- Research
- Open Access
- Published:

# Oscillation results for nonlinear second order difference equations with mixed neutral terms

*Advances in Difference Equations*
**volumeÂ 2020**, ArticleÂ number:Â 8 (2020)

## Abstract

In this paper, we establish new oscillation criteria for nonlinear second order difference equations with mixed neutral terms. The key idea of our approach is to compare with first order equations whose oscillatory behaviors are already known. The obtained results not only improve and extend existing results reported in the literature but also provide a new platform for the investigation of a wide class of nonlinear second order difference equations. The results are supported by examples to demonstrate the validity of the theoretical findings.

## 1 Introduction

The study of differential/difference equations has been the object of many researchers over the last decades. Different approaches and various techniques have been adopted to investigate the qualitative properties of their solutions. Recently and driven by their widespread applications, the investigation of differential/difference equations of fractional order has drawn significant attention. The existenceâ€“uniqueness, stability and oscillation of solutions have been the main features that have attracted consideration [1â€“18].

In spite of the increasing interest in the study of differential/difference equations of fractional order, the oscillation and nonoscillation of solutions for integer order second order difference equations are still considered as an open area to investigate [19â€“21]. Equations with neutral terms are of particular significance as they arise in many applications including systems of control, electrodynamics, mixing liquids, neutron transportation, networks and population models. In the qualitative analysis of such systems, indeed, the oscillatory behavior of solutions of equations, where the rate of the growth depends not only on the current and the past states but also on rate of change in the past, play an important role [22â€“24]. In the light of this motivation and justification, different results have been reported regarding the asymptotic behavior of second order difference equations with neutral terms [25â€“40]. For relevant results on the application of oscillation theory, the reader can consult [18, 41, 42].

Following this bias in this paper, we establish new oscillation criteria for all solutions of nonlinear second order difference equations with mixed neutral terms of the form

and

where \(y(t)=x(t)+p_{1}(t)x^{\beta}(t-k)-p_{2}x^{\delta}(t-k)\) and under the conditions:

- (i)
*Î±*,*Î²*,*Î³*,*Î¼*and*Î´*are the ratios of positive odd integers, \(\alpha\ge1\), - (ii)
\(\{p_{1}(t)\}\), \(\{p_{2}(t)\}\), \(\{q(t)\}\) and \(\{c(t)\}\) are sequences of positive real numbers,

- (iii)
*k*,*m*, \(m^{*}\) are positive real numbers with \(h(t)=t-m+k+1\) and \(h^{*}(t)=t+m^{*}+k+1\).

Let \(\theta=\max\{k,m-1,m^{*}+1\}\). By a solution of Eq.Â (1) (respectively, (2)), we mean a real sequence \(\{ x(t)\}\) defined for all \(t\ge t_{0}-\theta\) and satisfies Eq.Â (1) (respectively, (2)) for all \(t \ge t_{0}\).

A solution of Eq.Â (1) (respectively, (2)) is called oscillatory if its terms are neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory. If all solutions of the equation are oscillatory, then we say the equation itself is called oscillatory.

The objective of the present paper is to provide sufficient conditions for the oscillation of Eqs.Â (1) and (2) whenever \(\beta<1\) and \(\delta>1\) and subject to the assumption

The key idea of our approach is to conduct a comparison with first order equations whose oscillatory behaviors are already known. In view of the theorems established in the literature, the results of this paper are new and have merit in the sense that no existing results can provide criteria which ensure the oscillation of all solutions of Eq.Â (1) or (2). Thus, we claim that the obtained results not only improve and extend existing results reported in the literature but also provide a new platform for the investigation of a wide class of nonlinear second order difference equations.

The rest of the paper is organized as follows: Sect.Â 2 is devoted to the main results of the paper. We present our investigations in two folds for Eqs.Â (1) and (2). Meanwhile, a relevant result on the existence of positive solutions for first order difference equations is stated. The proofs rely on some mathematical inequalities which are given for the sake of completeness. In Sect.Â 3, we provide two examples with specific parameters to illustrate the applicability of our theorems. We end the paper by a concluding remark.

## 2 Main results

For the sake of convenience, we use the notations

for \(t \ge t_{1}\) for some \(t_{1} \ge t_{0}\) where \(\{p(t)\}\) is a sequence of positive real numbers.

Prior to proceeding to the main results, we start by stating two fundamental lemmas.

