Abstract
In this paper, new oscillation results for nonlinear third-order difference equations with mixed neutral terms are established. Unlike previously used techniques, which often were based on Riccati transformation and involve limsup or liminf conditions for the oscillation, the main results are obtained by means of a new approach, which is based on a comparison technique. Our new results extend, simplify, and improve existing results in the literature. Two examples with specific values of parameters are offered.
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1 Introduction and preliminaries
Oscillation of solutions for third-order difference equations has received comparably little attention, although such equations are of importance in many fields of science such as economics, physics, mathematical biology, and other areas of mathematics [3, 4, 6, 7, 10, 11, 13, 14, 24, 27, 31, 32, 35–37]. It is worth to mention that third-order difference equations may have totally different behavior from corresponding third-order differential equations; see the explicit example in [9]. On the other hand, oscillation of solutions for difference equations of first and second order have been extensively investigated in the literature; see the monographs [1, 2, 8] and the papers [5, 12, 15–19, 21, 23, 25, 26, 30, 33].
In this study, we consider a nonlinear third-order difference equation with mixed nonlinear neutral terms. We obtain conditions guaranteeing oscillation of solutions of this equation. The main results are proved by using a comparison technique with first-order equations. Such an approach was effectively used for other types of equations in [20, 22]. To demonstrate this, we present two examples, which cannot be discussed using any of the previously established results.
We consider the equation
where \(y(t)=x(t)+{p_{4}}(t)x^{\alpha _{4}}(t-k)-{p_{5}}(t)x^{\alpha _{5}}(t-k)\), and subject to the assumptions:
-
(i)
\({\alpha _{1}}\), \({\alpha _{2}}\), \({\alpha _{3}}\), \({\alpha _{4}}\), \({\alpha _{5}}\) are the ratios of positive odd integers, \({\alpha _{1}}\geq 1\),
-
(ii)
\({p_{1}},{p_{2}},{p_{3}},{p_{4}},{p_{5}}:{\mathbb{Z}}\to (0,\infty )\) are sequences,
-
(iii)
\(m,m^{*},k\in {\mathbb{N}}\) are such that \(m>2\), \(m^{*}>2\), \(k< m-1\).
A solution of (1.1) is called oscillatory if it is neither eventually negative nor eventually positive. We call (1.1) oscillatory provided that all its solutions are oscillatory.
The objective of this paper is to offer conditions ensuring oscillation of (1.1) whenever
and subject to the assumption
In view of the results established in the literature and to the best of our observations, there are no oscillation results for (1.1).
This paper is organized as follows: In Sect. 2, we give some auxiliary results and introduce some notation. Sect. 3 features the main results of the paper. We present our investigations under two cases for (1.1). The first case is when \({\alpha _{4}}<1<{\alpha _{5}}\), and the other case is when \({\alpha _{4}}<{\alpha _{5}}\leq 1\). Our approach is based on a comparison technique with first-order difference equations. In Sect. 4, two examples are provided in order to illustrate our main theorems.
2 Auxiliary results and notations
We start with the following fundamental result. See [22, Lemma 1], and for the proof of (I), see [29, Lemma 2.2].
Lemma 2.1
-
(I)
If the first-order delay difference inequality
$$ \Delta y(t)+{p_{2}}(t)y^{\gamma }(t-m+1)\leq 0 $$has an eventually positive solution, then so does the corresponding delay difference equation.
-
(II)
If the first-order advanced difference inequality
$$ \Delta y(t)-{p_{2}}(t)y^{\gamma } \bigl(t+m^{*} \bigr)\geq 0 $$has an eventually positive solution, then so does the corresponding advanced difference equation.
We also need the following lemmas.
