Oscillation Criteria for Delay Difference Equations with Continuous Time, Piecewise Linear Delay Functions, and Oscillatory Coefficients

Abstract

This paper considers difference equations with continuous time, piecewise linear delay functions, and oscillatory coefficients. We present new conditions on the coefficients that provide the oscillatory property of the solutions of the considered difference equations. The given criteria are compared to the existing oscillatory conditions in the literature using examples.

1. Introduction

Applications of delay difference equations with continuous time, and evidence of their importance, can be found in mechanical and electrical systems, as is stated in [1], as well as in modeling distributed chaos, as is pointed out in [2,3]. Analyses of the oscillatory properties of solutions of delay difference equations with continuous time were the subject of the papers [4,5,6,7,8] and also the papers [9,10,11], but in the form of special cases of iterative functional equations. The literature on the oscillatory properties of solutions of differential and discrete difference equations can be divided into two groups: one with nonnegative or positive coefficients and the other with oscillatory coefficients. Studies on the oscillatory properties of solutions of differential equations with nonnegative coefficients can be found, for example, in the papers [12,13,14], while those with oscillating coefficients are found in [15,16]. For example, the papers [13,17,18,19,20,21] are devoted to discrete difference equations with positive coefficients and oscillatory coefficients, respectively. Since many papers dealing with difference equations with continuous time have analyzed equations with positive or nonnegative coefficients (all of those mentioned above) but only (to the best of our knowledge) the paper [22] has considered oscillatory coefficients, our aim is to expand the set of known oscillatory conditions for solutions of delay difference equations with continuous time and oscillatory coefficients.

In this paper, we continue the research from our paper [22], so we analyze the oscillatory property of the solutions of the delay difference equation

$$\Delta x\left(t\right)+\sum _{i=1}^m{p}_i\left(t\right)x\left({\tau }_i\left(t\right)\right)=0,t\ge t_0,$$

(1)

where $\Delta x\left(t\right)=x(t+1)-x\left(t\right)$, $m\in \mathbb{N}$, and $t_0\in \mathbb{R}$. For $i=1,2,\dots ,m$, $p_i:[t_0,\infty )\to \mathbb{R}$ is a piecewise continuous and oscillatory function, but the delay argument ${\tau }_i$ is in the form

$${\tau }_i\left(t\right)=t-k_i\left(t\right),$$

(2)

where the function $k_i:[t_0,\infty )\to \mathbb{N}$ is piecewise constant. Also, the delay arguments have the following properties:

$${\tau }_i\left(t\right)<t\mathrm{for}\mathrm{all}t\ge t_0\mathrm{and}i=1,2,\dots ,m,$$

(3)

and

$$\underset{t\to \infty }{lim}{\tau }_i\left(t\right)=\infty ,i=1,2,\dots ,m.$$

(4)

We present two oscillatory criteria and compare them to the oscillatory conditions in [22]. The comparison, using the presented examples, shows that there is a set of difference equations for which our new oscillatory criteria prove that their solutions oscillate, while the known criteria are not applicable to them. This confirms that we have extended the set of difference equations in form (1) for which conditions verifying their oscillatory properties exist.

A solution of delay difference Equation (1) is a real-valued function x defined on the interval $[t_{-1},\infty )$, with

$$t_{-1}=\underset{1\le i\le m}{min}\left\{inf\{{\tau }_i\left(s\right):s\ge t_0\}\right\},$$

(5)

that satisfies Equation (1). If a solution of (1) changes the sign on the interval $(s,\infty )$ for any s, it is oscillatory. Otherwise, it is nonoscillatory. As in [7], for any bounded real-valued function $\phi$ defined on the interval $[t_{-1},t_0+1)$, there exists a unique solution x of (1) that satisfies the initial condition $x\left(t\right)=\phi \left(t\right)$, $t\in [t_{-1},t_0+1)$ since it can be represented by

$$\begin{array}{cc}\hfill x\left(t\right)& =\phi \left(t\right),t_{-1}\le t<t_0+1,\hfill \\ \hfill x\left(t\right)& =x(t-1)-\sum _{i=1}^m{p}_i(t-1)x\left({\tau }_i(t-1)\right),t\ge t_0+1.\hfill \end{array}$$

As a standard, for any $a\in \mathbb{R}$, $n\in {\mathbb{N}}_0$, and a real-valued function f,

$$\sum _{q=a}^{a-1}f\left(q\right)=0\mathrm{and}\sum _{q=a}^{a+n}f\left(q\right)=f\left(a\right)+f(a+1)+\dots +f(a+n)$$

and

$$f^n\left(t\right)=f\left(f^{n-1}\left(t\right)\right),\mathrm{with}{f}^0\left(t\right)=t.$$

2. The New Oscillation Criteria

We define a real-valued function

$$\tau \left(t\right)=t-\underset{t_0\le s\le t}{inf}\left(\underset{1\le i\le m}{min}{k}_i\left(s\right)\right)$$

(6)

and

$$\underline{\tau }\left(t\right)=\underset{s\ge t}{inf}\left(\underset{1\le i\le m}{min}{\tau }_i\left(s\right)\right).$$

(7)

These functions have the following properties:

$$t-\tau \left(t\right)\in \mathbb{N}\mathrm{for}\forall t\ge t_0,i=1,2,\dots ,m,$$

(8)

$$\tau \left(t\right)-{\tau }_i\left(t\right)\in {\mathbb{N}}_0\mathrm{for}\forall t\ge t_0,i=1,2,\dots ,m,$$

(9)

$$\tau (t+n)-\tau \left(t\right)\in \mathbb{N}\mathrm{for}\forall t\ge t_0\mathrm{and}\mathrm{any}n\in \mathbb{N},$$

(10)

$$\mathrm{if}t\in [a,b]\subset [t_0,\infty ),\mathrm{then}t,\tau \left(t\right),{\tau }_1\left(t\right),\dots ,{\tau }_m\left(t\right)\in [\underline{\tau }\left(a\right),b],$$

(11)

$$\tau \left(t\right)\ge \underline{\tau }\left(t\right)\mathrm{for}\forall t\in [t_0,\infty ).$$

(12)

For more details and illustrations with examples, see [22].

First, we give the following lemma.

Lemma 1.

If there exists a sequence of disjoint intervals ${\left\{[a_j,b_j]\right\}}_{j\in \mathbb{N}}$ with the properties $b_j-a_j\in \mathbb{N}$ for every $j\in \mathbb{N}$,

$$p_i\left(t\right)\ge 0fort\in \bigcup _{j\in \mathbb{N}}[a_j,b_j],i=1,2,\dots ,m,$$

(13)

and for some $c>0$,

$$\sum _{q=a_j}^{b_j}\sum _{i=1}^m{p}_i\left(q\right)\ge c\mathrm{for}\forall j\in \mathbb{N},$$

(14)

then there exists a sequence ${\left\{{\eta }_j\right\}}_{j\in \mathbb{N}}$ such that ${\eta }_j\in \{a_j,a_j+1,\dots ,b_j\}$ and

$$\sum _{q=a_j}^{{\eta }_j-1}\sum _{i=1}^m{p}_i\left(q\right)<{\displaystyle \frac{c}{2}}$$

(15)

but

$$\sum _{q=a_j}^{{\eta }_j}\sum _{i=1}^m{p}_i\left(q\right)\ge {\displaystyle \frac{c}{2}}.$$

(16)

Proof. 

