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Strong singularities of attractive and repulsive type to 2n-order neutral differential equation
Journal of Inequalities and Applications volume 2019, Article number: 259 (2019)
Abstract
This paper is devoted to the existence of a positive periodic solution for a kind of 2n-order neutral differential equation with a singularity, where nonlinear term \(g(t,x)\) has strong singularities of attractive and repulsive type at the origin. Our proof is based on coincidence degree theory.
1 Introduction
In this paper, we consider the following 2n-order p-Laplacian neutral differential equation with a singularity:
where \(\phi _{p}:\mathbb{R}\rightarrow \mathbb{R}\) is given by \(\phi _{p}(s)= \vert s \vert ^{p-2}s\), and \(p>1\) is a constant, c, τ are constants and \(\vert c \vert \neq 1\), \(\tau \in [0,T)\), \(\sigma \in C^{1}( \mathbb{R},\mathbb{R})\) is a T-periodic function, \(f: \mathbb{R} \times \mathbb{R}\to \mathbb{R}\) is a continuous T-periodic function about t and \(f(t,0)=0\), \(e\in C(\mathbb{R},\mathbb{R})\) is a T-periodic function, n is a positive integer, \(g:\mathbb{R}\times (0,+\infty )\to \mathbb{R}\) is a \(L^{2}\)-Carathéodory function, and \(g(t,\cdot )=g(t+T,\cdot )\). It is said that equation (1.1) is singularity of attractive type (resp. repulsive type) if \(g(t,x) \rightarrow +\infty \) (resp. \(g(t,x)\rightarrow -\infty \)) as \(x\rightarrow 0^{+}\) for \(t\in \mathbb{R}\).
Zhang [21] in 1995 first introduced the property of neutral operator \((Ax)(t):=x(t)-cx(t-\tau )\) and discussed a kind of neutral differential equation
The author has given some properties of the neutral operator A, i.e., if \(\vert c \vert \neq 1\), then A has continuous inverse on \(C_{T}:=\{x\mid x \in C(\mathbb{R},\mathbb{R}), x(t+T)\equiv x(t), \forall t\in \mathbb{R}\}\),
-
(i)
\(\Vert A^{-1}x \Vert \leq \frac{ \Vert x \Vert }{ \vert 1- \vert c \vert \vert }\), \(\forall x\in C_{T}\), here \(\Vert x \Vert :=\max_{x\in \mathbb{R}} \vert x(t) \vert \);
-
(ii)
\(\int ^{T}_{0} \vert (A^{-1}x)(t) \vert \,dt\leq \frac{1}{ \vert 1- \vert c \vert \vert }\int ^{T}_{0} \vert x(t) \vert \,dt\), \(\forall x\in C_{T}\).
Afterwards, using the above properties of the neutral operator A, a priori estimation and Leray–Schauder degree theory, Zhang proved that equation (1.2) has at least one periodic solution. Zhu and Lu [23] in 2007 discussed the existence of periodic solution for the following p-Laplacian neutral differential equation:
Since \((\phi _{p}(x'(t)))'\) is a nonlinear term (i.e., quasilinear), coincidence degree theory [5] does not apply directly. In order to get around this difficulty, Zhu and Lu translated the p-Laplacian neutral differential equation into a two-dimensional system
where \(\frac{1}{p}+\frac{1}{q}=1\), for which coincidence degree theory can be applied. Based on the works of Zhang and Lu, the Krasnoselskii fixed point theorem [1,2,3, 6], topological degree theory [4, 14, 15, 20], and the fixed point in a cone [12, 13, 17, 18], fixed point theorem of Leray–Schauder type [10] have been employed to discuss the existence of a periodic solution of neutral differential equations.
Nowadays, the existence of periodic solutions for neutral differential equations with singularity has been researched (see [7,8,9, 19]). Among these, a good deal of work has been performed on the existence of a positive periodic solution of fourth-order neutral Liénard equation with a singularity of repulsive type. Kong and Lu [7] in 2017 studied the following singular Liénard equation:
where c is a constant with \(\vert c \vert <1\), \(g(t,x(t-\delta (t)))=g_{0}(x(t))+g _{1}(t,x(t-\delta (t)))\), \(g_{0}\in C((0,+\infty ),\mathbb{R})\) has a strong singularity of repulsive type at \(x=0\), and \(\int ^{T}_{0}e(t)\,dt=0\). By applying coincidence degree theory, they proved that equation (1.3) has at least one positive T-periodic solution.