Lemma 2.2
(see [28])
If \(X,Y\geq 0\), then
and
Lemma 2.3
Assume (1.2). Then
implies that eventually one of the following four situations occur:
Proof
By (2.3), there exists \(t_{0}\in {\mathbb{N}}_{0}\) satisfying
We first assume that
Then, for all \(t\geq t_{1}\), we have
Hence,
Now, for \(t\geq t_{1}\), we get
due to (1.2). Thus,
Hence, for \(t\geq t_{2}\), we obtain
Therefore,
By (2.6), (2.7), and (2.8), we have
so Case PPP holds. Next, if (2.5) does not hold, then the only other possibility is
and hence
We assume that
Then, for all \(t\geq t_{1}\), we have
Thus,
Now, for \(t\geq t_{1}\), we get
Hence,
By (2.9), (2.11), and (2.12), we have
so Case NNN holds. Next, if (2.10) does not hold, then the only other possibility is
We assume that
Then, for all \(t\geq t_{1}\), we have
By (2.9), (2.13), and (2.15), we have
so Case PPN holds. Finally, if (2.14) does not hold, then the only other possibility is
By (2.9), (2.13), and (2.16), we have
so Case NPN holds. There are no other cases. See Table 1 for an illustration of the proof.
□
Throughout the remainder of the paper, we suppose that
For convenience, we introduce the notations
Remark 2.4
-
1.
Note that, due to the assumptions \(m>2\), \(m^{*}>2\), and \(k< m-1\), it is always possible to find \(k_{0}\), \(k_{1}\), \(k_{2}\), \(k_{3}\) such that (2.17) holds, e.g., one may pick
$$ k_{0}=k_{1}=k_{2}=1 \quad \text{and}\quad k_{3}=2. $$ -
2.
Note that \(\xi _{0}(t)>t\) holds always since \(m^{*}-2k_{0}>0\). Hence, equations involving \(\xi _{0}\) are of advanced type. Moreover, \(\xi _{1}(t)< t\), \(\xi _{2}(t)< t\), and \(\xi _{3}(t)< t\) always since \(m+1-k_{1}>0\), \(m+1-k>0\), and \(m+1-k-k_{3}>0\). Hence, equations involving \(\xi _{1}\), \(\xi _{2}\), \(\xi _{3}\) are of delay type.
3 Main results
Now we present our first oscillation result.
Theorem 3.1
Let \({\alpha _{4}}<1<{\alpha _{5}}\). Suppose that (i)–(iii), (1.2), and (2.17) hold. Assume that there exists \(p:{\mathbb{Z}}\to (0,\infty )\) such that
Let \(\theta _{0},\theta _{1}\in (0,1)\). If the first-order advanced difference equation
and the first-order delay difference equations
and
are oscillatory, then so is (1.1).
Proof
Assume that x is a nonoscillatory solution of (1.1), say
eventually. It follows from (1.1) that, eventually,
Hence (2.3) is satisfied, and thus, by Lemma 2.3, only the four Cases PPP, PPN, NNN, and NPN are possible. We now discuss each of these possible cases.
Cases PPP and PPN. Applying (2.1) with
we obtain
while applying (2.2) with
we get
Using these two inequalities, we have
Since y in both Cases PPP and PPN is positive and nondecreasing, there exists \(C>0\) satisfying \(y(t)\geq C\), and so we have
Next, due to (3.1), there exists \(\kappa \in (0,1)\) such that
Thus, we have
Case PPP. By (3.8), we get
Summing (3.9) from \(t-k_{0}\) to \(t-1\), we get
Therefore, we have
Summing (3.10) again from \(t-k_{0}\) to \(t-1\), we obtain
In summary, y is a positive and increasing solution of
Employing Lemma 2.1 (II), (3.2) also has an eventually positive solution, which is a contradiction.
Case PPN. We introduce
By (3.8), we obtain
First, we see that, eventually,
Since \(t_{1}/t\to 0\) as \(t\to \infty \), there exists \(\theta \in (0,1)\) so that
Now, we put
Then, eventually,
and hence
Altogether, eventually,
In summary, Z is a positive and decreasing solution of
Employing Lemma 2.1 (I), (3.3) also has an eventually positive solution, which is a contradiction.
Cases NNN and NPN. Throughout the remainder of the proof, we introduce Z again by (3.11). First note that, eventually,
Hence, eventually,
Thus, eventually,
Case NNN. First note that, eventually,
and therefore, eventually,
and thus, eventually,
In summary, Z is a positive and decreasing solution of
Employing Lemma 2.1 (I), (3.4) also has an eventually positive solution, which is a contradiction.