For any $j\in \mathbb{N}$, if

$$\sum _{i=1}^m{p}_i\left(a_j\right)\ge {\displaystyle \frac{c}{2}},$$

then

$$\sum _{q=a_j}^{a_j-1}\sum _{i=1}^m{p}_i\left(q\right)=0<{\displaystyle \frac{c}{2}}\mathrm{and}\sum _{q=a_j}^{a_j}\sum _{i=1}^m{p}_i\left(q\right)=\sum _{i=1}^m{p}_i\left(a_j\right)\ge {\displaystyle \frac{c}{2}},$$

so (15) and (16) hold for ${\eta }_j=a_j$.

For $a_j=b_j$, (14) gives

$$\sum _{q=a_j}^{b_j}\sum _{i=1}^m{p}_i\left(q\right)=\sum _{i=1}^m{p}_i\left(a_j\right)\ge c\ge {\displaystyle \frac{c}{2}},$$

that is, (15) and (16) hold for ${\eta }_j=a_j$. Besides this, if

$$\sum _{i=1}^m{p}_i\left(a_j\right)<{\displaystyle \frac{c}{2}},$$

then $a_j<b_j$, and there exists ${\eta }_j\in \{a_j+1,a_j+2,\dots ,b_j\}$ such that (15) and (16) hold.

Consequently, for every $j\in \mathbb{N}$, there exists ${\eta }_j\in \{a_j,a_j+1,\dots ,b_j\}$ such that (15) and (16) hold. □

Now, we prove an auxiliary lemma.

Lemma 2.

Let $t_{-1}$, τ, and $\underline{\tau }$ be defined by (5), (6), and (7), respectively. Assume that the function $x:[t_{-1},\infty )\to \mathbb{R}$ is a nonoscillatory solution of (1). If there exists a sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ such that

$$\underset{j\to \infty }{lim}{\xi }_j=\infty ,$$

(17)

$$\left[\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right]\subset [t_0,\infty )\mathit{are}\mathit{nonempty}\mathit{and}\mathit{disjoint}\mathit{for}j\in \mathbb{N},$$

(18)

$$p_i\left(t\right)\ge 0fort\in \bigcup _{j\in \mathbb{N}}\left[\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right],i=1,2,\dots ,m,$$

(19)

and

$$\alpha =\underset{j\to \infty }{lim\; inf}\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\sum _{q=\tau \left(s\right)}^{s-1}\sum _{i=1}^m{p}_i\left(q\right)\mathit{such}\mathit{that}0<\alpha <2,$$

(20)

then

$$\underset{j\to \infty }{lim\; inf}{\displaystyle \frac{x\left(\tau \left({\xi }_j\right)\right)}{x\left(\tau (\tau \left({\xi }_j\right)-1)\right)}}\ge {\displaystyle \frac{{\alpha }^2}{2(2-\alpha )}}.$$

(21)

Proof. 

Let $I_j$ and ${\underline{I}}_j$, for $j\in \mathbb{N}$, denote $\left[\tau (\tau \left({\xi }_j\right)-1),{\xi }_j-1\right]$ and $\left[\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right]$, respectively. Note that the properties (10) and (8) give

$$\tau (\tau \left({\xi }_j\right)-1)<\tau \left(\tau \left({\xi }_j\right)\right)<\tau \left({\xi }_j\right)\le {\xi }_j-1,$$

so $\{\tau \left(\tau \left({\xi }_j\right)\right),\dots ,\tau \left({\xi }_j\right),\tau \left({\xi }_j\right)+1,\dots ,{\xi }_j-1\}\subset I_j\subset {\underline{I}}_j$.

Due to the definition of limit inferior and (20), for an arbitrary $\epsilon >0$, so for $0<\epsilon <\alpha$,

$$\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\sum _{q=\tau \left(s\right)}^{s-1}\sum _{i=1}^m{p}_i\left(q\right)\ge \alpha -\epsilon \mathrm{for}\forall j\in \mathbb{N}.$$

Hence,

$$\sum _{q=\tau \left(s\right)}^{s-1}\sum _{i=1}^m{p}_i\left(q\right)\ge \alpha -\epsilon \mathrm{for}\forall s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\},\forall j\in \mathbb{N}$$

(22)

and consequently,

$$\sum _{q=\tau \left({\xi }_j\right)}^{{\xi }_j-1}\sum _{i=1}^m{p}_i\left(q\right)\ge \alpha -\epsilon \mathrm{for}\forall j\in \mathbb{N}.$$

(23)

Since $\{\tau \left({\xi }_j\right),\tau \left({\xi }_j\right)+1,\dots ,{\xi }_j-1\}\subset {\underline{I}}_j$, condition (19) ensures that $p_i\left(q\right)\ge 0$ for every $q\in \{\tau \left({\xi }_j\right),\tau \left({\xi }_j\right)+1,\dots ,{\xi }_j-1\}$ and $i=1,2,\dots ,m$. Besides this, (18), (19), and (23) ensure that the sequence of intervals ${\left\{[\tau \left({\xi }_j\right),{\xi }_j-1]\right\}}_{j\in \mathbb{N}}$ satisfies the conditions of Lemma 1 with $c=\alpha -\epsilon$. Therefore, there exists a sequence ${\left\{{\eta }_j\right\}}_{j\in \mathbb{N}}$ such that ${\eta }_j\in \{\tau \left({\xi }_j\right),\tau \left({\xi }_j\right)+1,\dots ,{\xi }_j-1\}$,

$$\sum _{q=\tau \left({\xi }_j\right)}^{{\eta }_j-1}\sum _{i=1}^m{p}_i\left(q\right)<{\displaystyle \frac{\alpha -\epsilon }{2}},$$

(24)

$$\sum _{q=\tau \left({\xi }_j\right)}^{{\eta }_j}\sum _{i=1}^m{p}_i\left(q\right)\ge {\displaystyle \frac{\alpha -\epsilon }{2}}.$$

(25)

According to (22),

$$\sum _{q=\tau \left({\eta }_j\right)}^{{\eta }_j-1}\sum _{i=1}^m{p}_i\left(q\right)\ge \alpha -\epsilon \mathrm{for}\forall j\in \mathbb{N},$$

so using (24), we obtain

$$\sum _{q=\tau \left({\eta }_j\right)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)=\sum _{q=\tau \left({\eta }_j\right)}^{{\eta }_j-1}\sum _{i=1}^m{p}_i\left(q\right)-\sum _{q=\tau \left({\xi }_j\right)}^{{\eta }_j-1}\sum _{i=1}^m{p}_i\left(q\right)>{\displaystyle \frac{\alpha -\epsilon }{2}}.$$

(26)

Property (10) secures that $\tau \left({\eta }_j\right)\le \tau \left({\xi }_j\right)-1$.