### Lemma 1

*Let*
\(\{q(t)\}\)*be a sequence of positive real numbers*, *m**and*
\(m^{*}\)*are positive real numbers and*
\(f:\mathbb{R} \to\mathbb{R}\)*is a continuous nondecreasing function such that*
\(xf(x)>0\)*for*
\(x\ne0\).

- (I)
*The first order delay difference inequality*$$\Delta y(t)+q(t)f\bigl(y(t-m+1)\bigr)\le0 $$*has an eventually positive solution*,*so does the delay difference equation*$$\Delta y(t)+q(t)f\bigl(y(t-m+1)\bigr)= 0. $$ - (II)
*The first order advanced difference inequality*$$\Delta y(t)-q(t)f\bigl(y\bigl(t+m^{*}+1\bigr)\bigr)\ge0 $$*has an eventually positive solution*,*and so does the advanced difference equation*$$\Delta y(t)-q(t)f\bigl(y\bigl(t+m^{*}+1\bigr)\bigr)= 0. $$

The above statement is the discrete analog of Lemma 6.2.2 in [21] and Corollary 1 in [35, 36, 43]. The proof is straightforward and hence is omitted.

### Lemma 2

([44])

*If**X**and**Y**are nonnegative*, *then*

*where equalities hold if and only if*
\(X=Y\).

### 2.1 Oscillation of Eq.Â (1) when \(\beta<1\) and \(\delta>1\)

In what follows, we present our first oscillation result.

### Theorem 1

*Let*
\(\beta<1\), \(\delta>1\), *conditions* (i)*â€“*(iv) *and* (3) *hold*. *Assume that there exist a positive sequence*
\(\{p(t)\}\)*and positive real numbers*
\(k_{1}\)*and*
\(k_{2}\)*such that*
\(k_{1}< m-k-1\)*and*
\(k_{2}< m^{*}+k+1\)*where*

*If the first order advanced equation*

*where*
\(\rho(t)=t+m^{*}+k+1-k_{2}\), *is oscillatory and and we may assume that there exists a number*
\(\theta\in(0,1)\)*such that the two delay equations*

*and*

*are oscillatory*, *then Eq*.Â (1) *is oscillatory*.

### Proof

Let \(\{x(t)\}\) be a nonoscillatory solution of Eq.Â (1) say \(x(t)>0\), \(x(t-k)>0\), \(x(t-m+1)>0\) and \(x(t+m^{*}+1)>0\), for \(t \ge t_{1}\) for some \(t_{1} \ge t_{0}\). It follows from Eq.Â (1) that

Hence \(a(t)(\Delta y(t))^{\alpha}\) is nonincreasing and of one sign. That is, there exists \(t_{2} \ge t_{1}\) such that \(\Delta y(t)>0\) or \(\Delta y(t) <0\) for \(t \ge t_{2}\). From this, we shall consider the following four cases:

*Case (I)*: Since \(\Delta y(t) <0\) for \(t \ge t_{2}\),

By condition (3), we conclude that \(\lim_{t \to\infty }y(t)=-\infty\). This is a contradiction to the fact that *y* is eventually positive.

*Case (II)*: From the definition of *y*, we get

or

If we apply (4) with \(\lambda=\delta>1\), \(X=p_{2}^{\frac {1}{\delta}}(t)x(t)\) and \(Y= (\frac{1}{\delta}p(t)p_{2}^{\frac {-1}{\delta}}(t) )^{\frac{1}{\delta-1}}\), we have

If we apply (5) with \(\lambda=\beta< 1\), \(X=p_{1}^{\frac {1}{\beta}}(t)x(t)\) and \(Y= (\frac{1}{\beta}p(t)p_{1}^{\frac {-1}{\beta}}(t) )^{\frac{1}{\beta-1}}\), we have

Thus, we see that

Since *y* is nondecreasing, there exists a constant \(C>0\) such that \(y(t) \ge C\). Thus, we have

Now, there exists a constant \(c_{1} \in(0,1)\) such that

Thus, we have

Clearly, we see that

If we let \(w(t)=a(t)(\Delta y(t))^{\alpha}\), then \(\Delta y(t)= ( \frac{w(t)}{a(t)} )^{\frac{1}{\alpha}} \). The above inequality becomes

It follows from LemmaÂ 1(I) that the corresponding difference equation (8) has a positive solution. This is a contradiction.

