 Research
 Open access
 Published:
On the qualitative behavior of the solutions to secondorder neutral delay differential equations
Journal of Inequalities and Applications volumeÂ 2020, ArticleÂ number:Â 256 (2020)
Abstract
Differential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a secondorder neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented.
1 Introduction
Consider the class of nonlinear neutral delay differential equations of the form
where \(w(y)=u(y)+b(y)u(\vartheta (y))\) and Î¼ is the ratio of two odd positive integers. We assume the following conditions hold.

(A1)
\(a, c, \vartheta, \varsigma \in C (\mathbb{R_{+}},\mathbb{R_{+}})\) such that \(\vartheta (y)\leq y\), \(\varsigma (y)\leq y\) for \(y \geq y_{0}\), \(\vartheta (y) \to \infty \), \(\varsigma (y) \to \infty \) as \(y \to \infty \).

(A2)
\(g \in C(\mathbb{R,\mathbb{R}})\) is nondecreasing and odd with \(ug(u)>0\) for \(u\neq 0\).

(A3)
\(a(y)>0\) and \(\int _{0}^{\infty } (a(\eta ) )^{1/\mu }\,d\eta =\infty \). By letting \(A(y)=\int _{0}^{y} (a(\eta ) )^{1/\mu }\,d\eta \), we have \(\lim_{y \to \infty } A(y)=\infty \).

(A4)
\(b \in C(\mathbb{R_{+}},\mathbb{R_{}})\) with \(1+(2/3)^{1/\mu } \leq b_{0} \leq b(y) \leq 0 \) for \(y \in \mathbb{R_{+}}\).

(A5)
\(b \in C(\mathbb{R_{+}},\mathbb{R_{}})\) with \(1 <b_{0} \leq b(y) \leq 0 \) for \(y \in \mathbb{R_{+}}\).
In 1978, Brands [1] showed that the solutions to
are oscillatory, if and only if, the solutions to \(u''(y)+c(y)u(y) =0\) are oscillatory. Baculikova et al. [2] considered (1) and studied the oscillatory behavior of (1) for \(g(u)=u\), \(0\leq {}b(y)\leq {}b_{0}<\infty \) and (A3). They obtained sufficient conditions for the oscillation of the solutions of the linear counterpart of (1), using comparison techniques. Chatzarakis et al. [3] considered the equation
Also, Chatzarakis et al. [4] studied (2) to obtain new oscillation criteria. DÅ¾urina [5] studied the linear counterpart of (1) when \(0\leq b(y)\leq b_{0}<\infty \) and (A3) and established sufficient conditions for the oscillation of the solutions of the linear counterpart of (1) by comparison techniques. Karpuz et al. [6] studied (1) for various ranges of the neutral coefficient b. Pinelas and Santra [7] studied necessary and sufficient conditions for the solutions of
Wong [8] obtained necessary and sufficient conditions for the oscillation of
where the constant b satisfies \(1< b<0\). Grace et al. [9] studied (1) and established sufficient conditions for \(0 \leq b(y) <1\). For further work on this type of equations, we refer the reader to [10â€“36] and the references cited therein. We may note that most of the authors considered only sufficient conditions, and only a few considered necessary and sufficient conditions. Hence, the objective of this work is to establish both necessary and sufficient conditions for oscillation of (1) without using comparison techniques.
In Sect.Â 2 some preliminary results are presented, Sect.Â 3 deals with main results, Sect.Â 4 represents the conclusion and the final section includes open problems.
2 Preliminary results
In this section, two lemmas are presented which we need for our work in the sequel.
Lemma 2.1
Under the assumptions (A1)â€“(A3) and (A4) or (A5) and the solution u of (1) is an eventually positive solution, we have

(i)
\(w(y)<0\), \(w^{\prime }(y)>0\) and \((a(w^{\prime })^{\mu })^{\prime }(y)<0\);

(ii)
\(w(y)>0\), \(w^{\prime }(y)>0\) and \((a(w^{\prime })^{\mu })^{\prime }(y)<0\),
for sufficiently large y.
