Theory and Modern Applications

# Oscillation of solutions of some generalized nonlinear α-difference equations

## Abstract

In this paper, the authors discuss the oscillation of solutions of some generalized nonlinear α-difference equation

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)+q\left(k\right)f\left(u\left(k-\tau \left(k\right)\right)\right)=0,$
(1)

$k\in \left[a,\mathrm{\infty }\right)$, where the functions p, q, f and τ are defined in their domain of definition and $\alpha >1$, is a positive real. Further, $uf\left(u\right)>0$ for $u\ne 0$, $p\left(k\right)>0$ and ${lim}_{k\to \mathrm{\infty }}\left(k-\tau \left(k\right)\right)=\mathrm{\infty }$, where ${R}_{k}={\sum }_{r=0}^{\left[\frac{k-\ell }{\ell }\right]}\frac{1}{{\alpha }^{r}p\left(r\ell \right)}\to \mathrm{\infty }$ as $k\to \mathrm{\infty }$ and $u\left(k\right)$ is defined for $k\ge {min}_{i\ge 0}\left(i-\tau \left(i\right)\right)$ for all $k\in \left[a,\mathrm{\infty }\right)$ for some $a\in \left[0,\mathrm{\infty }\right)$.

MSC:39A12.

## 1 Introduction

The basic theory of difference equations is based on the operator Δ defined as $\mathrm{\Delta }u\left(k\right)=u\left(k+1\right)-u\left(k\right)$, $k\in \mathbb{N}=\left\{0,1,2,3,\dots \right\}$. Even though many authors  have suggested the definition of Δ as

$\mathrm{\Delta }u\left(k\right)=u\left(k+\ell \right)-u\left(k\right),\phantom{\rule{1em}{0ex}}k\in \mathbb{R},\ell \in \mathbb{R}-\left\{0\right\},$
(2)

there was no significant progress in this area. But recently,  considered the definition of Δ as given in (2) and developed the theory of difference equations in a different direction. For convenience, the operator Δ defined by (2) is labeled as ${\mathrm{\Delta }}_{\ell }$, and by defining its inverse ${\mathrm{\Delta }}_{\ell }^{-1}$, many interesting results and applications in number theory (see ) were obtained. By extending the study related to the sequences of complex numbers and being real, some new qualitative properties of the solutions like rotatory, expanding, shrinking, spiral and weblike were obtained for difference equation involving ${\mathrm{\Delta }}_{\ell }$. The results obtained using ${\mathrm{\Delta }}_{\ell }$ can be found in . Popenda and Szmanda [8, 9] defined Δ as

${\mathrm{\Delta }}_{\alpha }u\left(k\right)=u\left(k+1\right)-\alpha u\left(k\right),$
(3)

and based on this definition, they studied the qualitative properties of a particular difference equation, and no one else has handled this operator. Recently Manuel et al. [6, 10] considered the definition of ${\mathrm{\Delta }}_{\ell }$ as given in (3), and by defining its inverse, some interesting results on number theory were obtained.

In , Szafranski and Szmanda obtained sufficient conditions for the oscillation of a similar difference equation involving Δ. In this paper the theory is extended from Δ to ${\mathrm{\Delta }}_{\alpha \left(\ell \right)}$ for all real $k\in \left[a,\mathrm{\infty }\right)$, and we discuss the oscillatory behavior of solutions of generalized nonlinear α-difference equation (1).

Throughout this paper, we make use of the following assumptions.

1. (a)

$\mathbb{N}=\left\{0,1,2,3,\dots \right\}$, $\mathbb{N}\left(a\right)=\left\{a,a+1,a+2,\dots \right\}$;

2. (b)

${\mathbb{N}}_{\ell }\left(a\right)=\left\{a,a+\ell ,a+2\ell ,\dots \right\}$;

3. (c)

$⌈x⌉$ and $\left[x\right]$ denote upper integer and integer part of x, respectively;

4. (d)

$j=k-{k}_{i}-\left[\frac{k-{k}_{i}}{\ell }\right]\ell$, ${k}_{i}\in \left[0,\mathrm{\infty }\right)$.

## 2 Preliminaries

In this section, we present some preliminaries which will be useful for future discussion.

Definition 2.1 

The inverse of the generalized α-difference operator denoted by ${\mathrm{\Delta }}_{\alpha \left(\ell \right)}^{-1}$ on $u\left(k\right)$ is defined as follows. If ${\mathrm{\Delta }}_{\alpha \left(\ell \right)}v\left(k\right)=u\left(k\right)$, then

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}^{-1}u\left(k\right)=v\left(k\right)-{\alpha }^{\left[\frac{k}{\ell }\right]}v\left(j\right),$
(4)

where $k\in {\mathbb{N}}_{\ell }\left(j\right)$, $j=k-\left[\frac{k}{\ell }\right]\ell$.

