^{(\alpha-1)}H(s) \Biggr\} =-\infty $$
(30)

and

$$ \limsup_{n\rightarrow\infty}(n-a)^{1-\alpha} \Biggl\{ \sum _{s=a}^{n-\alpha }(n-s-1)^{(\alpha-1)}H(s) \Biggr\} =+\infty, $$
(31)

where \(H(n)\) is defined as in (16), then every solution \(u(m,n)\) of problem (1)-(B1) is oscillatory in \(\Omega\times\mathbb{N}_{a}\).

Proof

Suppose to the contrary that there is a nonoscillatory solution \(u(m,n)\) of problem (1)-(B1) that has no zero in \(\Omega\times \mathbb{N}_{a}\) for some \(n^{*}\geq a\). Then \(u(m,n)>0\) or \(u(m,n)<0\) for \(n\geq n^{*}\).

Case 1. \(u(m,n)>0, n\geq n^{*}\). As in the proof of Theorem 3.1, we obtain (22). Applying the operator \(\Delta^{-\alpha}\) to inequality (22), we have

$$ \Delta^{-\alpha}\Delta_{n}^{\alpha}U(n)\leq \Delta^{-\alpha} H(n). $$
(32)

By Lemma 2.5 it follows from the left-hand side of (32) that

$$\begin{aligned} \Delta^{-\alpha}\Delta_{n}^{\alpha}U(n) =& \Delta^{-\alpha}\Delta \Delta ^{-(1-\alpha)}U(n) \\ =&\Delta\Delta^{-\alpha}\Delta^{-(1-\alpha)}U(n)- \frac {(n-a)^{(\alpha-1)}}{\Gamma(\alpha)} \Delta^{-(1-\alpha)}U(a) \\ =&U(n)- \frac{C_{0}}{\Gamma(\alpha)}(n-a)^{(\alpha-1)}, \end{aligned}$$
(33)

where \(\Delta^{-(1-\alpha)}U(a)=\Delta^{-(1-\alpha)}U(n) |_{n=a}=C_{0}\) is a constant. Applying Definition 2.1 to the right-hand side of (32), we have

$$ \Delta^{-\alpha}H(n)= \frac{1}{\Gamma(\alpha)}\sum _{s=a}^{n-\alpha}(n-s-1)^{(\alpha-1)} H(s). $$
(34)

Combining (32)-(34), we get

$$ U(n)\leq\frac{C_{0}}{\Gamma(\alpha)}(n-a)^{(\alpha-1)}+\frac{1}{\Gamma (\alpha)}\sum _{s=a}^{n-\alpha}(n-s-1)^{(\alpha-1)} H(s). $$
(35)

It follows from (35) that

$$\begin{aligned} \Gamma(\alpha) (n-a)^{1-\alpha }U(n) \leq& C_{0}(n-a)^{(\alpha-1)}(n-a)^{1-\alpha} \\ &{}+ (n-a)^{1-\alpha}\sum_{s=a}^{n-\alpha}(n-s-1)^{(\alpha-1)}H(s). \end{aligned}$$
(36)

Using Lemma 2.8, we obtain

$$\begin{aligned} & \lim_{n\rightarrow\infty}(n-a)^{1-\alpha }(n-a)^{(\alpha -1)} \\ &\quad= \lim_{n\rightarrow\infty}(n-a)^{1-\alpha} \frac{\Gamma(n-a+1)}{\Gamma(n-a+1+(1-\alpha))} \\ &\quad= \lim_{n\rightarrow\infty}(n-a)^{1-\alpha}\frac {(n-a)\Gamma(n-a)}{(n-a+1-\alpha)\Gamma(n-a+(1-\alpha))} \\ &\quad= \lim_{n\rightarrow\infty}\frac{n-a}{n-a+1-\alpha }\frac {\Gamma(n-a)(n-a)^{1-\alpha}}{\Gamma(n-a+(1-\alpha))} \\ &\quad=1. \end{aligned}$$
(37)

Noting (37) and taking \(n\rightarrow\infty\) in (36), we have

$$\liminf_{n\rightarrow\infty} \bigl\{ (n-a)^{1-\alpha}U(n) \bigr\} \leq - \infty, $$

which contradicts with \(U(n)>0\).