Proof
Assume there exists a \(y_{1} \geq {}y_{0}\) such that \(u(y)>0\), \(u(\vartheta (y))\), and \(u(\varsigma (y))>0\) for \(y\geq {}y_{1}\). From (1) and (A2), we have
which implies that \((a(w^{\prime })^{\mu } )(y)\) is nonincreasing on \([y_{1},\infty )\). We have \(a(y)>0\), and thus either \(w^{\prime }(y)<0\) or \(w^{\prime }(y)>0\) for \(y\geq {}y_{2}\), where \(y_{2}\geq {}y_{1}\).
If \(w^{\prime }(y)>0\) for \(y\geq {}y_{2}\), then we have (i) and (ii). We prove now that \(w^{\prime }(y)<0\) cannot occur.
If \(w^{\prime }(y)<0\) for \(y\geq {}y_{2}\), then there exists \(\kappa _{1}>0\) such that \((a(w^{\prime })^{\mu } )(y)\leq \kappa _{1}\) for \(y\geq {}y_{2}\), which yields upon integration over \([y_{2},y)\subset [y_{2},\infty )\) after dividing through by a
By virtue of condition (A3), \(\lim_{t\to \infty }w(y) =\infty \). We consider the following possibilities:
Let the solution u be unbounded. There exists a sequence \(\{y_{k}\}\) such that \(\lim_{k \to \infty } y_{k} = \infty \) and \(\lim_{k\to \infty } u(y_{k}) =\infty \), where \(u(y_{k}) = \max \{u(\eta ): y_{0} \leq \eta \leq y_{k}\}\). Since \(\lim_{y \to \infty } \vartheta (y) = \infty \), \(\vartheta (y_{k}) > y_{0}\) for all sufficiently large k. By \(\vartheta (y) \leq y\),
Therefore, for all large k,
which contradicts \(\lim_{y \to \infty } w(y) = \infty \).
Let the solution u be bounded, then w is bounded, from which one concludes \(\lim_{y \to \infty } w(y) = \infty \), a contradiction. Hence, w satisfies one of the cases (i) or (ii). This completes the proof.â€ƒâ–¡
Lemma 2.2
Under the assumptions (A1)â€“(A3), (A4) or (A5), (i) and u is an eventually positive solution of (1), we have \(\lim_{ y \to \infty }u(y)=0\).
Proof
Assume that there exists a \(y_{1} \geq {}y_{0}\) such that \(u(y)>0\), \(u(\vartheta (y))\), and \(u(\varsigma (y))>0\) for \(y\geq {}y_{1}\). Then LemmaÂ 2.1 holds and w satisfies one of the cases (i) or (ii) for \(y_{2} \geq y_{1}\), where \(y \geq y_{2}\). Let w satisfy (i) for \(y \geq y_{2}\). Therefore,
which implies that \(\limsup_{y \to \infty } u(y)=0\) and hence \(\lim_{y \to \infty }u(y)=0\).â€ƒâ–¡
Remark 1
In view of (ii) of LemmaÂ 2.1, it is obvious that \(\lim_{y\to \infty }w(y)>0\), i.e., there exists \(\kappa _{1}>0\) such that \(w(y)\geq \kappa _{1}\) for all large y.
3 Main results
In this section, we establish the necessary and sufficient conditions for the oscillation of the solution of (1) by considering the two cases when \(g(v)/v^{\mu _{1}}\) is nonincreasing and \(g(v)/v^{\mu _{1}}\) is nondecreasing.
3.1 The case when \(g(v)/v^{\mu _{1}}\) is nonincreasing
Suppose that there exists \({\mu _{1}}\) such that \(0<{\mu _{1}}<\mu \) and
For example the function \(g(u)=u^{\mu _{2}} \operatorname{sgn}(u)\) with \(0<{\mu _{2}}<{\mu _{1}}<\mu \) satisfying (5).