Lemma 2.2 

If the real-valued function $u\left(k\right)$ is defined for all $k\in \left[a,\mathrm{\infty }\right)$ and $\alpha >1$, then

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}^{-1}u\left(k\right)=\sum _{r=0}^{\left[\frac{k-a-j-\ell }{\ell }\right]}\frac{u\left(a+j+r\ell \right)}{{\alpha }^{⌈\frac{a+j+\ell -k+r\ell }{\ell }⌉}}+{\alpha }^{⌈\frac{k-a}{\ell }⌉}u\left(a+j\right)$
(5)

for all $k\in {\mathbb{N}}_{\ell }\left(j\right)$, $j=k-a-\left[\frac{k-a}{\ell }\right]\ell$.

Definition 2.3 

The solution $u\left(k\right)$ of (1) is called oscillatory if for any ${k}_{1}\in \left[a,\mathrm{\infty }\right)$ there exists ${k}_{2}\in {\mathbb{N}}_{\ell }\left({k}_{1}\right)$ such that $u\left({k}_{2}\right)u\left({k}_{2}+\ell \right)\le 0$. The difference equation itself is called oscillatory if all its solutions are oscillatory. If the solution $u\left(k\right)$ is not oscillatory, then it is said to be nonoscillatory (i.e., $u\left(k\right)u\left(k+\ell \right)>0$ for all $k\in \left[{k}_{1},\mathrm{\infty }\right)$).

## 3 Main results

In this section we present conditions for the oscillation of equation (1).

Theorem 3.1 Assume that

1. (i)

$q\left(k\right)\ge 0$ and ${\sum }_{r=0}^{\mathrm{\infty }}{\alpha }^{r}q\left(r\ell \right)=\mathrm{\infty }$,

2. (ii)

${lim inf}_{|u\left(k\right)|\to \mathrm{\infty }}|f\left(u\left(k\right)\right)|>0$.

Then every solution of Equation (1) is oscillatory.

Proof Assume that Equation (1) has a nonoscillatory solution $u\left(k\right)$, and we assume that $u\left(k\right)$ is eventually positive. Then there is a positive integer ${k}_{1}$ such that

(6)

From Equation (1) we have

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)=-q\left(k\right)f\left(u\left(k-\tau \left(k\right)\right)\right),\phantom{\rule{1em}{0ex}}k\ge {k}_{1},$

and so $p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)$ is eventually nonincreasing. We first show that

In fact, if there is ${k}_{2}\ge {k}_{1}$ such that $p\left({k}_{2}\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left({k}_{2}\right)=c<0$ and $p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\le c$ for $k\ge {k}_{2}$, that is,

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\le \frac{c}{p\left(k\right)},$

hence by Lemma 2.2,

which contradicts the fact that $u\left(k\right)>0$ for $k\ge {k}_{2}$. Hence, $p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\ge 0$ for $k\ge {k}_{1}$. Therefore we obtain

Let $L={lim}_{k\to \mathrm{\infty }}u\left(k\right)$.

Then $L>0$ is finite or infinite.

Case 1. $L>0$ is finite.

From the function $f\left(k\right)$ defined in its domain of definition, we have

$\underset{k\to \mathrm{\infty }}{lim}f\left(u\left(k-\tau \left(k\right)\right)\right)=f\left(L\right)>0.$

Thus, we may choose a positive integer ${k}_{4}$ ($\ge {k}_{1}$) such that

$f\left(u\left(k-\tau \left(k\right)\right)\right)>\frac{1}{2}f\left(L\right),\phantom{\rule{1em}{0ex}}k\ge {k}_{4}.$
(7)

By substituting (7) in Equation (1) we obtain

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)+\frac{1}{2}f\left(L\right)q\left(k\right)\le 0,\phantom{\rule{1em}{0ex}}k\ge {k}_{4}.$
(8)