Since x is a nonoscillatory solution of (1), without loss of generality, we can assume that

$$x\left(t\right)>0\mathrm{for}\mathrm{every}t\ge t_1,\mathrm{where}{t}_0\le t_1\le {\underline{\tau }}^2\left(\tau (\tau \left({\xi }_1\right)-1)\right).$$

(27)

For every $t\in {\underline{I}}_j$, property (11) ensures that for every $i=1,2,\dots ,m$, ${\tau }_i\left(t\right)\in \left[{\underline{\tau }}^2\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right]$; thus, ${\tau }_i\left(t\right)\ge t_1$, so (1), (19), and (27) provide

$$x(t+1)-x\left(t\right)=-\sum _{i=1}^m{p}_i\left(t\right)x\left({\tau }_i\left(t\right)\right)\le 0\mathrm{for}\forall t\in {\underline{I}}_j,$$

i.e.,

$$x(t+1)\le x\left(t\right)\mathrm{for}\forall t\in {\underline{I}}_j.$$

Consequently,

$$\mathrm{if}t,t+n\in {\underline{I}}_j\mathrm{for}\mathrm{some}n\in \mathbb{N},\mathrm{then}x\left(t\right)\ge x(t+n).$$

(28)

Also, (11) ensures that ${\tau }_1\left(t\right),{\tau }_2\left(t\right),\dots ,{\tau }_m\left(t\right),\tau \left(t\right)\in {\underline{I}}_j$ for every $t\in I_j$. Therefore, (9) and (28) provide that

$$x\left({\tau }_i\left(t\right)\right)\ge x\left(\tau \left(t\right)\right)\mathrm{for}\mathrm{every}t\in I_j.$$

(29)

Summing up (1) from $t=\tau \left({\xi }_j\right)$ to $t={\eta }_j$, we obtain

$$\begin{array}{cc}\hfill x\left(\tau \left({\xi }_j\right)\right)& =x({\eta }_j+1)+\sum _{q=\tau \left({\xi }_j\right)}^{{\eta }_j}\sum _{i=1}^m{p}_i\left(q\right)x\left({\tau }_i\left(q\right)\right)\hfill \\ \hfill & \left(q\in [\tau \left({\xi }_j\right),{\eta }_j]\subset [\tau \left({\xi }_j\right),{\xi }_j-1]\subset I_j,\mathrm{so}\mathrm{according}\mathrm{to}\left(29\right)\mathrm{and}\left(19\right),\right)\hfill \\ \hfill & \ge x({\eta }_j+1)+\sum _{q=\tau \left({\xi }_j\right)}^{{\eta }_j}\sum _{i=1}^m{p}_i\left(q\right)x\left(\tau \left(q\right)\right)\hfill \\ \hfill & (\tau \left({\eta }_j\right)-\tau \left(q\right)\in {\mathbb{N}}_0\mathrm{according}\mathrm{to}\left(10\right)\mathrm{and}\tau \left(q\right)\in {\underline{I}}_j\mathrm{according}\mathrm{to}\left(11\right),\hfill \\ \hfill & \mathrm{and}\mathrm{thus}\left(28\right)\mathrm{gives}x\left(\tau \left(q\right)\right)\ge x\left(\tau \left({\eta }_j\right)\right),\mathrm{so}\mathrm{according}\mathrm{to}\left(19\right),)\hfill \\ \hfill & \ge x({\eta }_j+1)+x\left(\tau \left({\eta }_j\right)\right)\sum _{q=\tau \left({\xi }_j\right)}^{{\eta }_j}\sum _{i=1}^m{p}_i\left(q\right).\hfill \end{array}$$

Hence, according to (25),

$$x\left(\tau \left({\xi }_j\right)\right)\ge x({\eta }_j+1)+x\left(\tau \left({\eta }_j\right)\right){\displaystyle \frac{\alpha -\epsilon }{2}}.$$

(30)

Similarly, summing up (1) from $t=\tau \left({\eta }_j\right)$ to $t=\tau \left({\xi }_j\right)-1$, we obtain

$$\begin{array}{cc}\hfill x\left(\tau \left({\eta }_j\right)\right)& =x\left(\tau \left({\xi }_j\right)\right)+\sum _{q=\tau \left({\eta }_j\right)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)x\left({\tau }_i\left(q\right)\right)\hfill \\ \hfill & \left(q\in [\tau \left({\eta }_j\right),\tau \left({\xi }_j\right)-1]\subset [\tau \left(\tau \left({\xi }_j\right)\right),\tau \left({\xi }_j\right)-1]\subset I_j\right)\hfill \\ \hfill & \ge x\left(\tau \left({\xi }_j\right)\right)+\sum _{q=\tau \left({\eta }_j\right)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)x\left(\tau \left(q\right)\right)\hfill \\ \hfill & \left(\tau (\tau \left({\xi }_j\right)-1)-\tau \left(q\right)\in {\mathbb{N}}_0\mathrm{and}\tau \left(q\right)\in {\underline{I}}_j\right)\hfill \\ \hfill & \ge x\left(\tau \left({\xi }_j\right)\right)+x\left(\tau (\tau \left({\xi }_j\right)-1)\right)\sum _{q=\tau \left({\eta }_j\right)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right).\hfill \end{array}$$

Therefore, (26) implies

$$x\left(\tau \left({\eta }_j\right)\right)\ge x\left(\tau \left({\xi }_j\right)\right)+x\left(\tau (\tau \left({\xi }_j\right)-1)\right){\displaystyle \frac{\alpha -\epsilon }{2}}.$$

(31)

Combining the inequalities (30) and (31), we obtain

$$x\left(\tau \left({\xi }_j\right)\right)\ge x({\eta }_j+1)+x\left(\tau \left({\xi }_j\right)\right){\displaystyle \frac{\alpha -\epsilon }{2}}+x\left(\tau (\tau \left({\xi }_j\right)-1)\right){\left({\displaystyle \frac{\alpha -\epsilon }{2}}\right)}^2,$$

i.e.,

$$x\left(\tau \left({\xi }_j\right)\right)\left(1-{\displaystyle \frac{\alpha -\epsilon }{2}}\right)\ge x({\eta }_j+1)+x\left(\tau (\tau \left({\xi }_j\right)-1)\right){\displaystyle \frac{{\left(\alpha -\epsilon \right)}^2}{4}}.$$