Theorem 3.1
Assume that (A1)â€“(A4) and (5) hold. Then each unbounded solution of (1) is oscillatory if and only if
Proof
On the contrary, we assume that there exists a nonoscillatory unbounded solution \(u(y)\) of (1). Suppose that the solution \(u(y)\) is eventually positive. Then there exists \(y_{1} \geq y_{0}\) such that \(u(y) > 0\), \(u(y)>0\), \(u(\vartheta (y))>0\) and \(u(\varsigma (y))>0\) for \(y\geq {}y_{1}\). Proceeding as in the proof of LemmaÂ 2.1, we see that \((a(w')^{\mu } )(y)\) is nonincreasing, and w satisfies one of the cases (i) or (ii) on \([y_{2},\infty )\), where \(y_{2}\geq {}y_{1}\). Then we have the following two possible cases.
Case 1. Let w satisfy (i) for \(y\geq {}y_{2}\). As u is the unbounded solution, there exists \(y\geq {}y_{2}\) such that \(u(y)=\max \{u(s): y_{2}\leq s\leq {}T\}\). Since \(w(y)=u(y)+b(y)u(\vartheta (y))\), we have \(u(y)\leq {}w(y)+\{1(2/3)^{1/\mu }\}u(\vartheta (y))< u(y)\), which leads a contradiction.
Case 2. Let w satisfy (ii) for \(y\geq y_{2}\). Note that \(\lim_{y \to \infty } (a(w')^{\mu } )(y)\) exists. Using \(w(y) \leq u(y)\) in (1) and integrating the new inequality from y to +âˆž, we obtain
That is,
for \(y\geq y_{3}\). Let \(y_{4}> y_{3}\) be a point such that
Then integrating (7) from \(y_{3}\) to y, we get
i.e.,
Since \((a(w')^{\mu } )(y)\) is nonincreasing on \([y_{4},\infty )\), there exist \(\kappa >0\) and \(y_{5}> y_{4}\) such that \((a(w')^{\mu } )(y) \leq \kappa \) for \(y\geq y_{5}\). Integrating the inequality \(w'(y) \leq (\kappa / a(y))^{1/\mu }\), we have
Since \(\lim_{t\to \infty }A(y)=\infty \), the last inequality becomes
On the other hand, (5) implies that
Consequently, (8) becomes
If we define
then \(w^{\mu _{1}} / (\kappa ^{1/\mu }A )^{\mu _{1}} \geq \Upsilon ^{{ \mu _{1}}/\mu }/ (2\kappa ^{1/\mu } )^{\mu _{1}}\). Taking the derivative of Ï’ we get
Therefore, \(\Upsilon (y)\) is nonincreasing on \([y_{5}, \infty )\) so \(\Upsilon ^{{\mu _{1}}/\mu }(\varsigma (y))/\Upsilon ^{{\mu _{1}}/\mu }(y) \geq 1\), and
We have \({\mu _{1}}/\mu <1\) and \(\Upsilon (y)\) is positive and nonincreasing. Integrating the last inequality, from \(y_{5}\) to y, we have
which contradicts (6).
If \(u(y)<0\) for \(y\geq {}y_{1}\), then we set \(y(y):=u(y)\) for \(y\geq {}y_{1}\) in (1). Using (A2), we find
where \(\overline{w}(y)=y(y)+b(y)y(\vartheta (y))\) and \(\overline{g}(u):=g(u)\) for \(u\in \mathbb{R}\). Clearly, gÌ… satisfies (A2). Then, proceeding as above, we can find the same contradiction.
To prove the condition (6) is necessary, assume that (6) does not hold; so for some \(\kappa > 0\) and \(y \geq y_{0}\) we have
We set
We define the operator \(\Omega: S \to C([y_{0},+\infty ),\mathbb{R})\) by
For every \(u \in S\) and \(y \geq Y\), we have
For every \(u \in S\) and \(y \geq Y\), we have \(u(y)\leq \kappa ^{1/\mu } A(y)\) and \(g(u(y))\leq g(\kappa ^{1/\mu } A(y))\). Then
which implies that \((\Omega u)(y) \in S\). Let us define now a sequence of continuous function \(v_{n}: [y_{0}, +\infty )\to \mathbb{R}\) by the recursive formula
Inductively, it is easy to verify that, for \(n>1\),
Therefore the pointwise limit of the sequence exists. Let \(\lim_{y \to \infty }u_{n}(y)=v(y)\) for \(y \geq y_{0}\). By Lebesgueâ€™s dominated convergence theorem, \(u \in S\) and \((\Omega u)(y) =u(y)\), where \(u(y)\) is a solution of (1) on \([Y,\infty )\) such that \(u(y)>0\). Hence, (6) is necessary. This completes the proof.â€ƒâ–¡
Example 3.2
Consider the delay differential equation
Here \(\mu = 3/5\), \(a(y)=e^{y}\), \(1 < b(y)=e^{y} \leq 0\), \(\vartheta (y)=y1\), \(\varsigma (y)=y2\), \(A(y)=\int _{0}^{y} e^{5s/3} \,ds= \frac{3}{5} (e^{5y/3}1 )\), \(g(v)=v^{1/3}\). For \({\mu _{1}}=1/2\), we have a decreasing function \(g(v)/v^{\mu _{1}}=v^{1/6}\). Now
So, all the conditions of TheoremÂ 3.1 hold, and therefore every unbounded solution of (9) is oscillatory.