By Lemma 2.2, we obtain

$\begin{array}{r}p\left(k+\ell \right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k+\ell \right)-{\alpha }^{⌈\frac{k-{k}_{4}}{\ell }⌉}p\left({k}_{4}+j\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left({k}_{4}+j\right)\\ \phantom{\rule{1em}{0ex}}+\frac{1}{2}f\left(L\right)\sum _{r=0}^{\frac{k-{k}_{4}-\ell -j}{\ell }}\frac{p\left({k}_{4}+j+r\ell \right)}{{\alpha }^{⌈\frac{{k}_{4}+j+\ell -k+r\ell }{\ell }⌉}}\le 0,\end{array}$

and so

$\frac{1}{2}f\left(L\right)\sum _{r=0}^{\frac{k-{k}_{4}-\ell -j}{\ell }}\frac{p\left({k}_{4}+j+r\ell \right)}{{\alpha }^{⌈\frac{{k}_{4}+j+\ell -k+r\ell }{\ell }⌉}}\le {\alpha }^{⌈\frac{k-{k}_{4}}{\ell }⌉}p\left({k}_{4}+j\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left({k}_{4}+j\right),\phantom{\rule{1em}{0ex}}k\ge {k}_{4},$

Case 2. $L=\mathrm{\infty }$. For this case, from condition (ii) we have

$\underset{k\to \mathrm{\infty }}{lim inf}f\left(u\left(k-\tau \left(k\right)\right)\right)>0,$

and so we may choose a positive constant c and a positive integer ${k}_{5}$ sufficiently large such that

(9)

Substituting (9) into Equation (1) we have

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)+cq\left(k\right)\le 0,\phantom{\rule{1em}{0ex}}k\le {k}_{5}.$

Using a similar argument as in Case 1, we obtain a contradiction to condition (i). This completes the proof. □

Example 3.2 For the generalized α-difference equation

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(\frac{1}{k}{\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)=\left(\frac{4{k}^{2}+6k\ell +{\ell }^{2}}{{\left(k+\ell \right)}_{\ell }^{\left(2\right)}}\right){\left(-\alpha \right)}^{⌈\frac{k+2\ell }{\ell }⌉},$

all the conditions of Theorem 3.1 hold and hence all the solutions are oscillatory. In fact $u\left(k\right)={\left(-\alpha \right)}^{⌈\frac{k}{\ell }⌉}k$ is one such solution.

Theorem 3.3 Assume that

1. (iii)

$q\left(k\right)\ge 0$ and ${\sum }_{r=0}^{\mathrm{\infty }}{\alpha }^{r}R\left(r\ell \right)q\left(r\ell \right)=\mathrm{\infty }$.

Then every bounded solution of (1) is oscillatory.

Proof Proceeding as in the proof of Theorem 3.1, with the assumption that $u\left(k\right)$ is a bounded nonoscillatory solution of (1), we get inequality (8), and so we obtain

$R\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)+\frac{1}{2}f\left(L\right)R\left(k\right)q\left(k\right)\le 0,\phantom{\rule{1em}{0ex}}k\ge {k}_{4}.$
(10)

It is easy to see that

$\begin{array}{r}R\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)\\ \phantom{\rule{1em}{0ex}}\ge {\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(R\left(k\right)p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)-\alpha p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}R\left(k\right).\end{array}$
(11)

Using (10) in (11), (11) reduces to

$\begin{array}{r}R\left(k\right)p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)-{\alpha }^{⌈\frac{k-{k}_{2}}{\ell }⌉}R\left({k}_{2}+j\right)p\left({k}_{2}+j\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left({k}_{2}+j\right)\\ \phantom{\rule{1em}{0ex}}-\alpha \sum _{r=0}^{\frac{k-{k}_{4}-j-\ell }{\ell }}\frac{p\left({k}_{2}+j+r\ell \right)}{{\alpha }^{⌈\frac{{k}_{2}-k+j+\ell +r\ell }{\ell }⌉}}{\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left({k}_{2}+j+r\ell \right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}R\left({k}_{2}+j+r\ell \right)\\ \phantom{\rule{1em}{0ex}}+\frac{1}{2}f\left(L\right)\sum _{r=0}^{\frac{k-{k}_{4}-j-\ell }{\ell }}\frac{R\left({k}_{2}+j+r\ell \right)}{{\alpha }^{⌈\frac{{k}_{2}-k+j+\ell +r\ell }{\ell }⌉}}q\left({k}_{2}+j+r\ell \right)\le 0,\end{array}$

which implies

$\begin{array}{r}\frac{1}{2}f\left(L\right)\sum _{r=0}^{\frac{k-{k}_{4}-j-\ell }{\ell }}\frac{R\left({k}_{2}+j+r\ell \right)}{{\alpha }^{⌈\frac{{k}_{2}-k+j+\ell +r\ell }{\ell }⌉}}q\left({k}_{2}+j+r\ell \right)\\ \phantom{\rule{1em}{0ex}}\le u\left(k+\ell \right)+{\alpha }^{⌈\frac{k-{k}_{4}}{\ell }⌉}R\left({k}_{4}+j\right)p\left({k}_{4}+j\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left({k}_{4}+j+r\ell \right)-{\alpha }^{⌈\frac{k-{k}_{4}}{\ell }⌉}u\left({k}_{4}+j\right),\phantom{\rule{1em}{0ex}}k\ge {k}_{4}.\end{array}$