Case NPN. We let
First, we have, eventually,
so,
Next, we have, eventually,
so,
Thus, we see that
In summary, Z is a positive and decreasing solution of
Employing Lemma 2.1 (I), (3.5) also has an eventually positive solution, which is a contradiction. □
We now prove the following consequence of Theorem 3.1.
Theorem 3.2
Let \({\alpha _{4}}<1<{\alpha _{5}}\). Suppose that (i)–(iii), (1.2), (2.17), and (3.1) hold. Let \(\theta _{0},\theta _{1}\in (0,1)\). If the first-order advanced difference equation (3.2) and the first-order delay difference equations (3.3) and
are oscillatory, then so is (1.1).
Proof
We claim that oscillation of (3.19) implies oscillation of both (3.4) and (3.5). As all the other assumptions are the same as in Theorem 3.1, the statement then follows from Theorem 3.1. So assume that (3.19) is oscillatory. First, suppose that (3.4) is not oscillatory, say, there exists eventually positive Z satisfying
From the equality in (3.20), we see that Z is eventually decreasing, and since
we obtain \(Z(\xi _{3}(t))\leq Z(\xi _{2}(t))\) eventually. Using this in (3.20), we get
By Lemma 2.1 (I), (3.19) also has an eventually positive solution, a contradiction, showing that (3.4) is indeed oscillatory. Next, suppose that (3.5) is not oscillatory, say, there exists eventually positive Z satisfying
By Lemma 2.1 (I), (3.19) also has an eventually positive solution, a contradiction, showing that (3.5) is indeed oscillatory as well. Thus, the proof is complete. □
Remark 3.3
Observe that the minimum occurring in (3.19) may be calculated as follows:
The following theorem gives some further criteria for a special case.
Theorem 3.4
Let \({\alpha _{4}}<1<{\alpha _{5}}\) and \({\alpha _{2}}\leq {\alpha _{1}}\leq {\alpha _{3}}\). Suppose that (i)–(iii), (1.2), (2.17), and (3.1) hold. If
and
then (1.1) is oscillatory.
Proof
We claim that under the additional assumption \({\alpha _{2}}\leq {\alpha _{1}}\leq {\alpha _{3}}\), (3.21), (3.22), (3.23), and (3.24) imply oscillation of (3.2), (3.3), (3.4), and (3.5), respectively. As all the other assumptions are the same as in Theorem 3.1, the statement then follows from Theorem 3.1. First, suppose that (3.2) is not oscillatory, say, there exists eventually positive y satisfying (3.2). As can be seen from (3.2), y is eventually increasing and thus bounded below by some \(C>0\). Summing (3.2) from \(\tau =t\) to \(\tau =\xi _{0}(t)-1\), we obtain, eventually,
and thus, as \({\alpha _{3}}\geq {\alpha _{1}}\),
contradicting (3.21). Next, suppose that (3.3) is not oscillatory, say, there exists eventually positive Z satisfying (3.3). As can be seen from (3.3), Z is eventually decreasing and thus bounded above by some \(C>0\). Summing (3.3) from \(\tau =\xi _{1}(t)\) to \(\tau =t\), we obtain, eventually,
and thus, as \({\alpha _{1}}\geq {\alpha _{2}}\),
contradicting (3.22). Next, suppose that (3.4) is not oscillatory, say, there exists eventually positive Z satisfying (3.4). As can be seen from (3.4), Z is eventually decreasing and thus bounded above by some \(C>0\). Summing (3.4) from \(\tau =\xi _{2}(t)\) to \(\tau =t\), we obtain, eventually,
and thus, as \({\alpha _{1}}{\alpha _{5}}\geq {\alpha _{1}}\geq {\alpha _{2}}\),
contradicting (3.23). The proof that (3.24) implies oscillation of (3.5) follows exactly like the proof that (3.23) implies oscillation of (3.4), only with \(\xi _{2}\) and \(\Lambda _{2}\) replaced by \(\xi _{3}\) and \(\Lambda _{3}\), respectively. □
Remark 3.5
Note that Tang and Liu [34] have developed other oscillation criteria for sublinear delay difference equations, which could be used in place of (3.22), (3.23), and (3.24) to examine the oscillation of (3.3), (3.4), and (3.5). Similar criteria for superlinear advanced difference equations could not be found in the literature.