Theorem 3.3
Let assumptions (A1)â€“(A4) hold. Then each unbounded solution of (1) oscillates if and only if (6) holds for every \(\kappa >0\).
Proof
To prove sufficiency by contradiction, assume that the solution u of (1) is eventually positive and unbounded. So, there exists \(y_{1}\geq {}y_{0}\) such that \(u(y)>0\), \(u (\vartheta (y) )>0\) and \(u (\varsigma (y) )>0\) for \(y\geq {}y_{1}\). Proceeding as in the proof of LemmaÂ 2.1, \((a(w')^{\mu } )(y)\) is nonincreasing, w satisfies one of the cases (i) or (ii) on \([y_{2},\infty )\), where \(y_{2}\geq {}y_{1}\). We have the following two possible cases.
Case 1. Let w satisfy (i) for \(y \geq y_{2}\). This case is similar to the proof of TheoremÂ 3.1.
Case 2. Let w satisfy (ii) for \(y \geq y_{2}\). Since \(w(y)\) is unbounded and monotonically increasing, it follows that
If \(c =0\), then \(\lim_{t\to \infty }A(y)=+\infty \) implies that \(\lim_{t\to \infty }w(y)< +\infty \), which is invalid (\(\because w(y)\) is unbounded). Hence \(c\neq 0\). Therefore, there exist a constant \(\kappa > 0\) and a \(y_{2} > y_{1}\) such that \(w(y)\geq \kappa ^{1/\mu } A(y)\) for \(y\geq y_{2}\). Consequently, \(u(y) \geq w(y) \geq \kappa ^{1/\mu } A(y)\) for \(y \geq y_{2}\). Using \(u(y)\geq \kappa ^{1/\mu } A(y)\) in (1) and then integrating the final inequality from \(y_{2}\) to +âˆž, we obtain a contradiction to (6) for every \(\kappa >0\).
By using the same transformation as in the proof of TheoremÂ 3.1 we can get a contradiction for an eventually negative unbounded solution, so we omit it here.
One can prove the necessary part by following the proof of TheoremÂ 3.1. So we omit it here. The proof of the theorem is complete.â€ƒâ–¡
Theorem 3.4
Assume that (A1)â€“(A4) and (5) hold. Then each solution of (1) is oscillatory or \(\lim_{y \to \infty }u(y)=0\) if and only if (6) holds for every \(\kappa >0\).
Proof
On the contrary, we assume that the solution u of (1) is eventually positive. Then there exists \(y_{1}\geq {}y_{0}\) such that \(u(y)>0\), \(u(\vartheta (y))>0\) and \(u(\varsigma (y))>0\) for \(y\geq {}y_{1}\). Proceeding as in the proof of LemmaÂ 2.1, we see \((a(w')^{\mu } )(y)\) is nonincreasing, and w satisfies one of the cases (i) or (ii) on \([y_{2},\infty )\), where \(y_{2}\geq {}y_{1}\). Thus, we have the following two possible cases.
Case 1. Let w satisfy (i) for \(y\geq y_{2}\). Then, by LemmaÂ 2.2, we have \(\lim_{y \to \infty }u(y)=0\).
Case 2. Let w satisfy (ii) for \(y\geq y_{2}\). The case follows from the proof of TheoremÂ 3.1.
