- Research
- Open access
- Published:
The distribution of generalized zeros of oscillatory solutions for second-order nonlinear neutral delay difference equations
Advances in Difference Equations volume 2019, Article number: 282 (2019)
Abstract
In this paper, the distributions of generalized zeros of oscillatory solutions for second-order nonlinear neutral delay difference equations are studied. By means of inequality techniques, specific function sequences and non-increasing solutions for corresponding first-order difference inequality, some new estimates for the distribution of the zeros of oscillatory solutions are presented, which extend and improve some well-known results.
1 Introduction
In recent years, the study of oscillation of differential equations has become more and more perfect, including various sufficient conditions, necessary conditions, the existence of non-oscillatory solutions, and even the zeros distribution of oscillatory solutions.
In 2017, Li et al. [1] studied the distribution of zeros of oscillatory solutions for second-order nonlinear neutral delay differential equation
and obtained a sufficient condition for oscillation of differential equation.
However, most of references about oscillation of difference equations are concerned with sufficient or necessary conditions for oscillation; see [2,3,4,5,6,7,8]. We will also naturally ask some questions of difference equations: Are there any bounds for the distance between adjacent generalized zeros of oscillatory solutions when equations show oscillation? And how do we estimate these bounds? Therefore, we obtain the oscillation criteria of difference equations by studying the distribution of zeros.
The distribution of generalized zeros of oscillation solutions for first-order dynamic equations and second-order non-neutral dynamic equations on time scale can be found in [9,10,11]. However, most oscillatory results for second-order neutral dynamic equations are sufficient conditions for oscillation; see [12,13,14,15,16,17,18,19]. To the best of our knowledge, there is no paper on the generalized zero distribution of oscillation solutions for second-order neutral dynamic equations on time scale.
Motivated by the above papers, we consider the second-order neutral difference equation of the following form:
where Δ denotes the forward difference operator \(\Delta x(t)=x(t+1)-x(t)\), \(z(t)=x(t)+p(t)x(t-\tau)\), \(\mathbb{Z}\) represents the set of all integers and
Throughout this paper, we assume that the following hypotheses are satisfied:
- \((H_{1})\) :
-
\(a(t),q(t),p(t)\in(0,\infty)\), where \(t\in[ t_{0},\infty )_{\mathbb{Z}}\).
- \((H_{2})\) :
-
\(\tau, \sigma\in\mathbb{R^{+}}\), where \(\mathbb{R^{+}}\) represents the set of all positive real numbers, and \(\sigma>\tau\).
- \((H_{3})\) :
-
There exists a positive constant k such that \(\frac {f(u)}{u}\geqslant k\) for all \(u\neq0\).
- \((H_{4})\) :
-
There exists a function \(H(t)\) which satisfies \(H(t)\geqslant\frac{p(t-\sigma)q(t)}{q(t-\tau)}\) and \(\Delta H(t)\leqslant0\), \(t\geqslant t_{1}\) for some \(t_{1}\geqslant t_{0}+\sigma\), where \(t\in\mathbb{Z}\).
In this paper, we relate the distance between adjacent generalized zeros of an oscillation solution of (1.1) to a positivity problem of certain solution for a first-order delay difference inequality
where \(P(t)\in[ 0,1)\) which define by (2.1), \(r_{1}\) is a constant satisfying \(r_{1}\geqslant2\).
2 Preliminaries
In order to prove our main results, we establish some fundamental results in this section.
For convenience, we define a sequence \(\{F_{n}(t)\}\in[ 0,1)\) by
where \(T_{0}\) satisfies \(x(t)>0\), \(t\geqslant T_{0}\) when \(x(t)\) is eventually positive solution.
If \(t_{n}\) is a generalized zero of solution of (1.1), then it satisfies \(x([t_{n}])\cdot x([t_{n}]+1)\leqslant0\). Let \(d_{s}(x)\) be the least upper bound of the distance between adjacent generalized zeros of a solution \(x(t)\) of Eq. (1.1) on \([s,\infty)\).
Lemma 2.1
Assume that \(x(t)\) is an eventually positive solution of (1.1), and \((H_{1})\sim(H_{3})\) hold. Then \(z(t)\) satisfies \(z(t)>0\), \(\Delta z(t)>0\), \(\Delta(a(t)\Delta z(t))<0\).