Since $x({\eta }_j+1)>0$,

$$x\left(\tau \left({\xi }_j\right)\right){\displaystyle \frac{2-\left(\alpha -\epsilon \right)}{2}}\ge x\left(\tau (\tau \left({\xi }_j\right)-1)\right){\displaystyle \frac{{\left(\alpha -\epsilon \right)}^2}{4}}.$$

Given that $\alpha <2$ and $\epsilon >0$, we have

$${\displaystyle \frac{x\left(\tau \left({\xi }_j\right)\right)}{x\left(\tau (\tau \left({\xi }_j\right)-1)\right)}}\ge {\displaystyle \frac{{\left(\alpha -\epsilon \right)}^2}{2\left(2-\left(\alpha -\epsilon \right)\right)}}.$$

Consequently,

$$\underset{j\to \infty }{lim\; inf}{\displaystyle \frac{x\left(\tau \left({\xi }_j\right)\right)}{x\left(\tau (\tau \left({\xi }_j\right)-1)\right)}}\ge {\displaystyle \frac{{\left(\alpha -\epsilon \right)}^2}{2\left(2-\left(\alpha -\epsilon \right)\right)}}.$$

(32)

Since (32) holds for an arbitrarily small $\epsilon$, this implies (21). The proof of the lemma is complete. □

Now, we are ready to formulate and prove a new criterion that shows that all of the solutions of the observed difference equation are oscillatory.

Theorem 1.

For $t_{-1}$, τ, and $\underline{\tau }$, defined by (5), (6), and (7), respectively, assume that there exists a sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ such that conditions (17)–(20) are satisfied. If

$$\underset{j\to \infty }{lim\; sup}\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)>1-{\displaystyle \frac{{\alpha }^2}{2\left(2-\alpha \right)}},$$

(33)

then all of the solutions of (1) are oscillatory.

Proof. 

As in the proof of the previous lemma, for $j\in \mathbb{N}$, $I_j$ and ${\underline{I}}_j$ denote the intervals $\left[\tau (\tau \left({\xi }_j\right)-1),{\xi }_j-1\right]$ and $\left[\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right]$, respectively.

Supposing the opposite, let the function $x:[t_{-1},\infty )\to \mathbb{R}$ be a nonoscillatory solution of (1). Without loss of generality, we can assume that

$$x\left(t\right)>0\mathrm{for}\mathrm{every}t\ge t_1,\mathrm{where}{t}_0\le t_1\le {\underline{\tau }}^2\left(\tau (\tau \left({\xi }_1\right)-1)\right).$$

(34)

Therefore, (11), (1), (19), and (34) imply

$$x(t+1)-x\left(t\right)=-\sum _{i=1}^m{p}_i\left(t\right)x\left({\tau }_i\left(t\right)\right)\le 0\mathrm{for}\forall t\in {\underline{I}}_j.$$

Consequently, for every $j\in \mathbb{N}$,

$$\mathrm{if}t,t+n\in {\underline{I}}_j\mathrm{for}\mathrm{some}n\in \mathbb{N},\mathrm{then}x\left(t\right)\ge x(t+n).$$

(35)

Since (11) ensures that ${\tau }_1\left(t\right),{\tau }_2\left(t\right),\dots ,{\tau }_m\left(t\right),\tau \left(t\right)\in {\underline{I}}_j$ for every $t\in I_j$, (9) and (35) provide

$$x\left({\tau }_i\left(t\right)\right)\ge x\left(\tau \left(t\right)\right)\mathrm{for}\forall t\in I_j.$$

(36)

For any $j\in \mathbb{N}$, summing up (1) from $t=\tau (\tau \left({\xi }_j\right)-1)$ to $t=\tau \left({\xi }_j\right)-1$, then given that $\left\{\tau (\tau \left({\xi }_j\right)-1),\dots ,\tau \left({\xi }_j\right)-1\right\}\subset I_j$ and applying (19) and (36), afterwards, according to (19), (10), and (35), we have

$$\begin{array}{cc}\hfill x\left(\tau (\tau \left({\xi }_j\right)-1)\right)& =x\left(\tau \left({\xi }_j\right)\right)+\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)x\left({\tau }_i\left(q\right)\right)\hfill \\ \hfill & \ge x\left(\tau \left({\xi }_j\right)\right)+\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)x\left(\tau \left(q\right)\right)\hfill \\ \hfill & \ge x\left(\tau \left({\xi }_j\right)\right)+x\left(\tau (\tau \left({\xi }_j\right)-1)\right)\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right).\hfill \end{array}$$

Dividing the obtained inequality by $x\left(\tau (\tau \left({\xi }_j\right)-1)\right)$, which is positive, we obtain

$$1\ge {\displaystyle \frac{x\left(\tau \left({\xi }_j\right)\right)}{x\left(\tau (\tau \left({\xi }_j\right)-1)\right)}}+\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right),$$

i.e.,

$$\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)\le 1-{\displaystyle \frac{x\left(\tau \left({\xi }_j\right)\right)}{x\left(\tau (\tau \left({\xi }_j\right)-1)\right)}}.$$

Hence,

$$\underset{j\to \infty }{lim\; sup}\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)\le 1-\underset{j\to \infty }{lim\; inf}{\displaystyle \frac{x\left(\tau \left({\xi }_j\right)\right)}{x\left(\tau (\tau \left({\xi }_j\right)-1)\right)}}.$$

Applying (21) from Lemma 2, we have

$$\underset{j\to \infty }{lim\; sup}\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)\le 1-{\displaystyle \frac{{\alpha }^2}{2(2-\alpha )}},$$

(37)

which is a contradiction to condition (33). The proof of the theorem is complete. □

Due to the following corollary, the nonnegativity of the coefficients in the definition of $\alpha$ and the coefficients in condition (33), combined with property (10), yield the conclusion that for $\alpha >3-\sqrt{5}$, there is no need to check the condition (33).

Corollary 1.

For $t_{-1}$, τ, and $\underline{\tau }$ as defined by (5), (6), and (7), respectively, assume that there exists a sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ such that conditions (17)–(20) are satisfied. If $\alpha >3-\sqrt{5}$, then all of the solutions of (1) are oscillatory.

Proof. 

Since $\tau (\tau \left({\xi }_j\right)-1)\le \tau \left(\tau \left({\xi }_j\right)\right)$, condition (19) and definition (20) give

$$\alpha \le \underset{j\to \infty }{lim\; inf}\sum _{q=\tau \left(\tau \left({\xi }_j\right)\right)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right)\le \underset{j\to \infty }{lim\; sup}\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^m{p}_i\left(q\right).$$

(38)

For $2>\alpha >3-\sqrt{5}$, the inequality

$$\alpha >1-{\displaystyle \frac{{\alpha }^2}{2(2-\alpha )}}$$

holds, so (38) ensures that condition (33) is satisfied, and based on Theorem 1, it implies that all of the solutions of (1) are oscillatory. □

Notice that the oscillatory conditions in Theorem 1 and Corollary 1 require nonnegativity of the coefficients in the disjoint intervals and do not use the intervals where the coefficients are negative. Therefore, the presented oscillatory conditions can be applied to establishing the oscillatory property of solutions of the difference equations in form (1) with oscillating coefficients, but they also apply to equations with positive coefficients.