Inspired by the above paper [7], in this paper, we further consider the existence of a positive T-periodic solution for equation (1.1) with strong singularities of attractive and repulsive type. Applying coincidence degree theory, we obtain the following conclusions.
Theorem 1.1
Assume that the following conditions hold:
- \((H_{1})\) :
-
There exists a positive constant N such that
$$ \bigl\vert f(t,u) \bigr\vert \leq N, \quad \textit{for } (t,u)\in [0,T]\times \mathbb{R}. $$ - \((H_{2})\) :
-
There exist two positive constants \(D_{1}\), \(D_{2}\) with \(D_{1}< D_{2}\) such that \(g(t,x)-e(t)<-N\) for all \((t,x) \in [0,T] \times (0,D_{1})\), and \(g(t,x)-e(t)>N\) for all \((t,x) \in [0,T]\times (D_{2},+\infty )\).
- \((H_{3})\) :
-
There exist positive constants a, b, p and \(1\leq p<+ \infty \) such that
$$ g(t,x)\leq ax^{p-1}+b, \quad \textit{for all } (t,x)\in [0,T]\times (0,+ \infty ). $$ - \((H_{4})\) :
-
\(g(t,x)=g_{0}(x)+g_{1}(t,x)\), where \(g_{0}\in C((0,\infty ); \mathbb{R}) \) and \(g_{1}:[0,T]\times [0,\infty )\rightarrow \mathbb{R}\) is an \(L^{2}\)-Carathéodory function.
- \((H_{5})\) :
-
(Strong singularity of repulsive type)
$$ \lim_{x\to 0^{+}} g_{0}(x)=- \infty , \quad \textit{and} \quad \lim_{x\to 0^{+}} \int ^{1}_{x}g_{0}(s)\,ds=+\infty . $$
Then (1.1) has at least one positive T-periodic solution if
where \(\pi _{p}=2\int ^{(p-1)/p}_{0} \frac{ds}{(1-\frac{s^{p}}{p-1})^{1/p}}=\frac{2\pi (p-1)^{1/p}}{p \sin (\pi /p)}\), \(\sigma ':=\max_{t\in [0,T]} \vert \sigma (t) \vert \).
Remark 1.2
It is worth mentioning that the friction term \(f(x)x'(t)\) in equation (1.3) satisfies \(\int ^{T}_{0}f(x(t))x'(t)\,dt=0\), which is crucial to estimating a priori bounds of a positive T-periodic solution for equation (1.3). However, in this paper, the friction term \(f(t,x')\) may not satisfy \(\int ^{T}_{0}f(t,x'(t))\,dt=0\). For example, let
Obviously, \(\int ^{T}_{0}(\sin ^{2}(2t)+5)\cos x'(t)\,dt\neq 0\). This implies that our methods to estimate a priori bounds of a positive T-periodic solution for equation (1.1) are more difficult than equation (1.3).
Remark 1.3
From [7], the condition composed on \(e(t)\) is \(\int ^{T}_{0}e(t)\,dt=0\). However, the paper is unnecessary. For example, let \(e(t)=e^{\sin 4\pi t}\). Obviously, \(\int ^{T}_{0} \frac{1}{4}e^{\sin 2\pi t}\neq 0\). Moreover, coefficient c of neutral operator A satisfies \(\vert c \vert <1\) in [7]; in this paper, coefficient c satisfies \(\vert c \vert <1\) and \(\vert c \vert >1\). At last, the singular term \(g_{0}\) of equation (1.3) has not a deviating argument (i.e., \(\sigma \equiv 0\)). The singular term \(g_{0}\) of this paper satisfies time-dependent deviating argument (see condition \((H_{4})\)). It is easy to verify that the work on estimating lower bounds of a positive periodic solution for equation (1.1) is more complex than equation (1.3). Therefore, our result can be more general.
Remark 1.4
If equation (1.1) satisfies singularity of attractive type, i.e., \(\lim_{x\to 0^{+}}g_{0}(x)=+\infty \) and \(\lim_{x\to 0^{+}}\int ^{1}_{x}g_{0}(s)\,ds=-\infty \). Obviously, attractive condition and \((H_{2})\), \((H_{3})\), \((H_{5})\) are contradictions. Therefore, the above method and conditions are no longer applicable to prove the existence of a positive periodic solution for equation (1.1) with singularity of attractive type. Next, we have to find another way and conditions to get over these problems.
