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

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

(1)

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

MSC:39A12.

1 Introduction

The basic theory of difference equations is based on the operator Δ defined as $\mathrm{\Delta }u(k)=u(k+1)-u(k)$, $k\in \mathbb{N}=\{0,1,2,3,\dots \}$. Even though many authors [1–4] have suggested the definition of Δ as

$$\mathrm{\Delta }u(k)=u(k+\ell )-u(k),k\in \mathbb{R},\ell \in \mathbb{R}-\{0\},$$

(2)

there was no significant progress in this area. But recently, [5] 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 [5–7]) were obtained. By extending the study related to the sequences of complex numbers and ℓ 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 [5–7]. Popenda and Szmanda [8, 9] defined Δ as

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

(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 [11], 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 ${\mathrm{\Delta }}_{\alpha (\ell )}$ for all real $k\in [a,\mathrm{\infty })$, 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.

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

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

• (c) $\lceil x\rceil$ and $[x]$ denote upper integer and integer part of x, respectively;

• (d) $j=k-k_i-[\frac{k-k_i}{\ell }]\ell$, $k_i\in [0,\mathrm{\infty })$.

2 Preliminaries

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

Definition 2.1 [12]

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

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

(4)

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

Lemma 2.2 [13]

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

$${\mathrm{\Delta }}_{\alpha (\ell )}^{-1}u(k)=\sum _{r=0}^{[\frac{k-a-j-\ell }{\ell }]}\frac{u(a+j+r\ell )}{{\alpha }^{\lceil \frac{a+j+\ell -k+r\ell }{\ell }\rceil }}+{\alpha }^{\lceil \frac{k-a}{\ell }\rceil }u(a+j)$$

(5)

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

Definition 2.3 [7]

The solution $u(k)$ of (1) is called oscillatory if for any $k_1\in [a,\mathrm{\infty })$ there exists $k_2\in {\mathbb{N}}_{\ell }(k_1)$ such that $u(k_2)u(k_2+\ell )\le 0$. The difference equation itself is called oscillatory if all its solutions are oscillatory. If the solution $u(k)$ is not oscillatory, then it is said to be nonoscillatory (i.e., $u(k)u(k+\ell )>0$ for all $k\in [k_1,\mathrm{\infty })$).

3 Main results

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

Theorem 3.1 Assume that

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

• (ii) ${lim\hspace{0.17em}inf}_{|u(k)|\to \mathrm{\infty }}|f(u(k))|>0$.

Then every solution of Equation (1) is oscillatory.

Proof Assume that Equation (1) has a nonoscillatory solution $u(k)$, and we assume that $u(k)$ is eventually positive. Then there is a positive integer $k_1$ such that

$$u(k-\tau (k))>0\text{for }k\ge k_1.$$

(6)

From Equation (1) we have

$${\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))=-q(k)f\left(u(k-\tau (k))\right),k\ge k_1,$$

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

$$p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)\ge 0\text{for }k\ge k_1.$$

In fact, if there is $k_2\ge k_1$ such that $p(k_2){\mathrm{\Delta }}_{\alpha (\ell )}u(k_2)=c<0$ and $p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)\le c$ for $k\ge k_2$, that is,

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

hence by Lemma 2.2,

$$\begin{array}{r}u(k)={\alpha }^{\lceil \frac{k-k_2}{\ell }\rceil }u(k_2+j)+c\sum _{r=0}^{\frac{k-k_2-\ell -j}{\ell }}\frac{1}{{\alpha }^{\lceil \frac{k_2+j+\ell -k+r\ell }{\ell }\rceil }p(k_2+j+\ell +r\ell )}\to -\mathrm{\infty }\\ \text{as }k\to \mathrm{\infty },\end{array}$$

which contradicts the fact that $u(k)>0$ for $k\ge k_2$. Hence, $p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)\ge 0$ for $k\ge k_1$. Therefore we obtain

$$u(k-\tau (k))>0,{\mathrm{\Delta }}_{\alpha (\ell )}u(k)\ge 0,{\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))\le 0\text{for }k\ge k_2.$$

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

Then $L>0$ is finite or infinite.

Case 1. $L>0$ is finite.