Hence, there exists a constant c such that

which is a contradiction to condition (iii) which completes the proof. □

Example 3.4 For the generalized α-difference equation

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(k{\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)=\left(\frac{\left(\alpha {2}^{\ell }+1\right)\left(\alpha k+k+\ell \right)}{{2}^{k+2\ell }}\right){\left(-1\right)}^{⌈\frac{k+2\ell }{\ell }⌉},$

all the conditions of Theorem 3.3 hold and hence all the solutions are oscillatory. In fact $u\left(k\right)=\frac{{\left(-1\right)}^{⌈\frac{k}{\ell }⌉}}{{2}^{k}}$ is one such solution.

Theorem 3.5 Assume that

1. (iv)

$\left(k-\tau \left(k\right)\right)$ is nondecreasing, where $\tau \left(k\right)\in \left[0,\mathrm{\infty }\right)$,

2. (v)

there exists $p\left({k}_{n}\right)$ such that $p\left({k}_{n}\right)\le 1$ for ${k}_{n}\in \left[0,\mathrm{\infty }\right)$,

3. (vi)

${\sum }_{r=0}^{\mathrm{\infty }}{\alpha }^{r}q\left(r\ell \right)=\mathrm{\infty }$,

4. (vii)

f is nondecreasing and there is a nonnegative constant M such that

$\underset{s\to 0}{lim sup}\frac{s}{f\left(s\right)}=M.$
(12)

Then the difference ${\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)$ of every solution $u\left(k\right)$ of Equation (1) oscillates.

Proof If not, then Equation (1) has a solution $u\left(k\right)$ such that its difference ${\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)$ is nonoscillatory. Assume first that the sequence ${\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)$ is eventually negative. Then there is a positive integer ${k}_{1}$ such that

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)<0,\phantom{\rule{1em}{0ex}}k>{k}_{1},$

and so $u\left(k\right)$ is decreasing for $k\ge {k}_{1}$, which implies that $u\left(k\right)$ is also nonoscillatory. Set

$w\left(k\right)=\frac{p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)}{f\left(u\left(k-\tau \left(k\right)\right)\right)}k\ge {k}_{2}\ge {k}_{1}.$
(13)

Then

$\begin{array}{rl}{\mathrm{\Delta }}_{\alpha \left(\ell \right)}w\left(k\right)=& \frac{p\left(k+\ell \right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k+\ell \right)}{f\left(u\left(k+\ell -\tau \left(k+\ell \right)\right)\right)}-\alpha \frac{p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)}{f\left(u\left(k-\tau \left(k\right)\right)\right)}=\frac{{\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)}{f\left(u\left(k-\tau \left(k\right)\right)\right)}\\ +p\left(k+\ell \right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k+\ell \right)\frac{f\left(u\left(k-\tau \left(k\right)\right)\right)-f\left(u\left(k+\ell -\tau \left(k+\ell \right)\right)\right)}{f\left(u\left(k+\ell -\tau \left(k+\ell \right)\right)\right)f\left(u\left(k-\tau \left(k\right)\right)\right)}\\ \le & \frac{{\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)}{f\left(u\left(k-\tau \left(k\right)\right)\right)}=-q\left(k\right),\phantom{\rule{1em}{0ex}}k\ge {k}_{2}.\end{array}$
(14)

By Lemma 2.2, we have

$w\left(k+\ell \right)-{\alpha }^{⌈\frac{k-{k}_{2}}{\ell }⌉}w\left({k}_{2}+j\right)\le -\sum _{r=0}^{\frac{k-{k}_{2}-j}{\ell }}\frac{q\left({k}_{2}+j+r\ell \right)}{{\alpha }^{⌈\frac{{k}_{2}+j-k+r\ell }{\ell }⌉}},$

and by (vi) we get

$\underset{k\to \mathrm{\infty }}{lim}w\left(k\right)=-\mathrm{\infty },$
(15)

which implies that eventually

(16)

By (15), we can choose ${k}_{3}$ ($\ge {k}_{2}$) such that

$w\left(k\right)\le -\left(M+\ell \right),\phantom{\rule{1em}{0ex}}k\ge {k}_{3}.$

That is,

$p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)+\left(M+\ell \right)f\left(u\left(k-\tau \left(k\right)\right)\right)\le 0,\phantom{\rule{1em}{0ex}}k\ge {k}_{3}.$
(17)