Let

or

*Case (III)*: From the above arguments, we have \(\Delta z(t)=-\Delta y(t) <0\) for \(t \ge t_{1}\). We have

In this case, we consider

For \(t_{1} \le u \le v\), we may write

If we let \(u=h(t)\) and \(v=\xi(t)\) in the above inequality, we obtain

Setting \(W(t)=a(t)(-\Delta z(t))^{\alpha}\), we get

The rest of the proof is similar to that of Case (I) and hence is omitted.

*Case (IV)*: From (15), we have the inequality

Summing (18) from \(t-k_{2}\) to \(t-1\), one can easily get

It follows from Lemma 1(II) that the corresponding differential equation (7) also has a positive solution. This contradiction completes the proof.â€ƒâ–¡

### Corollary 1

*Let*
\(\beta<1\), \(\delta>1\), *conditions* (i)*â€“*(iv) *and* (3) *hold*. *Assume that there exist a positive sequence*
\(\{p(t)\}\)*and positive real numbers*
\(k_{1}\)*and*
\(k_{2}\)*such that*
\(k_{1} < m-k-1\)*and*
\(k_{2} < m^{*}+k+1\)*and* (7) *hold*. *If*

*and*

*for some*
\(t_{1} \ge t_{0}\), *then Eq*.Â (1) *is oscillatory*.

### Corollary 2

*Let*
\(\beta<1\), \(\delta>1\), *conditions* (i)*â€“*(iv) *and* (3) *hold*. *Assume that there exist a positive sequence*
\(\{p(t)\}\)*and positive real numbers*
\(k_{1}\)*and*
\(k_{2}\)*such that*
\(k_{1} < m-k-1\)*and*
\(k_{2} < m^{*}+k+1\), (7) *hold and conditions* (20) *and* (22) *are satisfied*. *If*

*and*

*for some*
\(t_{1} \ge t_{0}\), *then Eq*.Â (1) *is oscillatory*.

### Proof

Let \(\{x(t)\}\) be a nonoscillatory solution of Eq.Â (1), say \(x(t) >0\), \(x(t-k)>0\) and \(y(t)>0\) for \(t \ge t_{1}\) for some \(t_{1} \ge t_{0}\). It is easy to see that \(\Delta y(t) >0\), \(t \ge t_{1}\). Proceeding as in the proof of Theorem 1, one conclude that Cases (I), (III) and (IV) are invalid.

For Case (II), we refer to (13) and (14). Summing (13) from *t* to *u* and letting \(u \to\infty\), we have

Using (14) in the above inequality, we get

or

or

Using this inequality in (13), we have

The remaining part of the proof follows by adopting similar arguments as in the proof of Theorem 1.â€ƒâ–¡

### 2.2 Oscillation of Eq.Â (2) when \(\beta<1\) and \(\delta>1\)

Unlike the previous subsection, the main theorem herein provides a criterion for the oscillation as well as the oscillatory behavior of Eq.Â (2).

### Theorem 2

*Let*
\(\beta<1\), \(\delta>1\), *conditions* (i)*â€“*(iv) *and* (3) *hold*. *Assume that there exist a positive sequence*
\(\{p(t)\}\), *a constant*
\(\theta\in(0,1)\)*and positive real numbers*
\(k_{1}\)*and*
\(k_{2}\)*such that*
\(k_{1}< m-k-1\)*and*
\(k_{2}< m^{*}+k+1\)*and* (7) *holds*. *Further*, *assume the following condition*:

*If the first order advanced equation*

*and the equation*

*are oscillatory*, *then either Eq*.Â (2) *is oscillatory or all solutions converge to zero*.

### Proof

Let \(\{x(t)\}\) be a nonoscillatory solution of Eq.Â (2), say \(x(t)>0\), \(x(t-k)>0\), \(x(t-m+1)>0\) for \(t \ge t_{1}\) for some \(t_{1} \ge t_{0}\). It follows from Eq.Â (2) that

Hence \(a(t)(\Delta y(t))^{\alpha}\) is of one sign. That is, there exists a \(t_{2} \ge t_{1}\) such that \(\Delta y(t)>0\) or \(\Delta y(t)< 0\) for \(t \ge t_{2}\). We shall study the following four cases:

*Case (I)*: We claim that \(\lim_{t \to\infty}x(t)=0\). To prove this, we assume that there exists a constant \(b >0\) such that \(x(t-m+1)>b\). Using this in (28), we get

Summing up the above inequality and using the first part of (25), we reach a contradiction. On the other hand, if we sum the above inequality from *t* to *u* and let \(u \to\infty\), we get

Summing up again and using the second part of (25), we arrive at the desired contradiction.