Proof
If \(x(t)\) is an eventually positive solution of Eq. (1.1), then there exists a \(t_{1}>t_{0}\) such that \(x(t)>0\), \(x(t-\tau)>0\) and \(x(t-\sigma)>0\) for all \(t\geqslant t_{1}\). Thus \(z(t)=x(t)+p(t)x(t-\tau)>0\). From (1.1) and condition \((H_{3})\), we obtain
so we can conclude \(a(t)\Delta z(t)\), \(t\geqslant t_{1}\) is decreasing. It can be seen that there exists a \(t_{2}>t_{1}\) such that \(\Delta z(t)>0\) or \(\Delta z(t)<0\) for \(t\geqslant t_{2}\). Now, we prove \(\Delta z(t)>0\), \(t\geqslant t_{2}\). If not, assume that \(\Delta z(t)<0\), \(t\geqslant t_{2}\), then also \(a(t)\Delta z(t)<-c<0\) and summing up it from \(t_{2}\) to \(t-1\), we have
Taking limits of both sides for the above inequality, we have \(\lim_{t\rightarrow\infty}z(t)=-\infty\), which is a contradiction. The proof is completed. □
In the following lemmas, let \(r=[r_{1}]:=\max\{a|a\leqslant r_{1},a\in \mathbb{Z}\}\), where \(r_{1}\) is the delay argument of (1.2). And δ is a constant satisfying \(|\delta|\leqslant r\).
Lemma 2.2
Let n be a positive integer such that
If \(x(t)\) is a non-increasing function on \([T_{1}-\delta,T]_{\mathbb {Z}}\) which satisfies (1.2) on \([T_{1},T]_{\mathbb{Z}}\), then \(x(t)\) cannot be positive on \([T_{1},T]_{\mathbb{Z}}\), where \(T>T_{1}+(3n+1)r+(n+1)-\delta\), \(T_{1}\geqslant t_{0}+(2n+1)r\).
Proof
Without loss of generality, we assume that \(x(t)\) is positive on \([T_{1},T]_{\mathbb{Z}}\). Summing up (1.2) from \(t-r\) to \(t-1\), we have
Multiplying this inequality by \(P(t)\) and using (1.2), we get
so
Using \(\Delta x(t)=x(t+1)-x(t)\), we get
Let \(y_{1}(t):=x(t)\prod_{\zeta=t_{0}}^{t-1}\frac{1}{1-P(\zeta)}\). Then \(y_{1}(t)>0\) on \([T_{1},T]_{\mathbb{Z}}\) and
i.e.
From the definition of \(y_{1}(t)\) and \(\Delta x(t)\leqslant0\), \(t\in [T_{1}+r+1-\delta,T]_{\mathbb{Z}}\) we obtain
Since \(\Delta x(t)+P(t)x(t-r_{1})\leqslant0\), we can conclude \(\Delta y_{1}(t)\leqslant0\), \(t\in[T_{1}+r+1-\delta,T]_{\mathbb{Z}}\), and from (2.8), we have
Repeating the above procedure to this inequality, we get
Let \(y_{2}(t):=y_{1}(t)\prod_{\zeta=t_{0}+2r}^{t-1}\frac {1}{1-F_{1}(\zeta)}\). It follows from (2.9) that
where \(\Delta y_{2}(t)\leqslant0\) for \(t\in[T_{1}+4r+2-\delta ,T]_{\mathbb{Z}}\) and hence
Repeating this argument n times, we obtain
where \(\Delta y_{n}(t)\leqslant0\) for \(t\in[T_{1}+(3n-2)r+n-\delta ,T]_{\mathbb{Z}}\). Now, summing up (2.10) from \(t-r\) to \(t-1\in [T_{1}+(3n+1)r+n-\delta,T]_{\mathbb{Z}}\), we have
Since \(y(t)\) is decreasing, we obtain
which is a contradiction with hypothesis (2.4). The proof of Lemma 2.2 is complete. □
Lemma 2.3
Assume that \(\sum_{s=t-r}^{t-2}P(s)\geqslant\beta\) for \(0<\beta<1\) and there exist \(T_{2}\geqslant t_{0}+r\), \(T\geqslant T_{2}+(1+n)r-\delta\), \(n=1,2,\ldots \) and a function \(x(t)\) satisfying inequality (1.2) on \([T_{2},T]_{\mathbb{Z}}\) with \(\Delta x(t)\leqslant0\) for \(t\in [T_{2}-\delta,T]_{\mathbb{Z}}\). If \(x(t)\) is positive on \([T_{2},T]_{\mathbb{Z}}\), then