3. Examples and Comparisons

The following examples illustrate the presented conditions for oscillatory solutions of the observed difference equation. For the first two examples, the oscillatory conditions from Theorem 1 are satisfied, but the oscillatory conditions from Corollary 1 and the oscillatory conditions from [22] are not fulfilled. The last example shows the application of the oscillatory conditions from Corollary 1 to a delay difference equation with unbounded delays. Moreover, the oscillatory conditions from [22] are not satisfied for that example either.

Theorem 2

([22] (Theorem 2.2)). For the functions τ and $\underline{\tau }$ defined by (6) and (7), respectively, assume that there exists a sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ such that condition (17) holds and the intervals of the sequence ${\left\{\left[\underline{\tau }\left(\tau \left({\xi }_j\right)\right),{\xi }_j\right]\right\}}_{j\in \mathbb{N}}$ are nonempty, disjoint, and contained in $[t_0,\infty )$. If, in addition,

$$p_i\left(t\right)\ge 0\mathrm{for}t\in \bigcup _{j\in \mathbb{N}}\left[\underline{\tau }\left(\tau \left({\xi }_j\right)\right),{\xi }_j\right],i=1,2,\dots ,m,$$

(39)

and

$$\underset{j\to \infty }{lim\; sup}\sum _{q=\tau \left({\xi }_j\right)}^{{\xi }_j}\sum _{i=1}^m{p}_i\left(q\right)>1,$$

(40)

then all of the solutions of (1) are oscillatory.

Theorem 3

([22] (Theorem 2.3)). For the functions τ and $\underline{\tau }$ defined by (6) and (7), respectively, assume that there exists a sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ such that condition (17) holds and the intervals of the sequence ${\left\{\left[\underline{\tau }\left({\tau }^j\left({\xi }_j\right)\right),{\xi }_j\right]\right\}}_{j\in \mathbb{N}}$ are nonempty, disjoint, and contained in $[t_0,\infty )$. If, moreover,

$$p_i\left(t\right)\ge 0fort\in \bigcup _{j\in \mathbb{N}}\left[\underline{\tau }\left({\tau }^j\left({\xi }_j\right)\right),{\xi }_j\right],i=1,2,\dots ,m,$$

(41)

and there exists a real number $c>1/e$ such that

$$\underset{j\to \infty }{lim}\underset{t\in I_j^{j-1}}{inf}\sum _{q=\tau \left(t\right)}^{t-1}\sum _{i=1}^m{p}_i\left(q\right)>c,\mathit{where}{I}_j^{j-1}=[{\tau }^{j-1}\left({\xi }_j\right),{\xi }_j],$$

(42)

then all of the solutions of (1) are oscillatory.

Example 1.

Consider the difference equation

$$\Delta x\left(t\right)+p_1\left(t\right)x(t-2)+p_2\left(t\right)x(t-3)=0,t\ge 0,$$

(43)

where the functions $p_1$ and $p_2$ are periodic, with the basic period 10, such that

$$\begin{array}{c}\hfill p_1\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{2}{5}}·{\left({\displaystyle \frac{7}{10}}\right)}^t,\hfill & t\in [0,7],\hfill \\ -{\displaystyle \frac{1}{5}}·{\left({\displaystyle \frac{7}{10}}\right)}^{10-t},\hfill & t\in (7,10),\hfill \end{array}\right.\end{array}$$

$$\begin{array}{c}\hfill p_2\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{3}},\hfill & t\in [0,1.5)\cup (2.5,4.5)\cup (5.5,7.5),\hfill \\ 0,\hfill & t\in [1.5,2.5]\cup [4.5,5.5],\hfill \\ -{\displaystyle \frac{1}{6}},\hfill & t\in [7.5,10).\hfill \end{array}\right.\end{array}$$

Here, ${\tau }_1\left(t\right)=t-2$, and ${\tau }_2\left(t\right)=t-3$; thus, $\tau \left(t\right)=t-2$, and $\underline{\tau }\left(t\right)=t-3$. Therefore, $\tau \left(\tau \right(t)-1)=t-5$, and $\underline{\tau }\left(\tau (\tau \left(t\right)-1)\right)=t-8$. For the sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in {\mathbb{N}}_0}$ such that ${\xi }_j=10j+8$,

$$\tau \left({\xi }_j\right)=10j+6,\tau (\tau \left({\xi }_j\right)-1)=10j+3,\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right)=10j,j\in {\mathbb{N}}_0,$$

so, for every $j\in {\mathbb{N}}_0$,

$$\begin{array}{cc}\hfill I_j& =\left[\tau (\tau \left({\xi }_j\right)-1),{\xi }_j-1\right]=[10j+3,10j+7],\hfill \\ \hfill {\underline{I}}_j& =\left[\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right]=[10j,10j+7].\hfill \end{array}$$

This means that conditions (17)–(19) are satisfied with

$$\begin{array}{cc}\hfill \alpha & =\underset{j\to \infty }{lim\; inf}\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\sum _{q=\tau \left(s\right)}^{s-1}\sum _{i=1}^2{p}_i\left(q\right)=\underset{j\to \infty }{lim\; inf}\underset{s\in \{10j+6,\dots ,10j+8\}}{min}\sum _{q=\tau \left(s\right)}^{s-1}\sum _{i=1}^2{p}_i\left(q\right)\hfill \\ \hfill & =\underset{j\to \infty }{lim\; inf}min\left\{\sum _{q=10j+4}^{10j+5}\sum _{i=1}^2{p}_i\left(q\right),\sum _{q=10j+5}^{10j+6}\sum _{i=1}^2{p}_i\left(q\right),\sum _{q=10j+6}^{10j+7}\sum _{i=1}^2{p}_i\left(q\right)\right\}\hfill \\ \hfill & \approx min\left\{0.496601,0.447621,0.746668\right\}=0.447621.\hfill \end{array}$$

Due to $1-{\displaystyle \frac{{\alpha }^2}{2(2-\alpha )}}\approx 0.935465$ and

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim\; sup}\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^2{p}_i\left(q\right)& =\underset{j\to \infty }{lim\; sup}\sum _{q=10j+3}^{10j+5}\sum _{i=1}^2{p}_i\left(q\right)\approx 0.967135>0.935465,\hfill \end{array}$$

condition (33) is also fulfilled; thus, all of the conditions of Theorem 1 are satisfied, and therefore all of the solutions of Equation (43) are oscillatory.