Case 2. \(u(m,n)<0, n\geq n_{0}\). As in the proof of Theorem 3.1, we obtain the fractional difference inequality (29). Then using the above-mentioned method, we easily obtain a contradiction. This completes the proof of Theorem 3.2. □

4 Oscillation of problem (1)-(B2)

Theorem 4.1

For \(n_{0}\in \mathbb{N}_{a}\), if

$$ \liminf_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1} \bigl(\Phi (s)+H(s)\bigr)=-\infty $$
(38)

and

$$ \limsup_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1} \bigl(\Phi (s)+H(s)\bigr)=+\infty, $$
(39)

where \(\Phi(n)\) and \(H(n)\) are defined as in (16), then every solution \(u(m,n)\) of problem (1)-(B2) is oscillatory in \(\Omega\times\mathbb{N}_{a}\).

Proof

Suppose to the contrary that there is a nonoscillatory solution \(u(m,n)\) of problem (1)-(B2) that has no zero in \(\Omega\times \mathbb{N}_{a}\) for some \(n^{*}\geq a\). Then \(u(m,n)>0\) or \(u(m,n)<0\) for \(n\geq n^{*}\).

Case 1. \(u(m,n)>0, n\geq n^{*}\). As in the proof of Theorem 3.1, we obtain (19). Using the discrete Gaussian formula and noting the boundary condition (B2), it follows from (19) that

$$ \sum_{m\in\Omega}Lu(m,n)=\sum _{m\in\partial\Omega}\Delta _{N}u(m-1,n)=\sum _{m\in\partial\Omega}\phi(m,n),\quad n\in\mathbb{N}_{a}. $$
(40)

Combing (19), (21), and (40), we have

$$ \Delta^{\alpha}U(n)+q(n)U(n)\leq\Phi(n)+H(n),\quad n\in \mathbb {N}_{a}. $$
(41)

The remainder of the proof is similar to that of Case 1 in Theorem 3.1. We omit it here.

Case 2. \(u(m,n)<0, n\geq n^{*}\). In this case, we easily obtain (19), (27), and (40). Then we have

$$ \Delta^{\alpha}U(n)+q(n)U(n)\geq\Phi(n)+H(n),\quad n\in\mathbb {N}_{a}. $$
(42)

The remainder of the proof is similar to that of Case 2 in Theorem 3.1. We omit it here, too. The proof of Theorem 4.1 is complete. □

Theorem 4.2

If

$$ \liminf_{n\rightarrow\infty}(n-a)^{1-\alpha} \Biggl\{ \sum _{s=a}^{n-\alpha }(n-s-1)^{(\alpha-1)}\bigl( \Phi(s)+H(s)\bigr) \Biggr\} =-\infty $$
(43)

and

$$ \limsup_{n\rightarrow\infty}(n-a)^{1-\alpha} \Biggl\{ \sum _{s=a}^{n-\alpha }(n-s-1)^{(\alpha-1)}\bigl( \Phi(s)+H(s)\bigr) \Biggr\} =+\infty, $$
(44)

where \(\Phi(n)\) and \(H(n)\) are defined as in (16), then every solution \(u(m,n)\) of problem (1)-(B2) is oscillatory in \(\Omega\times\mathbb{N}_{a}\).

5 Examples

Example 5.1

Consider the fractional partial difference equation

$$\begin{aligned} \Delta_{n}^{\frac{1}{2}}u(m,n) =& 2nLu(m,n) -\frac{2n}{m}u(m,n) \\ &{}+ \biggl\{ \frac{m}{3} +\frac{1}{3}\bigl[(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}-2 \bigr] \biggr\} , \quad (m,n)\in \mathbb{N}(1,3)\times\mathbb{N}_{0}, \end{aligned}$$
(45)

with boundary condition

$$ \Delta_{N}u(0,n)=\Delta_{N}u(4,n)=0,\quad n \in\mathbb{N}_{0}. $$
(46)