Set ${lim}_{k\to \mathrm{\infty }}u\left(k\right)=L$. Then $L\ge 0$. Now we prove that $L=0$. If $L>0$, then we have

$\underset{k\to \mathrm{\infty }}{lim}f\left(u\left(k-\tau \left(k\right)\right)\right)=f\left(L\right)>0$

since $f\left(k\right)$ is defined in its domain of definition. Choosing ${k}_{4}$ sufficiently large such that

$f\left(u\left(k-\tau \left(k\right)\right)\right)>\frac{1}{2}f\left(L\right),\phantom{\rule{1em}{0ex}}k\ge {k}_{4},$
(18)

and substituting (19) into (17), we obtain

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)+\frac{1}{2p\left(k\right)}\left(M+\ell \right)f\left(L\right)\le 0,\phantom{\rule{1em}{0ex}}k\ge {k}_{4}.$
(19)

From Lemma 2.2, we have

$u\left(k+\ell \right)-{\alpha }^{⌈\frac{k-{k}_{4}}{\ell }⌉}u\left({k}_{4}+j\right)+\frac{1}{2}\left(M+\ell \right)f\left(L\right)\sum _{r=0}^{\frac{k-{k}_{4}-\ell -j}{\ell }}\frac{1}{{\alpha }^{⌈\frac{{k}_{4}-k+j+r\ell }{\ell }⌉}p\left({k}_{4}-k+j+r\ell \right)}\le 0,$

which implies that ${lim}_{k\to \mathrm{\infty }}u\left(k\right)=-\mathrm{\infty }$.

This contradicts (16). Hence ${lim}_{k\to \mathrm{\infty }}u\left(k\right)=0$.

By the assumptions we have

$\underset{k\to \mathrm{\infty }}{lim sup}\frac{u\left(k-\tau \left(k\right)\right)}{f\left(u\left(k-\tau \left(k\right)\right)\right)}\le M.$

From this we can choose ${k}_{4}$ such that

$\frac{u\left(k-\tau \left(k\right)\right)}{f\left(u\left(k-\tau \left(k\right)\right)\right)}

That is, $u\left(k-\tau \left(k\right)\right)<\left(M+\ell \right)f\left(u\left(k-\tau \left(k\right)\right)\right)$, $k\ge {k}_{5}$, and so from (17) we get

$p\left(k\right){\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)+u\left(k-\tau \left(k\right)\right)<0,\phantom{\rule{1em}{0ex}}k\ge {k}_{5}.$

In particular, for a function $p\left({k}_{n}\right)$ satisfying condition (v), we have

$u\left({k}_{n}+l\right)-\alpha u\left({k}_{n}\right)+{x}_{{k}_{n}}-\tau \left({k}_{n}\right)\le p\left({k}_{n}\right)\left(u\left({k}_{n}+l\right)-\alpha u\left({k}_{n}\right)\right)+u\left({k}_{n}-\tau \left({k}_{n}\right)\right)<0$

for k sufficiently large, which implies that

$0

for all large k. This is a contradiction. The case that ${\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)$ is eventually positive can be treated in a similar fashion and this completes the proof of the theorem. □

Example 3.6 For the generalized α-difference equation

${\mathrm{\Delta }}_{\alpha \left(\ell \right)}\left(\frac{1}{k}{\mathrm{\Delta }}_{\alpha \left(\ell \right)}u\left(k\right)\right)=\left(\frac{\left({2}^{\ell }+1\right)\left(\left({2}^{\ell }+1\right)k+\ell \right)}{{\left(k+\ell \right)}_{\ell }^{\left(2\right)}}\right){\left(-\alpha \right)}^{⌈\frac{k+2\ell }{\ell }⌉}{2}^{k},$

all the conditions of Theorem 3.5 hold and hence all the solutions are oscillatory. In fact $u\left(k\right)={\left(-\alpha \right)}^{⌈\frac{k}{\ell }⌉}{2}^{k}$ is one such solution.

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## Acknowledgements

The authors express their sincere thanks to the referees for the careful and detailed reading of the manuscript and very helpful suggestions. Research supported by the National Board for Higher Mathematics, Department of Atomic Energy, Government of India, Mumbai. The second authors also gratefully acknowledges that part of this research was partially supported by the University Putra Malaysia under the GP-IBT Grant Scheme having project no. GP-IBT/2013/9420100.

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Manuel, M.M.S., Kılıçman, A., Srinivasan, K. et al. Oscillation of solutions of some generalized nonlinear α-difference equations. Adv Differ Equ 2014, 109 (2014). https://doi.org/10.1186/1687-1847-2014-109 