The graphs of the functions $p_1$ and $p_2$, the intervals ${\underline{I}}_j$, and the relevant points from conditions (20) and (33) are presented in Figure 1. Figure 2 shows the graphs of some of the solutions of Equation (43).

The conditions of Theorem 3 cannot be fulfilled since the length of the intervals of condition (41) are expanding as j increases, but the functions $p_1$ and $p_2$ are nonnegative only at the interval with a length of 7. Therefore, the oscillatory conditions of Theorem 3 cannot be satisfied.

The oscillatory conditions of Theorem 2 also cannot be satisfied. Namely, for any sequence of real numbers ${\left\{{\zeta }_j\right\}}_{j\in {\mathbb{N}}_0}$ such that condition (39) is fulfilled,

$$\left[\underline{\tau }\left(\tau \left({\zeta }_j\right)\right),{\zeta }_j\right]=\left[{\zeta }_j-5,{\zeta }_j\right].$$

Besides this, $p_1\left(t\right)\ge 0$ and $p_2\left(t\right)\ge 0$ are for $t\in [10j,10j+7]$ with $j\in {\mathbb{N}}_0$, so ${\zeta }_j\in [10j+5,10j+7]$. Therefore,

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim\; sup}\sum _{q=\tau \left({\zeta }_j\right)}^{{\zeta }_j}\sum _{i=1}^2{p}_i\left(q\right)& =\underset{j\to \infty }{lim\; sup}\sum _{q={\zeta }_j-2}^{{\zeta }_j}\sum _{i=1}^2{p}_i\left(q\right)\hfill \\ \hfill & \le \underset{j\to \infty }{lim\; sup}\underset{{\zeta }_j\in [10j+5,10j+7]}{max}\sum _{q={\zeta }_j-2}^{{\zeta }_j}\left(p_1\left(q\right)+p_2\left(q\right)\right)\hfill \\ \hfill & =\sum _{q=10j+3}^{10j+5}\left(p_1\left(q\right)+p_2\left(q\right)\right)\approx 0.967135<1.\hfill \end{array}$$

Consequently, condition (40) cannot be fulfilled.

Example 2.

The conditions of Theorem 1 are satisfied for equation

$$\Delta x\left(t\right)+p_1\left(t\right)x(t-1)+p_2\left(t\right)x(t-2)=0,t\ge 1,$$

(44)

with

$$p_1\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{5j}}t-{\displaystyle \frac{4j}{5}},\hfill & t\in \left[4j^2-4,4j^2+2j\right]\hfill \\ {\displaystyle \frac{2}{5}},\hfill & t\in \left[4j^2+2j,4j^2+2j+1\right]\hfill \\ {\displaystyle \frac{{(2j+1)}^2}{5j}}-{\displaystyle \frac{1}{5j}}t,\hfill & t\in \left[4j^2+2j+1,4{(j+1)}^2-4\right)\hfill \end{array}\right.\mathit{for}j\in \mathbb{N},$$

and

$$p_2\left(t\right)={\displaystyle \frac{{(2j+1)}^2}{20j+30}}-{\displaystyle \frac{1}{20j+30}}t\mathit{for}t\in \left[4j^2+2j-2,4{(j+1)}^2+2j\right),j\in \mathbb{N},$$

so all of the solutions of Equation (44) are oscillatory.

Namely, ${\tau }_1\left(t\right)=t-1$, and ${\tau }_2\left(t\right)=t-2$; thus, $\tau \left(t\right)=t-1$, and $\underline{\tau }\left(t\right)=t-2$. Therefore, $\tau \left(\tau \right(t)-1)=t-3$, and $\underline{\tau }\left(\tau (\tau \left(t\right)-1)\right)=t-5$. For every $j\in \mathbb{N}$,

$$p_1\left(t\right)\ge 0,p_2\left(t\right)\ge 0\mathit{for}\forall t\in \left[4j^2+2j-2,{(2j+1)}^2\right].$$

For the sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ such that ${\xi }_j=4j^2+2j+3$,

$$\tau \left({\xi }_j\right)=4j^2+2j+2,\tau (\tau \left({\xi }_j\right)-1)=4j^2+2j,\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right)=4j^2+2j-2,$$

so for $j\in \mathbb{N}$,

$$\begin{array}{cc}\hfill I_j& =\left[\tau (\tau \left({\xi }_j\right)-1),{\xi }_j-1\right]=[4j^2+2j,4j^2+2j+2],\hfill \\ \hfill {\underline{I}}_j& =\left[\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right]=[4j^2+2j-2,4j^2+2j+2].\hfill \end{array}$$

Hence, conditions (17)–(19) are satisfied, and

$$\begin{array}{cc}\hfill \alpha & =\underset{j\to \infty }{lim\; inf}\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\sum _{q=\tau \left(s\right)}^{s-1}\sum _{i=1}^2{p}_i\left(q\right)=\underset{j\to \infty }{lim\; inf}\underset{s\in \{{\xi }_j-1,{\xi }_j\}}{min}\sum _{i=1}^2{p}_i(s-1)\hfill \\ \hfill & =\underset{j\to \infty }{lim\; inf}min\left\{\sum _{i=1}^2{p}_i(4j^2+2j+1),\sum _{i=1}^2{p}_i(4j^2+2j+2)\right\}\hfill \\ \hfill & =\underset{j\to \infty }{lim\; inf}min\left\{{\displaystyle \frac{2}{5}}+{\displaystyle \frac{j}{10j+15}},{\displaystyle \frac{2}{5}}-{\displaystyle \frac{1}{5j}}+{\displaystyle \frac{2j-1}{20j+30}}\right\}={\displaystyle \frac{2}{5}}+{\displaystyle \frac{1}{10}}={\displaystyle \frac{1}{2}}.\hfill \end{array}$$

Since $1-{\displaystyle \frac{{\alpha }^2}{2(2-\alpha )}}={\displaystyle \frac{11}{12}}\approx 0.916667$ and

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim\; sup}\sum _{q=\tau (\tau \left({\xi }_j\right)-1)}^{\tau \left({\xi }_j\right)-1}\sum _{i=1}^2{p}_i\left(q\right)& =\underset{j\to \infty }{lim\; sup}\sum _{q={\xi }_j-3}^{{\xi }_j-2}\sum _{i=1}^2{p}_i\left(q\right)\hfill \\ \hfill & =\underset{j\to \infty }{lim\; sup}\left(\sum _{i=1}^2{p}_i(4j^2+2j)+\sum _{i=1}^2{p}_i(4j^2+2j+1)\right)\hfill \\ \hfill & =\underset{j\to \infty }{lim\; sup}\left({\displaystyle \frac{2}{5}}+{\displaystyle \frac{2j+1}{20j+30}}+{\displaystyle \frac{2}{5}}+{\displaystyle \frac{j}{10j+15}}\right)=1>{\displaystyle \frac{11}{12}},\hfill \end{array}$$

condition (33) also holds, so all of the conditions of Theorem 1 are fulfilled.

