 Research
 Open Access
New oscillation theorems for second order quasilinear difference equations with sublinear neutral term
 M. Nazreen Banu^{1}Email authorView ORCID ID profile and
 S. Mehar Banu^{2}
https://doi.org/10.1186/s1366201819320
© The Author(s) 2018
 Received: 14 October 2018
 Accepted: 13 December 2018
 Published: 22 December 2018
Abstract
Keywords
 Oscillation
 Quasilinear difference equations
 Sublinear neutral term
MSC
 39A10
1 Introduction
 (H_{1}):

\(\{ a_{n} \}\) is a positive real sequence such that \(\sum_{n = n_{0}}^{\infty} \frac{1}{a_{n}^{1/\beta}} = \infty\);
 (H_{2}):

\(\{ p_{n} \}\) and \(\{ q_{n} \}\) are positive real sequences for all \(n \in \mathbb{N} ( n_{0} )\) and \(p_{n} \to 0\) as \(n \to \infty\);
 (H_{3}):

k and ℓ are positive integers;
 (H_{4}):

\(\alpha \in (0,1],\beta\) and γ are ratio of odd positive integers.
Neutral type equations arise in a number of important applications in natural sciences and technology; see [7, 13]. Hence, in recent years there has been great interest in studying the oscillation of such type of equations. From the review of literature, one can see that many oscillation results are available for the equation when \(\alpha = 1\); see [1, 2, 5, 8–11, 14, 15, 18, 20], and the references cited therein. Also few results available for the oscillation of Eq. (1.1) while \(\beta = 1\); see [4, 12, 17, 19, 21, 22]. And as far as the authors knowledge there are no results available in the literature for the oscillatory behavior of Eq. (1.1).
Our purpose in this paper is to establish some new oscillation criteria for Eq. (1.1) which includes many of the known results as special cases when \(\alpha = 1\) or \(\alpha = 1\) and \(\beta = 1\) in Eq. (1.1). Further the methods used in this paper improve and extend some of the known results that are reported in the literature [3, 8–12, 14, 15, 17–21] and this is almost illustrated via examples.
2 Oscillation results
In this section, we obtain sufficient conditions for the oscillation of all solutions of Eq. (1.1). Due to the assumptions and the form of our equation, we need only to give proofs for the case of eventually positive solution since the proofs for eventually negative solutions would be similar.
Lemma 2.1
Proof
The proof of the lemma can be found in [3] and hence details are omitted. □
Lemma 2.2
Proof
Lemma 2.3
Proof
The proof of the lemma can be found in [21] and hence details are omitted. □
Lemma 2.4
If \(0 < \alpha < 1, \ell\) is a positive integer and \(\{ p_{n}\}\) is a positive real sequence with \(\sum_{n = n_{0}}^{\infty} p_{n} = \infty\), then every solution of eqution \(\Delta x_{n} + p_{n}x_{n  \ell}^{\alpha} = 0\), is oscillatory.
Lemma 2.5
If \(\alpha > 1\). If there exists a \(\lambda> \frac{1}{l} \ln \alpha\) such that \(\lim_{n \to \infty} \inf [ p_{n}\exp (  e^{\lambda n} ) ] > 0\), then every solution of eqution \(\Delta x_{n} + p_{n}x_{n  \ell}^{\alpha} = 0\) is oscillatory.
The proof of the Lemmas 2.4 and 2.5 can be found in [16] and hence details are omitted.
Next we state and prove some new oscillation results for Eq. (1.1).
Theorem 2.1
Proof
Corollary 2.2
Corollary 2.3
Then every solution of Eq. (1.1) is oscillatory.
Theorem 2.4
Proof
In the following by employing the Riccati substitution technique, we obtain new oscillation criteria for Eq. (1.1).
Theorem 2.5
Proof
Theorem 2.6
Proof
3 Examples
In this section, we present three examples to illustrate the main results.
Example 3.1
Example 3.2
Example 3.3
4 Conclusion
In this paper, by using a Riccati type transformation and the discrete mean value theorem we have established some new oscillation criteria for more general second order neutral difference equations. The obtained results include similar results to the ones established for second order difference equations with linear neutral terms or nonlinear neutral terms reported in the literature. Further none of the results in the papers [3–5, 8–12, 14, 15, 17–22] can be applied to Eqs. (3.1) to (3.3) to yield any conclusion.
Declarations
Acknowledgements
The authors thank the referee for carefully reading the manuscript and suggesting very useful comments which improve the content of the paper.
Availability of data and materials
Not Applicable.
Funding
Not Applicable.
Authors’ contributions
The authors have equally made the contributions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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