*Case (II)*: We proceed exactly as in Case (II) in the proof of Theorem 1 to obtain (12) and from Eq.Â (28) one can easily see that

The rest of the proof is similar to that of Case (IV) of Theorem 1 and hence is omitted.

Let

Therefore, we have

Thus, we get

or

*Case (III)*: Clearly, we see that \(\Delta z(t)=-\Delta y(t) <0\) for \(t \ge t_{1}\). However, this is impossible due to condition (3).

*Case (IV)*: It follows that

If we let \(w(t)=a(t) (\Delta z(t) )^{\alpha}\), then we obtain \(\Delta z(t)= (\frac{w(t)}{a(t)} )^{\frac{1}{\alpha}}\) and thus the above inequality becomes

It follows from Lemma 1(I) that the corresponding difference equation (27) also has a positive solution, which is a contradiction. This completes the proof.â€ƒâ–¡

### Corollary 3

*Let*
\(\beta<1\), \(\delta>1\), *conditions* (i)*â€“*(iv), (3) *and* (25) *hold*. *Assume that there exist a positive sequence*
\(\{p(t)\}\), *a constant*
\(\theta\in(0,1)\)*and positive real numbers*
\(k_{1}\)*and*
\(k_{2}\)*such that*
\(k_{1} < m-k-1\)*and*
\(k_{2} < m^{*}+k+1\)*and* (7) *hold*. *If*

*and*

*then either Eq*.Â (2) *is oscillatory or all solutions converge to zero*.

## 3 Examples and concluding remark

Two numerical examples are illustrated in this section to demonstrate the consistency to the theoretical findings. We end the paper by a concluding remark.

### Example 1

Corresponding to (1), we consider the equation

where \(a(t)=t\), \(p_{1}(t)=\frac{1}{t} \to0\) as \(t \to\infty\) and \(p_{2}(t)=1\). Let \(p(t)=1\), \(\alpha=1\), \(\beta=\frac{1}{3}\) and \(\delta =3=\gamma=\mu\). Assume that there exist positive real numbers, \(k_{1}\) and \(k_{2}\) such that \(k_{1} < m-k-1\) and \(k_{2} < m^{*}+k+1\) and let \(\rho (t)\) and \(\xi(t)\) be as in Theorem 1.

It is easy to see that all conditions of Corollary 1 are satisfied if

Hence, Eq.Â (33) is oscillatory.

### Example 2

Corresponding to (2), we consider the equation

where \(a(t)=t\), \(p_{1}(t)=\frac{1}{t} \to0\) as \(t \to\infty\) and \(p_{2}(t)=1\). Let \(p(t)=1\), \(\alpha=1\), \(\beta=\frac{1}{3}\) and \(\delta =3=\gamma=\mu\). Assume that there exist positive real numbers \(k_{1}\) and \(k_{2}\) such that \(k_{1} < m-k-1\) and \(k_{2} < m^{*}+k+1\) and let \(\rho(t)\) and \(\xi(t)\) be as in Theorem 2.

It is easy to see that all conditions of Corollary 3 are satisfied if the advanced equation

is oscillatory for \(\theta\in(0,1)\) and hence either Eq.Â (34) is oscillatory or all solutions converge to zero.

### Remark 1

In this paper, we study the oscillation of two classes of nonlinear second order difference equations involving nonlinear mixed neutral terms. The investigations are carried on under the canonical form of the equations, that is, when the main equations are subject to condition (3). Unlike the techniques most used in the literature, we employ a novel comparison technique that is based on comparing with the oscillatory behavior of first order delay difference equations.

The paper is presented under high degree of generality. Thus, it will be of interest to study its results for higher order nonlinear difference equations with mixed neutral terms of the form

and

In addition, the results of this paper can be easily obtained for a dynamic equation on time scales. We leave this for interested researchers as future work.

## References

Agarwal, R.P., Baleanu, D., Rezapour, S., Salehi, S.: The existence of solutions for some fractional finite difference equations via sum boundary conditions. Adv. Differ. Equ.

**2014**, Article ID 282 (2014)Rezapour, S., Salehi, S.: On the existence of solution for a k-dimensional system of three points nabla fractional finite difference equations. Bull. Iran. Math. Soc.

**41**(6), 1433â€“1444 (2015)Baleanu, D., Rezapour, S., Mohammadi, H.: Some existence results on nonlinear fractional differential equations. Philos. Trans. R. Soc.