The graphs of the functions $p_1$ and $p_2$, the intervals ${\underline{I}}_j$, and the relevant points from conditions (20) and (33) are presented in Figure 3. Figure 4 shows graphs of some of the solutions of Equation (44).

Now, we show that the oscillatory conditions of Theorem 2 cannot be fulfilled. For any sequence of real numbers ${\left\{{\zeta }_j\right\}}_{j\in \mathbb{N}}$ such that condition (39) holds,

$$\left[\underline{\tau }\left(\tau \left({\zeta }_j\right)\right),{\zeta }_j\right]=\left[{\zeta }_j-3,{\zeta }_j\right]\subset \left[4j^2+2j-2,{(2j+1)}^2\right],$$

so ${\zeta }_j\in \left[4j^2+2j+1,{(2j+1)}^2\right]$. Therefore, given that the functions $p_1$ and $p_2$ are nonincreasing on $\left[4j^2+2j,{(2j+1)}^2\right]$,

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim\; sup}\sum _{q=\tau \left({\zeta }_j\right)}^{{\zeta }_j}\sum _{i=1}^2{p}_i\left(q\right)& =\underset{j\to \infty }{lim\; sup}\sum _{q={\zeta }_j-1}^{{\zeta }_j}\sum _{i=1}^2{p}_i\left(q\right)\hfill \\ \hfill & \le \underset{j\to \infty }{lim\; sup}\underset{{\zeta }_j\in \left[4j^2+2j+1,{(2j+1)}^2\right]}{max}\sum _{q={\zeta }_j-1}^{{\zeta }_j}\sum _{i=1}^2{p}_i\left(q\right)\hfill \\ \hfill & \le \underset{j\to \infty }{lim\; sup}\left(\sum _{i=1}^2{p}_i(4j^2+2j)+\sum _{i=1}^2{p}_i(4j^2+2j+1)\right)=1.\hfill \end{array}$$

Consequently, condition (40) cannot be satisfied.

Finally, we show that the oscillatory conditions of Theorem 3 cannot be fulfilled. For any sequence of real numbers ${\left\{{\zeta }_j\right\}}_{j\in \mathbb{N}}$ such that condition (41) holds,

$$\left[\underline{\tau }\left({\tau }^j\left({\zeta }_j\right)\right),{\zeta }_j\right]=\left[{\zeta }_j-j-2,{\zeta }_j\right]\subset \left[4j^2+2j-2,{(2j+1)}^2\right],$$

so ${\zeta }_j\in \left[4j^2+3j,{(2j+1)}^2\right]$. Hence,

$$I_j^{j-1}=\left[{\tau }^{j-1}\left({\zeta }_j\right),{\zeta }_j\right]=\left[{\zeta }_j-j+1,{\zeta }_j\right]\subset \left[4j^2+2j+1,{(2j+1)}^2\right].$$

Given that the functions $p_1$ and $p_2$ are nonincreasing on $\left[4j^2+2j,{(2j+1)}^2\right]$ and the fact that $4j^2+3j-1\ge 4j^2+2j+1$ for $j>2$,

$$\begin{array}{cc}\hfill \underset{j\to \infty }{lim}\underset{t\in I_j^{j-1}}{inf}\sum _{q=\tau \left(t\right)}^{t-1}\sum _{i=1}^2{p}_i\left(q\right)& =\underset{j\to \infty }{lim}\underset{t\in I_j^{j-1}}{inf}\sum _{i=1}^2{p}_i(t-1)\hfill \\ \hfill & \le \underset{j\to \infty }{lim}\underset{t\in \left[4j^2+2j+1,4j^2+3j\right]}{inf}\sum _{i=1}^2{p}_i(t-1)\hfill \\ \hfill & =\underset{j\to \infty }{lim}\sum _{i=1}^2{p}_i(4j^2+3j-1)\hfill \\ \hfill & =\underset{j\to \infty }{lim}\left({\displaystyle \frac{j+2}{5j}}+{\displaystyle \frac{j+2}{20j+30}}\right)={\displaystyle \frac{1}{4}}<{\displaystyle \frac{1}{e}}.\hfill \end{array}$$

Consequently, condition (42) cannot be fulfilled.

Example 3.

Consider the difference equation

$$\Delta x\left(t\right)+p_1\left(t\right)x\left(t-\left[\sqrt{t}\right]\right)+p_2\left(t\right)x\left(t-2\left[\sqrt{t}\right]\right)=0,t\ge 2,$$

(45)

with

$$p_1\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{3}{14j}},\hfill & t\in \left[{(3j-2)}^2-1,{\left(3j\right)}^2-1\right]\hfill \\ -0.1,\hfill & t\in \left({\left(3j\right)}^2-1,{(3j+1)}^2-1\right)\hfill \end{array}\right.\mathit{for}j\in \mathbb{N},$$

and

$$p_2\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{14j}},\hfill & t\in \left[{(3j-2)}^2-1,{\left(3j\right)}^2-1\right]\hfill \\ -0.05,\hfill & t\in \left({\left(3j\right)}^2-1,{(3j+1)}^2-1\right)\hfill \end{array}\right.\mathit{for}j\in \mathbb{N},$$

where $[·]$ denotes the integer part.

The delay functions, ${\tau }_1\left(t\right)=t-\left[\sqrt{t}\right]$ and ${\tau }_2\left(t\right)=t-2\left[\sqrt{t}\right]$, are unbounded but satisfy conditions (3) and (4). $\tau \left(t\right)=t-\left[\sqrt{t}\right]$ and $\underline{\tau }\left(t\right)=t-2\left[\sqrt{t}\right]$; thus, for the sequence of real numbers ${\left\{{\xi }_j\right\}}_{j\in \mathbb{N}}$ such that ${\xi }_j={\left(3j\right)}^2$,

$$\begin{array}{cc}\hfill \tau \left({\xi }_j\right)& ={\left(3j\right)}^2-3j,\hfill \\ \hfill \tau \left(\tau \left({\xi }_j\right)\right)& ={\left(3j\right)}^2-3j-(3j-1)={\left(3j\right)}^2-6j+1={(3j-1)}^2,\hfill \\ \hfill \tau (\tau \left({\xi }_j\right)-1)& ={\left(3j\right)}^2-3j-1-(3j-1)={\left(3j\right)}^2-6j,\hfill \\ \hfill \underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right)& ={\left(3j\right)}^2-6j-2(3j-2)={\left(3j\right)}^2-12j+4={(3j-2)}^2.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill {\underline{I}}_j& =\left[\underline{\tau }\left(\tau (\tau \left({\xi }_j\right)-1)\right),{\xi }_j-1\right]=[{(3j-2)}^2,{\left(3j\right)}^2-1],\hfill \end{array}$$

so conditions (17)–(19) are satisfied. Equally,

$$[\tau \left(\tau \left({\xi }_j\right)\right),{\xi }_j-1]\subset {\underline{I}}_j\mathit{for}j\in \mathbb{N},$$

implying that

$$p_1\left(t\right)+p_2\left(t\right)={\displaystyle \frac{2}{7j}}\mathit{for}t\in [\tau \left(\tau \left({\xi }_j\right)\right),{\xi }_j-1].$$

