# New oscillation theorems for second order quasi-linear difference equations with sub-linear neutral term

- 308 Downloads

**Part of the following topical collections:**

## Abstract

## Keywords

Oscillation Quasi-linear difference equations Sub-linear 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, 9, 10, 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, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 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

*Let*\(\{ x_{n} \}\)

*be a positive solution of Eq*. (1.1)

*for all*\(n \in \mathbb{N} ( n_{0} )\).

*Then there exists a*\(n_{1} \in \mathbb{N} ( n_{0} )\)

*such that for all n*\(\ge n_{1}\)

### Proof

The proof of the lemma can be found in [3] and hence details are omitted. □

### Lemma 2.2

*Let*\(\{ x_{n} \}\)

*be a positive solution of Eq*. (1.1)

*for all*\(n \in \mathbb{N} ( n_{0} )\)

*and suppose Eq*. (2.1)

*holds*.

*Then there exists a*\(n_{1} \in \mathbb{N} ( n_{0} )\)

*such that*

*and*

### Proof

### Lemma 2.3

*Assume that*,

*for large*

*n*, \(( p_{n}, p_{n + 1},\ldots, p_{n + k - 1} ) \ne 0\).

*Then*

*has an eventually positive solution if and only if the corresponding inequality*

*has an eventually positive solution*.

### 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

*Let*\(\gamma \ge \beta\)

*be holds*.

*Assume that there exists a positive real sequence*\(\{ \mu_{n} \}\)

*tending to zero such that*\(B_{n} > 0\)

*for all*\(n \in \mathbb{N} ( n_{0} )\).

*If the first order delay difference equation*

*is oscillatory*,

*then every solution of Eq*. (1.1)

*is oscillatory*.

### Proof

### Corollary 2.2

*Let all conditions of Theorem*2.1

*hold with*\(\gamma = \beta\)

*for all*\(n \in \mathbb{N} ( n_{0} )\).

*If*

*then every solution of Eq*. (1.1)

*is oscillatory*.

### Corollary 2.3

*Let all conditions of Theorem*2.1

*hold with*\(\gamma > \beta\)

*for all*\(n \in \mathbb{N} ( n_{0} )\).

*If*\(\ell > k\)

*and there exists a*\(\lambda > \frac{1}{\ell - k}\ln \frac{\gamma}{\beta} \)

*such that*

*Then every solution of Eq*. (1.1) *is oscillatory*.

### Theorem 2.4

*Let*\(\gamma < \beta\)

*be holds*.

*Assume that there exists a positive decreasing real sequence*\(\{ \mu_{n} \}\)

*tending to zero such that*\(B_{n} > 0\)

*for all*\(n \in \mathbb{N} ( n_{0} )\).

*If for all*\(N \ge n_{0}\),

*for any constant*\(M > 0\),

*then every solution of Eq*. (1.1)

*is oscillatory*.

### Proof

In the following by employing the Riccati substitution technique, we obtain new oscillation criteria for Eq. (1.1).

### Theorem 2.5

*Let*\(\gamma \ge \beta\)

*hold*.

*Assume that there exists a positive decreasing real sequence*\(\{ \mu_{n} \}\)

*tending to zero*,

*such that*\(B_{n} > 0\)

*for all*\(n \in \mathbb{N} ( n_{0} )\).

*If there exists a positive*,

*nondecreasing a real sequence*\(\{ \rho_{n} \}\)

*such that*

*for sufficiently large*\(N > n_{1}\),

*then every solution of Eq*. (1.1)

*is oscillatory*.

### Proof

*N*to

*n*, we obtain

### Theorem 2.6

*Let*\(\gamma < \beta\)

*be holds*.

*Assume that there exists a positive*,

*nondecreasing real sequence*\(\{ \mu_{n} \}\)

*tending to zero*,

*such that*\(B_{n} > 0\)

*for all*\(n \in \mathbb{N} ( n_{0} )\).

*If there exists a positive*,

*nondecreasing real sequence*\(\{ \rho_{n} \}\)

*such that*,

*for some sufficiently large*\(N \ge n_{1}\),

*for any constant*\(M > 0\),

*then every solution of Eq*. (1.1)

*is oscillatory*.

### 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, 4, 5, 8, 9, 10, 11, 12, 14, 15, 17, 18, 19, 20, 21, 22] can be applied to Eqs. (3.1) to (3.3) to yield any conclusion.

## Notes

### 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.

### Authors’ contributions

The authors have equally made the contributions. All authors read and approved the final manuscript.

### Funding

Not Applicable.

### Competing interests

The authors declare that they have no competing interests.