for some integer \(n\geqslant0\), where \(f_{n}(\beta)\) is defined by
Proof
Since \(x(t)\) is non-increasing on \([T_{2}-\delta ,T]_{\mathbb{Z}}\), we find
Summing inequality (1.2) from \(t-r+1\) to \(t-1\), where \(t\in [T_{2}+2r-\delta,T]_{\mathbb{Z}}\), we obtain
Therefore
On the other hand, dividing inequality (1.2) by \(x(t)\),
because of \(\Delta x(t)<0\),
Multiplying from \(s-r\) to \(t-r-1\) where \(s\in[t-r+1,t-1]_{\mathbb {Z}}\), we find
According to (2.12), this yields
We can easily obtain
Combining (2.14), (2.15) with (2.13), and because of the fact
we have
Thus
Repeating this argument, it follows by induction that
The proof is complete. □
Remark
It can easily be seen that either \(f_{n}(\beta)\) satisfies \(\lim_{t\rightarrow\infty}f_{n}(\beta)=1\) or \(f_{n}(\beta)\) is nondecreasing and \(\lim_{t\rightarrow\infty} f_{n}(\beta)=\infty\) or \(f_{n}(\beta)\rightarrow\infty\) after finite number of terms or \(f_{n}(\beta)\) is negative.
Lemma 2.4
Assume that \(\sum_{s=t-r}^{t-2}P(s)\geqslant\beta\), \(t\geqslant t_{0}\) holds for some \(0<\beta<1\) and there exists a function \(x(t)\) satisfying inequality (1.2) on \([T_{2},T+Nr+1]_{\mathbb{Z}}\) for some positive integer N such that \(\Delta x(t)\leqslant0\) on \([T_{2}-\delta ,T_{2}+Nr+1]_{\mathbb{Z}}\) where \(T_{2}\geqslant t_{0}+r\). If \(x(t)\) is positive on \([T_{2},T_{2}+Nr+1]_{\mathbb{Z}}\), then
where m is a positive integer, \(N\geqslant m+2-\frac{\delta}{r}\), and \(g_{m}(\beta)\) is defined by
Proof
From \(\sum_{s=t-r}^{t-1}P(s)\geqslant\beta\), \(t\geqslant t_{0}\), we see that \(\sum_{s=t}^{t+r-1}P(s)\geqslant\beta \) for \(t\geqslant T_{2}\). Summing both sides of (1.2) from t to \(t+r-1\), we obtain
Since \(T_{2}+r\leqslant t\leqslant s\leqslant t+r-1\), it follows \(T_{2}\leqslant t-r\leqslant s-r\leqslant t-1\). Again, summing (1.2) from \(s-r\) to t yields
It is clear that \(x(u-r_{1})\) is non-increasing on \([s-r,t+1]_{\mathbb {Z}}\subseteq[T_{2}+r-\delta,T_{2}+(N-1)r+1]_{\mathbb{Z}}\). Thus,
In view of the last inequality and (2.17), we obtain
for all \(t\in[T_{2}+2r-\delta,T_{2}+(N-1)r+1]_{\mathbb{Z}}\). As is well known, we have the identity
Consequently,
Substituting into (2.18),
Since \(x(t+r)>0\) on \([T_{2}+2r-\delta,T_{2}+(N-1)r]_{\mathbb{Z}}\), we get
On the other hand, when \(t\in[T_{2}+2r-\delta,T_{2}+(N-2)r+1]_{\mathbb {Z}}\), we have \(T_{2}+2r-\delta\leqslant t\leqslant t+r\leqslant T_{2}+(N-1)r\). So (2.20) leads to
Since \(x(t)\) is non-increasing on \([T_{2}-\delta,T+Nr+1]\), it follows that
From this inequality and (2.19), we obtain
Rearranging,
Repeating the above procedure, we get
The proof of Lemma 2.4 is complete. □
Remark
Wu and Xu [18] proved that \(g_{m}(\beta)\) is decreasing. They found also that \(g_{m+1}(\beta)>\frac{1-\beta}{\beta_{2}}\) for \(m=1,2,\ldots \) . So when \(0<\beta\leqslant\sqrt{2}-1\), there exists a function \(g(\beta)=\frac{2(1-\beta-\frac{1}{g(\beta)})}{\beta_{2}}\) such that \(\lim_{m\rightarrow\infty}g_{m}(\beta)=g(\beta)\).