The necessary part is similar to TheoremÂ 3.1. The proof of the theorem is complete.â€ƒâ–¡
3.2 The case when \(g(u)/u^{\mu _{1}}\) is nondecreasing
Suppose that there exists \({\mu _{1}}>\mu \) such that
For example we might consider the function \(g(u)=u^{\mu _{2}} \operatorname{sgn}(u)\) with \(\mu <{\mu _{1}}<{\mu _{2}}\) satisfying (10).
Theorem 3.5
Assume that (A1)â€“(A3), (A5), (10), \(\varsigma ^{\prime }(y) \geq 1\) hold. Then each solution of (1) oscillates or \(\lim_{y \to \infty }u(y)=0\) if and only if
Proof
Proceeding in the proof of TheoremÂ 3.4, we can conclude that \(\lim_{y \to \infty }u(y)=0\) when z satisfies (i). Let us consider Case 2, for \(y\geq y_{2}\). By RemarkÂ 1, there exist a constant \(\kappa > 0\) and \(y_{2} >y_{1}\) such that \(z (\varsigma (y) )\geq \kappa \) for \(y\geq y_{2}\). Consequently,
for \(y\geq y_{2}\). Using \(w(y) \leq u(x)\) and (12) in (1), and then integrating the final inequality we have
Since \((a(w')' )(y)\) is nonincreasing and positive, we have
for all \(y \geq y_{2}\). Therefore,
implies that
Integrating the final inequality from \(y_{2}\) to +âˆž, we have
which contradicts (11).
Next, we show that (11) is necessary. Assume that (11) does not hold and let there exist \(y \geq y_{0}\) such that
where \(\kappa > 0\) is a constant. We set
We define the operator \(\Omega: S \to C([y_{0},\infty ),\mathbb{R})\) by
For every \(u \in S\) and \(y \geq Y\), \((\Omega u)(y)\geq \frac{1b_{0}}{5}\) and
which implies that \(\Omega u \in S\). The remaining proof follows from TheoremÂ 3.1. This completes the proof.â€ƒâ–¡
Example 3.6
Consider the differential equation
Here \(\mu = 1/5\), \(a(y)=1\), \(\varsigma (y)=y2\), \(g(v)=v^{\frac{7}{3}}\). For \({\mu _{1}}=4/3\), we have \(g(v)/v^{\mu _{1}}=v\), which is an increasing function. To check (11) we have
So, all conditions of TheoremÂ 3.5 hold, and therefore each solution of (13) oscillates or converges to zero.
4 Conclusion
It is worth noting that we have established the necessary and sufficient conditions when \(1 < b(y) \leq 0\). These conditions do not hold in all ranges of \(b(y)\).
Remark 2
Theorems 3.1â€“3.5 also hold for the following equation:
where \(b, a, c_{j}, g_{j}, \varsigma _{j}\) \((j =1,2,\dots,m)\) satisfy assumptions (A1)â€“(A5). In order to extend Theorems 3.1â€“3.5, we can find an index i so that \(c_{j}, g_{j}, \varsigma _{j}\) satisfies (6) and (11).
Example 4.1
Consider the neutral differential equation
Here \(\mu = 3/5\), \(a(y)=e^{y}\), \(b(y)=e^{y}\), \(\varsigma _{1}(y)=u2\), \(\varsigma _{2}(y)=u1\), \(A(y)=\int _{0}^{y} e^{5s/3} \,ds= \frac{3}{5} (e^{5y/3}1 )\), \(g_{1}(v)=v^{1/3}\) and \(g_{2}(v)=v^{1/5}\). For \({\mu _{1}}=1/2\), we have decreasing functions \(g_{1}(v)/v^{\mu _{1}}=v^{1/6}\) and \(g_{2}(v)/v^{\mu _{1}}=v^{3/10}\). Now,
So, all the conditions of TheoremÂ 3.1 hold, and therefore every unbounded solution of (14) is oscillatory.