## References

- 1.Agarwal, R.P.: Difference Equations and Inequalities. Marcel Dekker, New York (2000) zbMATHGoogle Scholar
- 2.Agarwal, R.P., Bohner, M., Grace, S.R., O’Regan, D.: Discrete Oscillation Theory. Hindawi Publ. Corp., New York (2005) CrossRefGoogle Scholar
- 3.Chang, J.: Kamenev-type oscillation criteria for delay difference equations. Acta Math. Sci.
**27**, 574–580 (2007) MathSciNetCrossRefGoogle Scholar - 4.Dharuman, C., Thandapani, E.: Oscillation of solutions of nonlinear difference equations with a superlinear neutral term. Nonauton. Dyn. Syst.
**5**, 52–58 (2018) MathSciNetCrossRefGoogle Scholar - 5.Elizabeth, S., Graef, J.R., Sundaram, P., Thandapani, E.: Classifying nonoscillatory solutions and oscillation of a neutral difference equation. J. Differ. Equ. Appl.
**11**, 605–618 (2005) CrossRefGoogle Scholar - 6.Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendan Press, Oxford (1991) zbMATHGoogle Scholar
- 7.Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1997) Google Scholar
- 8.Jiang, J.: Oscillatory criteria for second order quasilinear neutral delay difference equations. Appl. Math. Comput.
**125**, 287–293 (2002) MathSciNetzbMATHGoogle Scholar - 9.Jiang, J.: Oscillation of second order nonlinear neutral delay difference equations. Appl. Math. Comput.
**146**, 791–801 (2003) MathSciNetzbMATHGoogle Scholar - 10.Li, H.J., Yeh, C.C.: Oscillation criteria for second order neutral delay difference equations. Comput. Math. Appl.
**36**, 123–132 (1998) MathSciNetCrossRefGoogle Scholar - 11.Li, W.T., Saker, S.H.: Oscillation of second order sub linear neutral delay difference equations. Appl. Math. Comput.
**146**, 543–551 (2003) MathSciNetzbMATHGoogle Scholar - 12.Liu, X.: Oscillation of solutions of neutral difference equations with a nonlinear neutral term. Comput. Math. Appl.
**52**, 439–448 (2006) MathSciNetCrossRefGoogle Scholar - 13.Mac Donald, N.: Biological Delay Systems: Linear Stability Theory. Cambridge University Press, Cambridge (1989) Google Scholar
- 14.Saker, S.H.: New oscillation criteria for second order nonlinear neutral delay difference equations. Appl. Math. Comput.
**142**, 99–111 (2003) MathSciNetzbMATHGoogle Scholar - 15.Sun, Y.G., Saker, S.H.: Oscillation for second order nonlinear neutral delay difference equation. Appl. Math. Comput.
**163**, 909–918 (2005) MathSciNetzbMATHGoogle Scholar - 16.Tang, X.H., Liu, Y.J.: Oscillation for nonlinear delay difference equations. Tamkang J. Math.
**32**, 275–280 (2001) MathSciNetzbMATHGoogle Scholar - 17.Thandapani, E., Pandian, S., Balasubramanian, R.K.: Oscillation of solutions of nonlinear neutral difference equations with nonlinear neutral term. Far East J. Appl. Math.
**15**, 47–62 (2004) MathSciNetzbMATHGoogle Scholar - 18.Tripathy, A.K.: On the oscillation of second order nonlinear neutral delay difference equations. Electron. J. Qual. Theory Differ. Equ.
**2008**, 11 (2008) MathSciNetzbMATHGoogle Scholar - 19.Vidhya, K.S., Dharuman, C., Graef, J.R., Thandapani, E.: Oscillation of second order difference equations with a sublinear neutral term. J. Math. Appl.
**40**, 59–67 (2017) MathSciNetzbMATHGoogle Scholar - 20.Wang, D.M., Xu, Z.T.: Oscillation of second order quasilinear neutral delay difference equations. Acta Math. Appl. Sin.
**27**, 93–104 (2011) MathSciNetCrossRefGoogle Scholar - 21.Yildiz, M.K., Ogunmez, H.: Oscillation results of higher order nonlinear neutral delay difference equations with a nonlinear neutral term. Hacet. J. Math. Stat.
**43**, 809–814 (2014) MathSciNetzbMATHGoogle Scholar - 22.Zhang, Z., Chen, J., Zhang, C.: Oscillation of solutions for second order nonlinear difference equations with nonlinear neutral term. Comput. Math. Appl.
**41**, 1487–1491 (2001) MathSciNetCrossRefGoogle Scholar

## Copyright information

**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.