Here \(\alpha=\frac{1}{2}, a(n)=2n\), \(q(m,n)=\frac{2n}{m}\), \(h(m,n)=\frac{m}{3}+\frac{1}{3}[(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}-2]\). It is easy to see that \(q(n)=\frac{2}{3}n\) and

$$H(n)=\sum_{m\in\mathbb{N}(1,3)}h(m,n)=(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}. $$

Therefore,

$$ \sum_{s=n_{0}}^{n-1}H(s)=\sum _{s=n_{0}}^{n-1} \bigl\{ (-1)^{s+1}e^{s+1}-(-1)^{s}e^{s} \bigr\} =(-1)^{n}e^{n}-(-1)^{n_{0}}e^{n_{0}}, \quad n_{0}\in \mathbb{N}_{0}. $$
(47)

It follows from (47) that

$$\liminf_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1}H(s)=- \infty $$

and

$$\limsup_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1} H(s)=+\infty. $$

Using Theorem 3.1, we obtain that every solution of problem (45)-(46) is oscillatory in \(\mathbb{N}(1,3)\times\mathbb{N}_{0}\).

Example 5.2

Consider the fractional partial difference equation

$$\begin{aligned} \Delta_{n}^{\frac{1}{4}}u(m,n) =& 2\Gamma(n)Lu(m,n) - \frac{\Gamma(n+\frac{3}{4})}{2m\Gamma(n)}u(m,n) \\ &{}+ \frac{1}{4}\Gamma\biggl(\frac{1}{4}\biggr)m+ \frac{n}{2}, \quad (m,n)\in \mathbb{N}(1,2)\times\mathbb{N}_{0}, \end{aligned}$$
(48)

with boundary condition

$$ \Delta_{N}u(0,n)=\Delta_{N}u(3,n)=0, \quad n \in\mathbb{N}_{0}. $$
(49)

Here \(\alpha=\frac{1}{4}, a(n)=2\Gamma(n)\), \(q(m,n)=\frac{\Gamma (n+\frac{3}{4})}{2m\Gamma(n)}\), \(h(m,n)=\frac{1}{4}\Gamma(\frac{1}{4})m+\frac{n}{2}\). It is easy to see that

$$q(n)=\frac{\Gamma(n+\frac{3}{4})}{4\Gamma(n)}, \qquad H(n)=\sum_{m\in\mathbb{N}(1,2)}h(m,n)= \frac{3}{4}\Gamma\biggl(\frac{1}{4}\biggr)+n. $$

Therefore,

$$ \sum_{s=0}^{n-\alpha}(n-s-1)^{(\alpha-1)}H(s)= \sum_{s=0}^{n-\frac{1}{4}}(n-s-1)^{(-\frac{3}{4})} \biggl( \frac{3}{4}\Gamma\biggl(\frac{1}{4}\biggr)+s \biggr)>0, \quad n\in \mathbb{N}_{0}, $$
(50)

which shows that condition (30) of Theorem 3.2 does not hold. Indeed, \(u(m,n)=mn^{(\frac{1}{4})}\) is a nonoscillatory solution of problem (48)-(49).

Example 5.3

Consider the fractional partial difference equation

$$ \begin{aligned}[b] \Delta_{n}^{\frac{1}{3}}u(m,n)={}& \frac{1}{2}Lu(m,n)- \frac{2\Gamma(\frac{1}{3})\Gamma(n+\frac{2}{3})}{ 3n\Gamma(n)}u(m,n)\\ &{} +\Gamma\biggl(\frac{1}{3}\biggr)m^{2}-\frac{n\Gamma(n)}{\Gamma (n+\frac{2}{3})}, \quad (m,n)\in \mathbb{N}(1,2)\times\mathbb{N}_{0}, \end{aligned} $$
(51)

with boundary condition

$$ \Delta_{N}u(0,n)=\Delta_{N}u(3,n)= \frac{2n\Gamma (n)}{\Gamma(n+\frac{2}{3})}, \quad n\in \mathbb{N}_{0}. $$
(52)