Example 4.2
Consider the differential equation
Here \(\mu = 5/7\), \(a(y)=1\), \(\varsigma _{1}(y)=y2\), \(\varsigma _{2}(y)=y1\), \(g_{1}(v)=v^{5/3}\) and \(g_{2}(v)=v^{3}\). For \({\mu _{1}}=4/3\), we have decreasing functions \(g_{1}(v)/v^{\mu _{1}}=v^{1/3}\) and \(g_{2}(v)/v^{\mu _{1}}=v^{5/3}\). Clearly, all the conditions of TheoremÂ 3.5 hold. Thus, each solution of (15) oscillates or \(\lim_{y \to \infty }u(y)=0\).
Remark 3
Examples 4.1 and 4.2 prove the feasibility and effectiveness of RemarkÂ 2.
5 Open problem
This work leads to some open problems:

1.
Can we find necessary and sufficient conditions for the oscillation of solutions to secondorder differential equation (1) for the other ranges of the neutral coefficient b?

2.
Is it possible to generalize this work to fractional order?
Availability of data and materials
Not applicable.
References
Brands, J.J.M.S.: Oscillation theorems for secondorder functionaldifferential equations. J. Math. Anal. Appl. 63(1), 54â€“64 (1978)
Baculikova, B., Dzurina, J.: Oscillation theorems for second order neutral differential equations. Comput. Math. Appl. 61, 94â€“99 (2011)
Chatzarakis, G.E., Dzurina, J., Jadlovska, I.: New oscillation criteria for secondorder halflinear advanced differential equations. Appl. Math. Comput. 347, 404â€“416 (2019)
Chatzarakis, G.E., Jadlovska, I.: Improved oscillation results for secondorder halflinear delay differential equations. Hacet. J. Math. Stat. 48(1), 170â€“179 (2019)
DÅ¾urina, J.: Oscillation theorems for second order advanced neutral differential equations. Tatra Mt. Math. Publ. 48, 61â€“71 (2011)
Karpuz, B., Santra, S.S.: Oscillation theorems for secondorder nonlinear delay differential equations of neutral type. Hacet. J. Math. Stat. 48(3), 633â€“643 (2019)
Pinelas, S., Santra, S.S.: Necessary and sufficient condition for oscillation of nonlinear neutral firstorder differential equations with several delays. J. Fixed Point Theory Appl. 20(1), 27 (2018)
Wong, J.S.W.: Necessary and suffcient conditions for oscillation of second order neutral differential equations. J. Math. Anal. Appl. 252(1), 342â€“352 (2000)
Grace, S.R., DÅ¾urina, J., Jadlovska, I., Li, T.: An improved approach for studying oscillation of secondorder neutral delay differential equations. J. Inequal. Appl. 2018, 193 (2018)
Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation of second order differential equations with a sublinear neutral term. Carpath. J. Math. 30, 1â€“6 (2014)
Abdalla, B., Abdeljawad, T.: On the oscillation of Caputo fractional differential equations with MittagLeffler nonsingular kernel. Chaos Solitons Fractals 127, 173â€“177 (2019)
Abdalla, B., Abodayeh, K., Abdeljawad, T., Alzabut, J.: New oscillation criteria for forced nonlinear fractional difference equations. Vietnam J. Math. 45, 609â€“618 (2017)
Abdalla, B., Abdeljawad, T.: On the oscillation of Hadamard fractional differential equations. Adv. Differ. Equ. 409, 1â€“12 (2018)
Abdalla, B., Alzabut, J., Abdeljawad, T.: On the oscillation of higher order fractional difference equations with mixed nonlinearities. Hacet. J. Math. Stat. 47(2), 207â€“217 (2018)
Baculikova, B., Dzurina, J.: Oscillation theorems for second order nonlinear neutral differential equations. Comput. Math. Appl. 62, 4472â€“4478 (2011)
Baculikova, B., Li, T., Dzurina, J.: Oscillation theorems for second order neutral differential equations. Electron. J. Qual. Theory Differ. Equ. 74, 1 (2011)
Bazighifan, O., Elabbasy, E.M.: Oscillation of higherorder differential equations with distributed delay. J. Inequal. Appl. 2019, 55 (2019)
Bazighifan, O., Dassios, I.: Riccati technique and asymptotic behavior of fourthorder advanced differential equations. Mathematics 8, 1â€“11 (2020)
Bazighifan, O., Ruggieri, M., Santra, S.S., Scapellato, A.: Qualitative properties of solutions of secondorder neutral differential equations. Symmetry 12(9), 1â€“10 (2020)