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

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

Thus, we may choose a positive integer $k_4$ ($\ge k_1$) such that

$$f\left(u(k-\tau (k))\right)>\frac{1}{2}f(L),k\ge k_4.$$

(7)

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

$${\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))+\frac{1}{2}f(L)q(k)\le 0,k\ge k_4.$$

(8)

By Lemma 2.2, we obtain

$$\begin{array}{r}p(k+\ell ){\mathrm{\Delta }}_{\alpha (\ell )}u(k+\ell )-{\alpha }^{\lceil \frac{k-k_4}{\ell }\rceil }p(k_4+j){\mathrm{\Delta }}_{\alpha (\ell )}u(k_4+j)\\ +\frac{1}{2}f(L)\sum _{r=0}^{\frac{k-k_4-\ell -j}{\ell }}\frac{p(k_4+j+r\ell )}{{\alpha }^{\lceil \frac{k_4+j+\ell -k+r\ell }{\ell }\rceil }}\le 0,\end{array}$$

and so

$$\frac{1}{2}f(L)\sum _{r=0}^{\frac{k-k_4-\ell -j}{\ell }}\frac{p(k_4+j+r\ell )}{{\alpha }^{\lceil \frac{k_4+j+\ell -k+r\ell }{\ell }\rceil }}\le {\alpha }^{\lceil \frac{k-k_4}{\ell }\rceil }p(k_4+j){\mathrm{\Delta }}_{\alpha (\ell )}u(k_4+j),k\ge k_4,$$

which contradicts (i).

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

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}f\left(u(k-\tau (k))\right)>0,$$

and so we may choose a positive constant c and a positive integer $k_5$ sufficiently large such that

$$f\left(u(k-\tau (k))\right)\ge c\text{for }k\ge k_5.$$

(9)

Substituting (9) into Equation (1) we have

$${\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))+cq(k)\le 0,k\le k_5.$$

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

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

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

Theorem 3.3 Assume that

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

Then every bounded solution of (1) is oscillatory.

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

$$R(k){\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))+\frac{1}{2}f(L)R(k)q(k)\le 0,k\ge k_4.$$

(10)

It is easy to see that

$$\begin{array}{r}R(k){\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))\\ \ge {\mathrm{\Delta }}_{\alpha (\ell )}(R(k)p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))-\alpha p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k){\mathrm{\Delta }}_{\alpha (\ell )}R(k).\end{array}$$

(11)

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

$$\begin{array}{r}R(k)p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)-{\alpha }^{\lceil \frac{k-k_2}{\ell }\rceil }R(k_2+j)p(k_2+j){\mathrm{\Delta }}_{\alpha (\ell )}u(k_2+j)\\ -\alpha \sum _{r=0}^{\frac{k-k_4-j-\ell }{\ell }}\frac{p(k_2+j+r\ell )}{{\alpha }^{\lceil \frac{k_2-k+j+\ell +r\ell }{\ell }\rceil }}{\mathrm{\Delta }}_{\alpha (\ell )}u(k_2+j+r\ell ){\mathrm{\Delta }}_{\alpha (\ell )}R(k_2+j+r\ell )\\ +\frac{1}{2}f(L)\sum _{r=0}^{\frac{k-k_4-j-\ell }{\ell }}\frac{R(k_2+j+r\ell )}{{\alpha }^{\lceil \frac{k_2-k+j+\ell +r\ell }{\ell }\rceil }}q(k_2+j+r\ell )\le 0,\end{array}$$

which implies

$$\begin{array}{r}\frac{1}{2}f(L)\sum _{r=0}^{\frac{k-k_4-j-\ell }{\ell }}\frac{R(k_2+j+r\ell )}{{\alpha }^{\lceil \frac{k_2-k+j+\ell +r\ell }{\ell }\rceil }}q(k_2+j+r\ell )\\ \le u(k+\ell )+{\alpha }^{\lceil \frac{k-k_4}{\ell }\rceil }R(k_4+j)p(k_4+j){\mathrm{\Delta }}_{\alpha (\ell )}u(k_4+j+r\ell )-{\alpha }^{\lceil \frac{k-k_4}{\ell }\rceil }u(k_4+j),k\ge k_4.\end{array}$$

Hence, there exists a constant c such that

$$\sum _{r=0}^{\frac{k-k_4-j-\ell }{\ell }}\frac{R(k_4+j+r\ell )}{{\alpha }^{\lceil \frac{k_2-k+j+\ell +r\ell }{\ell }\rceil }}q(k_4+j+r\ell )\le c\text{for all }k\ge k_4,$$

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

Example 3.4 For the generalized α-difference equation

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

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

Theorem 3.5 Assume that

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

• (v) there exists $p(k_n)$ such that $p(k_n)\le 1$ for $k_n\in [0,\mathrm{\infty })$,

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

• (vii) f is nondecreasing and there is a nonnegative constant M such that

  $$\underset{s\to 0}{lim\hspace{0.17em}sup}\frac{s}{f(s)}=M.$$

  (12)