Now, we focus on the case \({\alpha _{4}}<{\alpha _{5}}\leq 1\).
Theorem 3.6
Let \({\alpha _{4}}<{\alpha _{5}}\leq 1\). Suppose that (i)–(iii), (1.2), and (2.17) hold. Assume
Let \(\theta _{0},\theta _{1}\in (0,1)\). If (3.2), (3.3), (3.4), and (3.5) are oscillatory, then so is (1.1).
Proof
Inspecting the proof of Theorem 3.1, we see that \({\alpha _{5}}>1\) was needed only in the discussions under the headline “Cases PPP and PPN” leading to (3.7). The rest of the proof does not use \({\alpha _{5}}>1\) and remains unaffected. Thus, while we used (3.1) in Theorem 3.1 to show (3.7), we will now use (3.25) to show (3.7), hence completing the proof. Applying (2.1) with
we obtain
and thus
Since y is positive and nondecreasing, there exists \(C>0\) such that \(y(t)\geq C\), and so we have
Next, due to (3.25), there exists \(\kappa \in (0,1)\) such that (3.7) is satisfied. This completes the proof. □
As before in Theorem 3.2 and Theorem 3.4, we now obtain the following results.
Theorem 3.7
Let \({\alpha _{4}}<{\alpha _{5}}\leq 1\). Suppose that (i)–(iii), (1.2), (2.17), and (3.25) hold. Let \(\theta _{0},\theta _{1}\in (0,1)\). If (3.2), (3.3), and (3.19) are oscillatory, then so is (1.1).
Theorem 3.8
Let \({\alpha _{4}}<{\alpha _{5}}\leq 1\) and \({\alpha _{2}}/{\alpha _{5}}\leq {\alpha _{1}}\leq {\alpha _{3}}\). Suppose that (i)–(iii), (1.2), (2.17), and (3.25) hold. If (3.21), (3.22), (3.23), and (3.24) hold, then (1.1) is oscillatory.
4 Examples
We conclude this paper by giving two examples, illustrating our theoretical findings.
Example 4.1
We consider the equation
Then (4.1) is in the form (1.1), where
Next, (i)–(iii) are satisfied, and so is (1.2) due to
Now we may pick (see Remark 2.4)
and then (2.17) is satisfied, and we have
Moreover, we have \({\alpha _{4}}<1<{\alpha _{5}}\) and \({\alpha _{2}}<{\alpha _{1}}={\alpha _{3}}\), so we will apply Theorem 3.4. We pick \(p={p_{4}}\), and then
and thus, (3.1) is satisfied. We also calculate
and
Hence, (3.21), (3.22), (3.23), and (3.24) hold. Now all the conditions of Theorem 3.4 are fulfilled, and thus, (4.1) is oscillatory.
Example 4.2
We consider the equation
Note that all data in (4.2) are the same as in (4.1), except
Moreover, we have \({\alpha _{4}}<{\alpha _{5}}<1\) and \({\alpha _{2}}/{\alpha _{5}}<{\alpha _{1}}={\alpha _{3}}\), so we will apply Theorem 3.8. We calculate
and thus, (3.25) is satisfied. The fulfillment of all other conditions of Theorem 3.8 follows in the same way as in Example 4.1, and hence (4.2) is oscillatory.
Remark 4.3
The results of this paper may be extended to higher-order difference equations of the form
where \(y(t)=x(t)+{p_{4}}(t)x^{\alpha _{4}}(t-k)-{p_{5}}x^{\alpha _{5}}(t-k)\). We leave the details for future consideration.
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J. Alzabut would like to thank Prince Sultan University for supporting this work.
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Alzabut, J., Bohner, M. & Grace, S.R. Oscillation of nonlinear third-order difference equations with mixed neutral terms. Adv Differ Equ 2021, 3 (2021). https://doi.org/10.1186/s13662-020-03156-0
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DOI: https://doi.org/10.1186/s13662-020-03156-0