Santra, S.S., Bazighifan, O., Ahmad, H., Chu, Y.M.: Secondorder differential equation: oscillation theorems and applications. Math. Probl. Eng. 2020, Article ID 8820066 (2020). https://doi.org/10.1155/2020/8820066
Santra, S.S., Dassios, I., Ghosh, T.: On the asymptotic behavior of a class of secondorder nonlinear neutral differential equations with multiple delays. Axioms 9, 134 (2020). https://doi.org/10.3390/axioms9040134
Karpuz, B., Santra, S.: New criteria for the oscillation and asymptotic behavior of secondorder neutral differential equations with several delays. Turk. J. Math. 44, 1990â€“2003 (2020). https://doi.org/10.3906/mat2006103
Santra, S.S., Bazighifan, O., Ahmad, H., Yao, S.W.: Secondorder differential equation with multiple delays: oscillation theorems and applications. Complexity 2020, Article ID 8853745 (2020). https://doi.org/10.1155/2020/8853745
Santra, S.S., Ghosh, T., Baghifan, O.: Explicit criteria for the oscillation of secondorder differential equations with several sublinear neutral coefficients. Adv. Differ. Equ. 2020, 643 (2020). https://doi.org/10.1186/s13662020031011
Li, T., Rogovchenko, Y.V.: Oscillation theorems for second order nonlinear neutral delay differential eqquations. Abstr. Appl. Anal. 2014, Article ID 594190 (2014)
Qian, Y., Xu, R.: Some new oscillation criteria for higher order quasilinear neutral delay differential equations. Differ. Equ. Appl. 3, 323â€“335 (2011)
Pinelas, S., Santra, S.S.: Necessary and sufficient conditions for oscillation of nonlinear first order forced differential equations with several delays of neutral type. Analysis 39(3), 97â€“105 (2019)
Ragusa, M.A.: Elliptic boundary value problem in vanishing mean oscillation hypothesis. Comment. Math. Univ. Carol. 40(4), 651â€“663 (1999)
Ragusa, M.A., Tachikawa, A.: Regularity for minimizes for functional of double phase with variable exponents. Adv. Nonlinear Anal. 9, 710â€“728 (2020)
Santra, S.S.: Existence of positive solution and new oscillation criteria for nonlinear first order neutral delay differential equations. Differ. Equ. Appl. 8(1), 33â€“51 (2016)
Santra, S.S.: Oscillation analysis for nonlinear neutral differential equations of second order with several delays. Mathematica 59(82), 111â€“123 (2017)
Santra, S.S.: Oscillation analysis for nonlinear neutral differential equations of second order with several delays and forcing term. Mathematica 61(84), 63â€“78 (2019)
Santra, S.S.: Necessary and sufficient condition for oscillatory and asymptotic behavior of secondorder functional differential equations. Kragujev. J. Math. 44(3), 459â€“473 (2020)
Santra, S.S., Dix, J.G.: Necessary and sufficient conditions for the oscillation of solutions to a secondorder neutral differential equation with impulses. Nonlinear Stud. 27(2), 375â€“387 (2020)
Yang, Q., Xu, Z.: Oscillation criteria for second order quasilinear neutral delay differential equations on time scales. Comput. Math. Appl. 62, 3682â€“3691 (2011)
Ye, L., Xu, Z.: Oscillation criteria for second order quasilinear neutral delay differential equations. Appl. Math. Comput. 207, 388â€“396 (2009)
Acknowledgements
The authors are thankful to the the editors and and the referees for their valuable suggestions and comments, which improved the content of this paper.
Authorsâ€™ information
Not applicable.
Funding
The authors received no direct funding for this work.
Author information
Authors and Affiliations
Contributions
The authors declare that they read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
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
Santra, S.S., Alotaibi, H. & Bazighifan, O. On the qualitative behavior of the solutions to secondorder neutral delay differential equations. J Inequal Appl 2020, 256 (2020). https://doi.org/10.1186/s13660020025235
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660020025235