Lemma 2.5
Assume that \(\sum_{s=t-r}^{t-2}P(s)\geqslant\beta\) holds for some \(\beta>\sqrt{2}-1\) and \(x(t)\) is a function satisfying inequality (1.2) on \([T_{2},T]_{\mathbb{Z}}\) with \(\Delta x(t)\leqslant0\) for \([T_{2}-\delta,T]_{\mathbb{Z}}\), \(T\geqslant T_{2}+(k_{\beta}+1)r-\delta\), \(T_{2}\geqslant t_{0}+r\) and \(k_{\beta}\) is defined by
Then \(x(t)\) is positive on \([T_{2},T]_{\mathbb{Z}}\).
Proof
Suppose, for the sake of contradiction, that \(x(t)\) is positive on \([T_{2},T]\). We consider two cases:
Case 1: \(\beta\geqslant1\). In this case \(k_{\beta}=1\) and \(T\geqslant T_{2}+2r-\delta\). Since \(\Delta x(t)\leqslant0\) on \([T_{2}-\delta ,T]_{\mathbb{Z}}\), we obtain
Summing both sides of (1.2) from \(T_{2}+r-\delta\) to \(T_{2}+2r-\delta-1\) and using the above inequality, we obtain
which is a contradiction.
Case 2: \(\sqrt{2}-1<\beta<1\). If \(k_{\beta}=n^{\ast}+m^{\ast}\), then
From Lemma 2.3, it follows that
On the other hand, by Lemma 2.4 we find
So, when \(t=T_{2}+({n^{\ast}}+1)r-\delta\) in (2.23) and (2.24), it follows that
which contradicts (2.22). If
then Lemma 2.3 implies a contradiction and the proof is complete. □
3 Main results
In this section, we obtain sufficient oscillation conditions for Eq. (1.1) about the distribution of generalized zeros.
Theorem 3.1
Let \((H_{1})\)–\((H_{4})\) and (2.4) establish for some positive integer n with \(r_{1}=\sigma-\tau\). Then the equation (1.1) oscillates and \(d_{\tilde{t}}(x)\leqslant2\sigma +3n(\sigma-\tau)\), where \(\tilde{t}=t_{1}+(2n+1)(\sigma-\tau)\).
Proof
If Eq. (1.1) has a non-oscillatory solution \(x(t)\), and \(-x(t)\) is also the solution of Eq. (1.1), so we only consider the situation of the solution of (1.1) is eventually positive. We assume \(x(t)>0\) on \([T_{0},T]_{\mathbb{Z}}\) for some integer \(T_{0}\geqslant\tilde{t}\) where \(T>T_{0}+2\sigma+3n(\sigma-\tau)\). Since \(z(t)=x(t)+p(t)x(t-\tau)\) for \(t\in[T_{0}+\tau,T]\), \(z(t)>0\) on \([T_{0}+\tau,T]_{\mathbb{Z}}\). From inequality (2.2), we have
Also from (2.2), we obtain \(x(t-\tau-\sigma)\leqslant-\frac {\Delta(a(t-\tau)\Delta z(t-\tau))}{kq(t-\tau)}\), \(t\in[T_{0}+\sigma+\tau,T]_{\mathbb{Z}}\). Then inequality (3.1) can be rewritten as
i.e.
We can conclude from condition \((H_{4})\) and \(\Delta(a(t)\Delta z(t))<0\),
Let
So
Differentiating both sides of (3.3), and because of (3.2), and \(\Delta(a(t)\Delta z(t))<0\), we obtain
From (3.4), we get
Summing up the above form from \(T_{0}\) to \(t-1\), we have
therefore
Let
Then
Adding (3.3) and (3.5) to (3.8), we have
From (3.6), (3.7) and the decreasing of \(y(t)\), we get
Substituting the above inequality into (3.9), we obtain
Set \(r_{1}=\sigma-\tau\), \(T_{1}=T_{0}+\sigma+\tau\) and \(P(t)=\frac {kq(t)}{1+H(t+1)}\sum_{s=T_{0}}^{t-1}\frac{1}{a(s-\sigma)}\), we conclude
What is more, (2.1) holds and \(y(t)\) is decreasing. Then we can conclude from Lemma 2.2 that \(y(t)\) cannot be positive on \([T_{1},T]_{\mathbb{Z}}\) when \(r_{1}=\sigma-\tau\), where \(T>T_{1}+3n(\sigma-\tau)\). This is a contradiction with (3.7). The proof is completed. □
Assume the following condition holds:
- (\(H_{5}\)):
-
\(\sum_{s=t-r}^{t-1}\frac{q(s)}{1+H(s+1)}\sum_{v=T_{0}}^{s-1}\frac{1}{a(v-\sigma)}\geqslant\beta\), \(t\geqslant t_{2}\) for some \(t_{2}\geqslant t_{0}+2\sigma-\tau\).