Here \(\alpha=\frac{1}{3}, a(n)=\frac{1}{2}, q(m,n)=\frac{2\Gamma(\frac{1}{3})\Gamma(n+\frac{2}{3})}{ 3n\Gamma(n)}, h(m,n)=\Gamma(\frac{1}{3})m^{2}-\frac{n\Gamma (n)}{\Gamma (n+\frac{2}{3})}, \phi(m,n)=\frac{2n\Gamma(n)}{\Gamma(n+\frac{2}{3})}\). Therefore,

$$\begin{aligned}& q(n)=\frac{2\Gamma(\frac{1}{3})\Gamma(n+\frac{2}{3})}{ 3n\Gamma(n)}, \qquad H(n)=\sum_{m\in\mathbb{N}(1,2)}h(m,n)=5 \Gamma \biggl(\frac{1}{3}\biggr)-\frac{2n\Gamma(n)}{\Gamma(n+\frac{2}{3})}, \\& \Phi(n) =\sum_{m\in\{0,3\}}\phi(m,n)=\frac{4n\Gamma(n)}{\Gamma(n+\frac{2}{3})}. \end{aligned}$$

It is easy to see that

$$ \sum_{s=0}^{n-1}\bigl[ \Phi(s)+H(s)\bigr]=\sum_{s=0}^{n-1} \biggl[5 \Gamma\biggl(\frac{1}{3}\biggr)+\frac{2s\Gamma(s)}{\Gamma(s+\frac{2}{3})} \biggr] >0, \quad n\in \mathbb{N}_{0}. $$
(53)

Thus, this time, condition (38) of Theorem 4.1 is false. Indeed, we easily see that \(u(m,n)=m^{2}n^{(\frac{1}{3})}\) is a nonoscillatory solution of the problem (51)-(52).

Example 5.4

Consider the fractional partial difference equation

$$\begin{aligned} \Delta_{n}^{\frac{2}{3}}u(m,n) =& 3nLu(m,n) -\frac{n}{m}u(m,n) \\ &{} + \biggl\{ \frac{m}{3}+\frac{1}{2}\bigl[(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}-1 \bigr] \biggr\} ,\quad (m,n)\in \mathbb{N}(1,2)\times\mathbb{N}_{0}, \end{aligned}$$
(54)

with boundary condition

$$ \Delta_{N}u(0,n)=\Delta_{N}u(3,n)= \frac{1}{4}\bigl[(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n} \bigr],\quad n\in\mathbb{N}_{0}. $$
(55)

Here \(\alpha=\frac{2}{3}, a(n)=3n\), \(q(m,n)=\frac{n}{m}\), \(h(m,n)=\frac{m}{3}+\frac{1}{2}[(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}-1]\), and \(\phi (m,n)=\frac{1}{4}[(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}]\). It is easy to see that \(q(n)=\frac{n}{2}\),

$$H(n)=\sum_{m\in\mathbb{N}(1,2)}h(m,n)=(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}, $$

and

$$\Phi(n)=\sum_{m\in\{0,3\}}=\frac{1}{2} \bigl[(-1)^{n+1}e^{n+1}-(-1)^{n}e^{n}\bigr]. $$

Therefore,

$$ \begin{aligned}[b] \sum_{s=n_{0}}^{n-1}\bigl(H(s)+ \Phi (s)\bigr)&= \frac{3}{2}\sum_{s=n_{0}}^{n-1} \bigl\{ (-1)^{s+1}e^{s+1}-(-1)^{s}e^{s} \bigr\} \\ &= \frac{3}{2} \bigl\{ (-1)^{n}e^{n}-(-1)^{n_{0}}e^{n_{0}} \bigr\} ,\quad n_{0}\in\mathbb{N}_{0}. \end{aligned} $$
(56)

It follows from (56) that

$$\liminf_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1} \bigl(H(s)+\Phi (s)\bigr)=-\infty $$

and

$$\limsup_{n\rightarrow\infty}\sum_{s=n_{0}}^{n-1} \bigl(H(s)+\Phi (s)\bigr)=+\infty. $$

We easily see that the conditions of Theorem 4.1 are satisfied. Then every solution of problem (54)-(55) is oscillatory in \(\mathbb {N}(1,2)\times\mathbb{N}_{0}\).