Hence,

$$\begin{array}{cc}\hfill \alpha & =\underset{j\to \infty }{lim\; inf}\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\sum _{q=\tau \left(s\right)}^{s-1}\sum _{i=1}^2{p}_i\left(q\right)=\underset{j\to \infty }{lim\; inf}\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\sum _{q=\tau \left(s\right)}^{s-1}{\displaystyle \frac{2}{7j}}\hfill \\ \hfill & =\underset{j\to \infty }{lim\; inf}\left({\displaystyle \frac{2}{7j}}·\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\left(s-\tau \left(s\right)\right)\right)=\underset{j\to \infty }{lim\; inf}\left({\displaystyle \frac{2}{7j}}·\underset{s\in \{\tau \left({\xi }_j\right),\dots ,{\xi }_j\}}{min}\left[\sqrt{s}\right]\right)\hfill \\ \hfill & =\underset{j\to \infty }{lim\; inf}\left({\displaystyle \frac{2}{7j}}·\left[\sqrt{\tau \left({\xi }_j\right)}\right]\right)=\underset{j\to \infty }{lim\; inf}\left({\displaystyle \frac{2}{7j}}·\left(3j-1\right)\right)={\displaystyle \frac{6}{7}}>3-\sqrt{5}.\hfill \end{array}$$

Consequently, the conditions of Corollary 1 are satisfied, so all of the solutions of Equation (45) are oscillatory.

The graphs of the functions $p_1$ and $p_2$, the intervals ${\underline{I}}_j$, and the relevant points from condition (20) are presented in Figure 5. Figure 6 shows graphs of some of the solutions of Equation (45).

The oscillatory conditions of Theorem 2 cannot be satisfied since for any sequence of real numbers ${\left\{{\zeta }_n\right\}}_{n\in \mathbb{N}}$ such that condition (39) is fulfilled, $\left[\underline{\tau }\left(\tau \left({\zeta }_n\right)\right),{\zeta }_n\right]\subset \left[{(3j-2)}^2-1,{\left(3j\right)}^2-1\right]$ for some $j\in \mathbb{N}$, so

$$p_1\left(t\right)+p_2\left(t\right)={\displaystyle \frac{2}{7j}}\mathit{for}t\in \left[\underline{\tau }\left(\tau \left({\zeta }_n\right)\right),{\zeta }_n\right].$$

Hence,

$$\sum _{q=\tau \left({\zeta }_n\right)}^{{\zeta }_n}\sum _{i=1}^2{p}_i\left(q\right)={\displaystyle \frac{2}{7j}}\left({\zeta }_n-\tau \left({\zeta }_n\right)+1\right)={\displaystyle \frac{2}{7j}}\left(\left[\sqrt{{\zeta }_n}\right]+1\right).$$

Given that ${\zeta }_n\subset \left[{(3j-2)}^2-1,{\left(3j\right)}^2-1\right]$,

$$3j-3<\left[\sqrt{{\zeta }_n}\right]<3j$$

and therefore

$${\displaystyle \frac{2(3j-2)}{7j}}<\sum _{q=\tau \left({\zeta }_n\right)}^{{\zeta }_n}\sum _{i=1}^2{p}_i\left(q\right)<{\displaystyle \frac{2(3j+1)}{7j}}.$$

(46)

Since the intervals of sequence $\left\{\left[\underline{\tau }\left(\tau \left({\zeta }_n\right)\right),{\zeta }_n\right]\right\}$ are disjoint, $n\to \infty$ implies that $j\to \infty$, so inequality (46) gives

$$\underset{n\to \infty }{lim\; sup}\sum _{q=\tau \left({\zeta }_n\right)}^{{\zeta }_n}\sum _{i=1}^2{p}_i\left(q\right)={\displaystyle \frac{6}{7}}<1.$$

Consequently, condition (40) cannot be fulfilled.

The conditions of Theorem 3 also cannot be fulfilled since there is no sequence of real numbers ${\left\{{\zeta }_n\right\}}_{n\in \mathbb{N}}$ such that condition (41) is satisfied. Namely, according to condition (41), there is a sequence of real numbers ${\left\{{\zeta }_n\right\}}_{n\in \mathbb{N}}$ such that

$$\left[\underline{\tau }\left({\tau }^n\left({\zeta }_n\right)\right),{\zeta }_n\right]\subset \left[{(3j-2)}^2-1,{\left(3j\right)}^2-1\right],$$

(47)

for some $j\in \mathbb{N}$. The length of the interval $\left[\underline{\tau }\left({\tau }^n\left({\zeta }_n\right)\right),{\zeta }_n\right]$ is expanding as n increases, and the length of the interval $\left[{(3j-2)}^2-1,{\left(3j\right)}^2-1\right]$ is $12j-4$. Even for ${\zeta }_3={\left(3j\right)}^2-1$,

$$\begin{array}{cc}\hfill \tau \left({\zeta }_3\right)={\left(3j\right)}^2-3j,{\tau }^2\left({\zeta }_3\right)& ={(3j-1)}^2,{\tau }^3\left({\zeta }_3\right)=(3j-1)(3j-2),\hfill \\ \hfill \underline{\tau }\left({\tau }^3\left({\zeta }_3\right)\right)& =(3j-2)(3j-3),\hfill \end{array}$$

so the length of the interval $\left[\underline{\tau }\left({\tau }^3\left({\zeta }_3\right)\right),{\zeta }_3\right]$ is $15j-7$. Since $12j-4<15j-7$ for every $j>1$, the condition (47) cannot be fulfilled for any integer $j>1$ and $n\ge 3$. Hence, the oscillatory conditions of Theorem 3 cannot be satisfied.

4. Conclusions

Our study of the oscillatory properties of the solutions of first-order difference equations with continuous time, piecewise linear delay functions, and oscillatory coefficients has led us to a new condition that ensures the oscillatory solutions. According to the proposed results, the oscillatory property of all solutions of the considered difference equation is ensured with sufficiently positive coefficients in the equation in the sense that the sum of the values of the coefficients defined by $\alpha$ in (20) is in the interval $(3-\sqrt{5},2)$. For $\alpha \in \left(0,3-\sqrt{5}\right]$, the oscillatory property of all solutions is ensured when condition (33) is fulfilled.

We have shown, using examples, that there are difference equations for which the previously known oscillatory conditions for the same types of difference equations are not applicable, but the proposed criteria are satisfied. Therefore, we have extended the set of difference equations with oscillatory coefficients for which conditions verifying their oscillatory properties exist.