Then we can obtain some further conclusions by means of Theorem 3.1.
Corollary 3.1
Suppose conditions \((H_{1})\)–\((H_{5})\) hold and a sequence \(\{\beta _{n}\}\) is defined by
If there is some positive constant \(n_{0}\in\mathbb{N}\) such that \(1\leqslant\beta< r\), then Eq. (1.1) is oscillatory and \(d_{\tilde{t}}\leqslant2\sigma+3n_{0}(\sigma-\tau)\), where \(\tilde{t}=t_{1}+(2n_{0}+1)(\sigma-\tau)\).
Proof
According to condition \((H_{5})\), we have \(\sum_{s=t-r}^{t-1}F_{0}(s)\geqslant\beta\). In addition, from the iterative sequence \(\{F_{n}(t)\}\), we get
In the same way, continuing the calculation n times, we obtain \(\sum_{s=t-r}^{t-1}F_{n}(s)\geqslant\beta_{n}\) for \(n=2,3,\ldots \) . Therefore, by mathematical induction, we have
Let \(n=n_{0}\). According to Theorem 3.1, the proof is completed. □
Theorem 3.2
Let \((H_{1})\)–\((H_{5})\) hold. Then Eq. (1.1) oscillates and \(d_{t_{2}}(x)\leqslant2\sigma+k_{\beta}(\sigma-\tau)\), where \(k_{\beta}\) is defined by (2.21).
Proof
As usual, we assume (1.1) has a solution \(x(t)>0\) on \([T_{0},T]_{\mathbb{Z}}\) where \(T>T_{0}+2\sigma+k_{\beta}(\sigma -\tau)\) and \(T_{0}\geqslant t_{2}\). Proceed as in the proof of Theorem 3.1, when \(T_{1}=T_{0}+2\sigma\). It follows that
where
Also from (3.10) we obtain
Since \((H_{5})\) holds, we conclude from Lemma 2.5 with \(\delta =\sigma-\tau\) that \(y(t)\) cannot be positive on \([T_{1},T]_{\mathbb{Z}}\), where \(T>T_{1}+k_{\beta}(\sigma-\tau)\). This contradiction completes the proof. □
4 Example
In this section, we will present an example to illustrate main results.
Example 4.1
We consider the delay difference equation
Compared with (1.1), we denote \(a(t)=1\) where \(c>t-1\), \(p(t)=1\), \(q(t)=\frac{7}{48(t-2)}\), \(\tau=1\), \(\sigma=3\), \(r=\sigma-\tau=2\), \(f(u)=u\) for \(u\neq0\). It is easy to verify that conditions \((H_{1})\) and \((H_{2})\). Since \(f(u)=u\), we get \(\frac{f(u)}{u}=1\). \((H_{3})\) holds. Now take \(H(t)\equiv1\) which satisfies \((H_{4})\). According to (2.1), we obtain \(F_{0}(t)=\frac{kq(t)}{1+H(t+1)}\sum_{s=T_{0}}^{t-1}\frac {1}{a(s-\sigma)}=\frac{7}{2\times48(t-2)} \sum_{s=2}^{t-1}1=\frac{7}{24}\), \(t\geqslant1\) and \(\frac {1}{1-F_{0}(\zeta)}=\frac{24}{17}\), so
Therefore, \(\sum_{s=t-2}^{t-1}F_{1}(s)\,ds\geqslant1\) for all \(t\geqslant6\). Here, all conditions of Theorem 3.1 are satisfied with \(n=1\), then we derive that (4.1) shows oscillatory and \(d_{\tilde {t}}(x)\leqslant2\sigma+3n(\sigma-\tau)=12\), where \(\tilde{t}=t_{1}+(2n+1)(\sigma-\tau)=t_{1}+6\) and \(t_{1}\geqslant t_{0}+\sigma=4\).