**371**, Article ID 20120144 (2013)Baleanu, D., Mousalou, A., Rezapour, S.: On the existence of solutions for some infinite coefficient-symmetric Caputoâ€“Fabrizio fractional integro-differential equations. Bound. Value Probl.

**2017**, Article ID 145 (2017)Baleanu, D., Ghafarnezhad, K., Rezapour, S., Shabibi, M.: On the existence of solutions of a three steps crisis integro-differential equation. Adv. Differ. Equ.

**2018**, Article ID 135 (2018)Alzabut, J., Abdeljawad, T., Baleanu, D.: Nonlinear delay fractional difference equations with applications on discrete fractional Lotkaâ€“Volterra competition model. J. Comput. Anal. Appl.

**25**(5), 889â€“898 (2018)Zhou, H., Alzabut, J., Yang, L.: On fractional Langevin differential equations with anti-periodic boundary conditions. Eur. Phys. J. Spec. Top.

**226**, 3577â€“3590 (2017)Zada, A., Waheed, H., Alzabut, J., Wang, X.: Existence and stability of impulsive coupled system of fractional integrodifferential equations. Demonstr. Math.

**52**, 296â€“335 (2019)Matar, M.M., Abu Skhail, E.S., Alzabut, J.: On solvability of nonlinear fractional differential systems involving nonlocal initial conditions. Math. Methods Appl. Sci. (2019). https://doi.org/10.1002/mma.5910

Rajchakit, G., Pratap, A., Raja, R., Cao, J., Alzabut, J., Huang, C.: Hybrid control scheme for projective lag synchronization of Riemann Liouville sense fractional order memristive BAM neural networks with mixed delays. Mathematics

**7**, Article ID 759 (2019). https://doi.org/10.3390/math7080759Ahmad, M., Zada, A., Alzabut, J.: Stability analysis for a nonlinear coupled implicit switched singular fractional differential system with

*p*-Laplacian. Adv. Differ. Equ.**2019**, Article ID 436 (2019)Abdalla, B., Alzabut, J., Abdeljawad, T.: On the oscillation of higher order fractional difference equations with mixed nonlinearities. Hacet. J. Math. Stat.

**42**(2), 207â€“217 (2018). https://doi.org/10.15672/HJMS.2017.458Grace, S.R., Agarwal, R.P., Wong, P.J.Y., Zafer, A.: On the oscillation of fractional differential equations. Fract. Calc. Appl. Anal.

**15**(2), 222â€“231 (2012)Wang, Y.-Z., Han, Z.-L., Zhao, P., Sun, S.-R.: Oscillation theorems for fractional neutral differential equations. Hacet. J. Math. Stat.

**44**(6), 1477â€“1488 (2015)Abdalla, B., Abudaya, K., Alzabut, J., Abdeljawad, T.: New oscillation criteria for forced nonlinear fractional difference equations. Vietnam J. Math.

**45**, 609â€“618 (2017)Baleanu, D., Rezapour, S., Saberpour, Z.: On fractional integro-differential inclusions via the extended fractional Caputoâ€“Fabrizio derivation. Bound. Value Probl.

**2019**, Article ID 79 (2019)Baleanu, D., Mousalou, A., Rezapour, S.: A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputoâ€“Fabrizio derivative. Adv. Differ. Equ.

**2017**, Article ID 51 (2017)Baleanu, D., Asad, J.H., Jajarmi, A.: The fractional model of spring pendulum: new features within different kernels. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci.

**19**(3), 447â€“454 (2018)Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York (2000)

Agarwal, R.P., Bohner, M., Grace, S.R., Oâ€™Regan, D.: Discrete Oscillation Theory. Hindawi Publishing Corporation, New York (2005)

Agarwal, R.P., Grace, S.R., Oâ€™Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)

Smith, F.E.: Population dynamics in

*Daphnia magna*and a new model for population growth. Ecology**44**(4), 651â€“663 (1963)Hale, J.K.: Theory of Functional Differential Equations, 2nd edn. Springer, New York (1977)

Burton, T.A., Purnaras, I.K.: A unification theory of Krasnoselskii for differential equations. Nonlinear Anal.

**89**, 121â€“133 (2013)Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation of second-order differential equations with a sublinear neutral term. Carpath. J. Math.