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

Proof If not, then Equation (1) has a solution $u(k)$ such that its difference ${\mathrm{\Delta }}_{\alpha (\ell )}u(k)$ is nonoscillatory. Assume first that the sequence ${\mathrm{\Delta }}_{\alpha (\ell )}u(k)$ is eventually negative. Then there is a positive integer $k_1$ such that

$${\mathrm{\Delta }}_{\alpha (\ell )}u(k)<0,k>k_1,$$

and so $u(k)$ is decreasing for $k\ge k_1$, which implies that $u(k)$ is also nonoscillatory. Set

$$w(k)=\frac{p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)}{f(u(k-\tau (k)))}k\ge k_2\ge k_1.$$

(13)

Then

$$\begin{array}{rl}{\mathrm{\Delta }}_{\alpha (\ell )}w(k)=& \frac{p(k+\ell ){\mathrm{\Delta }}_{\alpha (\ell )}u(k+\ell )}{f(u(k+\ell -\tau (k+\ell )))}-\alpha \frac{p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)}{f(u(k-\tau (k)))}=\frac{{\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))}{f(u(k-\tau (k)))}\\ +p(k+\ell ){\mathrm{\Delta }}_{\alpha (\ell )}u(k+\ell )\frac{f(u(k-\tau (k)))-f(u(k+\ell -\tau (k+\ell )))}{f(u(k+\ell -\tau (k+\ell )))f(u(k-\tau (k)))}\\ \le & \frac{{\mathrm{\Delta }}_{\alpha (\ell )}(p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k))}{f(u(k-\tau (k)))}=-q(k),k\ge k_2.\end{array}$$

(14)

By Lemma 2.2, we have

$$w(k+\ell )-{\alpha }^{\lceil \frac{k-k_2}{\ell }\rceil }w(k_2+j)\le -\sum _{r=0}^{\frac{k-k_2-j}{\ell }}\frac{q(k_2+j+r\ell )}{{\alpha }^{\lceil \frac{k_2+j-k+r\ell }{\ell }\rceil }},$$

and by (vi) we get

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

(15)

which implies that eventually

$$f\left(u(k-\tau (k))\right)>0\text{ and therefore }k-\tau (k)>0.$$

(16)

By (15), we can choose $k_3$ ($\ge k_2$) such that

$$w(k)\le -(M+\ell ),k\ge k_3.$$

That is,

$$p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)+(M+\ell )f\left(u(k-\tau (k))\right)\le 0,k\ge k_3.$$

(17)

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

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

since $f(k)$ is defined in its domain of definition. Choosing $k_4$ sufficiently large such that

$$f\left(u(k-\tau (k))\right)>\frac{1}{2}f(L),k\ge k_4,$$

(18)

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

$${\mathrm{\Delta }}_{\alpha (\ell )}u(k)+\frac{1}{2p(k)}(M+\ell )f(L)\le 0,k\ge k_4.$$

(19)

From Lemma 2.2, we have

$$u(k+\ell )-{\alpha }^{\lceil \frac{k-k_4}{\ell }\rceil }u(k_4+j)+\frac{1}{2}(M+\ell )f(L)\sum _{r=0}^{\frac{k-k_4-\ell -j}{\ell }}\frac{1}{{\alpha }^{\lceil \frac{k_4-k+j+r\ell }{\ell }\rceil }p(k_4-k+j+r\ell )}\le 0,$$

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

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

By the assumptions we have

$$\underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{u(k-\tau (k))}{f(u(k-\tau (k)))}\le M.$$

From this we can choose $k_4$ such that

$$\frac{u(k-\tau (k))}{f(u(k-\tau (k)))}<M+\ell ,k\ge k_5.$$

That is, $u(k-\tau (k))<(M+\ell )f(u(k-\tau (k)))$, $k\ge k_5$, and so from (17) we get

$$p(k){\mathrm{\Delta }}_{\alpha (\ell )}u(k)+u(k-\tau (k))<0,k\ge k_5.$$

In particular, for a function $p(k_n)$ satisfying condition (v), we have

$$u(k_n+l)-\alpha u(k_n)+x_{k_n}-\tau (k_n)\le p(k_n)(u(k_n+l)-\alpha u(k_n))+u(k_n-\tau (k_n))<0$$

for k sufficiently large, which implies that

$$0<u(k_n+l)+(u(k_n-\tau (k_n))-\alpha u(k_n))<0$$

for all large k. This is a contradiction. The case that ${\mathrm{\Delta }}_{\alpha (\ell )}u(k)$ 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

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

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