5 Conclusion
In this paper, two theorems on the distribution of oscillation zeros for second-order nonlinear neutral delay difference equations are obtained by means of inequality techniques, specific function sequences and non-increasing solutions for corresponding first-order difference inequality. Comparing with the corresponding differential equation, it is more complex to deal with the lower bound of summation. Function \(\frac{1}{a(t)}\prod_{s=1}^{t-1} (1+a(s))\) is invariant after derivation in difference equation, which is equivalent to \(e^{x}\) in differential equation. That is the difficulty we address and the innovation of this paper. We study a second-order equation under the canonical form, and it is also of great significance for the study of non-canonical forms. Moreover, this paper can be extended to the dynamic equation on time scale.
References
Li, H., Han, Z.L., Sun, S.R.: The distribution of zeros of oscillatory solutions for second order nonlinear neutral delay differential equations. Appl. Math. Lett. 63, 14–20 (2017)
Erbe, L.H., Kong, Q.K., Zhang, B.G.: Oscillation Theory for Functional Differential Equations. Dekker, New York (1995)
Stavroulakis, I.P.: Oscillation criteria for first order delay difference equations. Mediterr. J. Math. 1, 231–240 (2014)
Chatzarakis, G.E., Stavroulakis, I.P.: Oscillation of first order linear delay difference equations. J. Math. Anal. Appl. 3, 1–11 (2006)
Chatzarakis, G.E., Stavroulakis, I.P.: Oscillation criteria of first order linear difference equations with delay arguments. Nonlinear Anal. 68, 994–1005 (2008)
Stavroulakis, I.P.: Oscillation criteria for delay and difference equations with non-monotone arguments. Appl. Math. Comput. 226, 661–672 (2014)
Chatzarakis, G.E., Leonid, S.: Oscillation criteria for difference equations with non-monotone arguments. Adv. Differ. Equ. 2017, 62 (2017)
Chatzarakis, G.E., Philos, C.G., Stavroulakis, I.P.: An oscillation criterion for linear difference equations with general delay argument. Port. Math. 66, 513–533 (2009)
Zhang, B.G., Lian, F.: The distribution of generalized zeros of solutions of delay differential equations on time scales. J. Differ. Equ. Appl. 10(8), 759–771 (2004)
Wu, H.: The distribution ofzZeros of solutions of advanced dynamic equations on time scales. Bull. Malays. Math. Sci. Soc. 41, 63–79 (2018)
Saker, S.H., Mahmoud, R.R.: Distribution of zeros of sublinear dynamic equations with a damping term on time scales. Hacet. J. Math. Stat. 45(2), 455–471 (2016)
Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Comparison theorems for oscillation of second-order neutral dynamic equations. Mediterr. J. Math. 11, 1115–1127 (2014)
Li, T., Agarwal, R.P., Bohner, M.: Some oscillation results for second-order neutral dynamic equations. Hacet. J. Math. Stat. 41, 715–721 (2012)
Li, T., Saker, S.H.: A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales. Commun. Nonlinear Sci. Numer. Simul. 19, 4185–4188 (2014)
Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: New oscillation results for second-order neutral delay dynamic equations. Adv. Differ. Equ. 2012, 227 (2012)
Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators. Bull. Malays. Math. Sci. Soc. 38, 761–778 (2015)
Bohner, M., Li, T.: Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient. Appl. Math. Lett. 37, 72–76 (2014)
Wu, H.W., Xu, Y.T.: The distribution of zeros of solutions of neutral differential equations. Appl. Math. Comput. 156, 665–677 (2004)
Baker, A.F., El-Morshedy, H.A.: The distribution of zeros of all solutions of first order neutral differential equations. Appl. Math. Comput. 259, 777–789 (2015)
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
Funding
This research is supported by the Natural Science Foundation of China (61703180, 61803176), and supported by Shandong Provincial Natural Science Foundation (ZR2017MA043).
Author information
Authors and Affiliations
Contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’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 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.
About this article
Cite this article
Feng, L., Han, Z. The distribution of generalized zeros of oscillatory solutions for second-order nonlinear neutral delay difference equations. Adv Differ Equ 2019, 282 (2019). https://doi.org/10.1186/s13662-019-2220-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-019-2220-3