**30**(1), 1â€“6 (2014)Bohner, M., Grace, S.R., JadlovskÃ¡, I.: Oscillation criteria for second-order neutral delay differential equations. Electron. J. Qual. Theory Differ. Equ.

**2017**, Article ID 60 (2017)Bohner, M., Li, T.: Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient. Appl. Math. Lett.

**37**, 72â€“76 (2014)Dharuman, C., Graef, J.R., Thandapani, E., Vidhyaa, K.S.: Oscillation of second order difference equation with a sub-linear neutral term. J. Math. Appl.

**40**, 59â€“67 (2017)El-Morshedy, H.A.: Oscillation and nonoscillation criteria for half-linear second order difference equations. Dyn. Syst. Appl.

**15**(3â€“4), 429â€“450 (2006)El-Morshedy, H.A.: New oscillation criteria for second order linear difference equations with positive and negative coefficients. Comput. Math. Appl.

**58**(10), 1988â€“1997 (2009)El-Morshedy, H.A., Grace, S.R.: Comparison theorems for second order nonlinear difference equations. J. Math. Anal. Appl.

**306**(1), 106â€“121 (2005)Grace, S.R., El-Morshedy, H.A.: Oscillation criteria of comparison type for second order difference equations. J. Appl. Anal.

**6**(1), 87â€“103 (2000)Grace, S.R., Graef, J.R.: Oscillatory behavior of second order nonlinear differential equations with a sublinear neutral term. Math. Model. Anal.

**23**(2), 217â€“226 (2018)Graef, J.R., Grace, S.R., TunÃ§, E.: Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term. Opusc. Math.

**39**(1), 39â€“47 (2019)Ladas, G., Stavroulakis, I.P.: Oscillation caused by several retarded and advanced arguments. J. Differ. Equ.

**44**, 134â€“152 (1982)Li, W.-T., Saker, S.H.: Oscillation of second-order sublinear neutral delay difference equations. Appl. Math. Comput.

**146**(2â€“3), 543â€“551 (2003)Selvarangam, S., Madhan, M., Thandapani, E.: Oscillation theorems for second order nonlinear neutral type difference equations with positive and negative coefficients. Rom. J. Math. Comput. Sci.

**7**(1), 1â€“10 (2017)Selvarangam, S., Thandapani, E., Pinelas, S.: Oscillation theorems for second order nonlinear neutral difference equations. J. Inequal. Appl.

**2014**, Article ID 417 (2014)Alzabut, J., Bolat, Y.: Oscillation criteria for nonlinear higher-order forced functional difference equations. Vietnam J. Math.

**43**, 583â€“594 (2015). https://doi.org/10.1007/s10013-014-0106-yYildiz, M.K., Ã–gÃ¼nmez, H.: Oscillation results of higher order nonlinear neutral delay difference equations with a nonlinear neutral term. Hacet. J. Math. Stat.

**43**(5), 809â€“814 (2014)Baleanu, D., Jajarmi, A., Asad, J.H.: Classical and fractional aspects of two coupled pendulums. Rom. Rep. Phys.

**71**(1), 103 (2019)Baleanu, D., Asad, J.H., Jajarmi, A.: New aspects of the motion of a particle in a circular cavity. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci.

**19**(2), 361â€“367 (2018)Philos, C.G.: On the existence of nonoscillatory solutions tending to zero at for differential equations with positive delays. Arch. Math. (Basel)

**36**(21), 168â€“178 (1981)Hardy, G.H., Littlewood, I.E., Polya, G.: Inequalities. Cambridge University Press, Cambridge (1959)

### Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions, which helped in improving the contents of the paper.

### Availability of data and materials

The authors state that they have not used data or materials in this work.

## Funding

The second author would like to thank Prince Sultan University for supporting this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

## Author information

### Authors and Affiliations

### Contributions

The main idea of this paper was initially proposed by SRG who with JA performed all steps of the proofs of the main results. The two authors read and approved the final version of the manuscript.

### Corresponding author

## Ethics declarations

### Competing interests

The two authors declare that there is no competing interest concerning this work.

## Additional information

### Publisherâ€™s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the articleâ€™s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the articleâ€™s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Grace, S.R., Alzabut, J. Oscillation results for nonlinear second order difference equations with mixed neutral terms.
*Adv Differ Equ* **2020**, 8 (2020). https://doi.org/10.1186/s13662-019-2472-y

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/s13662-019-2472-y

### Keywords

- Oscillation criteria
- Comparison techniques
- Nonlinear second order difference equations
- Mixed neutral terms