Theorem 1.5
Assume that conditions \((H_{1})\) and \((H_{4})\) hold. Suppose the following conditions are satisfied:
- \((H_{6})\) :
-
There exist two positive constants \(D_{3}\), \(D_{4}\) with \(D_{3}< D_{4}\) such that \(g(t,x)-e(t)>N\) for all \((t,x) \in [0,T] \times (0,D_{3})\), and \(g(t,x)-e(t)<-N\) for all \((t,x) \in [0,T] \times (D_{4},+\infty )\).
- \((H_{7})\) :
-
There exist positive constants \(a'\), \(b'\) such that
$$ -g(t,x)\leq a'x^{p-1}+b', \quad \textit{for all } (t,x)\in [0,T]\times (0,+\infty ). $$ - \((H_{8})\) :
-
(Strong singularity of attractive type)
$$ \lim_{x\to 0^{+}}g_{0}(x)=+ \infty , \quad \textit{and} \quad \lim_{x\to 0^{+}} \int ^{1}_{x}g_{0}(s)\,ds=-\infty . $$
Then (1.1) has at least one positive T-periodic solution if
2 Preparation
We first recall the coincidence degree theory.
Lemma 2.1
(Gaines and Mawhin [5])
Suppose that X and Y are two Banach spaces, and \(L:D(L)\subset X\rightarrow Y\) is a Fredholm operator with index zero. Let \(\varOmega \subset X\) be an open bounded set and \(N:\overline{\varOmega }\rightarrow Y \) be L-compact on Ω̅. Assume that the following conditions hold:
-
(1)
\(Lx\neq \lambda Nx\), \(\forall x\in \partial \varOmega \cap D(L)\), \(\lambda \in (0,1)\);
-
(2)
\(Nx\notin \operatorname{Im} L\), \(\forall x\in \partial \varOmega \cap \operatorname{Ker} L\);
-
(3)
\(\deg \{JQN,\varOmega \cap \operatorname{Ker} L,0\}\neq 0\), where \(J:\operatorname{Im} Q\rightarrow \operatorname{Ker} L\) is an isomorphism.
Then the equation \(Lx=Nx\) has a solution in \(\overline{\varOmega }\cap D(L)\).
Lemma 2.2
(see [11])
If \(\vert c \vert \neq 1\), then \((Ax)(t):=x(t)-cx(t- \tau )\) has continuous bounded inverse on \(C_{T}:=\{x\in C(\mathbb{R}, \mathbb{R})\mid x(t+T)-x(t)\equiv 0\}\) and
Lemma 2.3
(see [22])
If \(\nu \in C^{1}(\mathbb{R}, \mathbb{R})\) and \(\nu (0)=\nu (T)=0\), then
Similar to Zhu and Lu [23], we rewrite (1.1) in the form:
Let
with the norm \(\Vert x \Vert :=\max \{ \Vert x_{1} \Vert , \Vert x_{2} \Vert \}\);
with the norm \(\Vert x \Vert _{\infty }:=\max \{ \Vert x \Vert , \Vert x' \Vert \}\). Clearly, X and Y are both Banach spaces. Meanwhile, define
where \(D(L)=\{x=(x_{1},x_{2})^{\top }\in C^{n}(\mathbb{R},\mathbb{R} ^{2}): x(t+T)-x(t)\equiv 0, t \in \mathbb{R}\}\). Define a nonlinear operator \(N: X\rightarrow Y\) as follows:
Then (2.1) can be converted to the abstract equation \(Lx=Nx\).
From the definition of L, one can easily see that
So L is a Fredholm operator with index zero. Let \(P:X\rightarrow \operatorname{Ker} L\) and \(Q:Y\rightarrow \operatorname{Im} Q\subset \mathbb{R}^{2}\) be defined by
then \(\operatorname{Im} P=\operatorname{Ker} L\), \(\operatorname{Ker} Q=\operatorname{Im} L\). Let K denote the inverse of \(L| _{\operatorname{Ker} p\cap D(L)}\). It is easy to see that \(\operatorname{Ker} L=\operatorname{Im}Q=\mathbb{R}^{n}\) and
where
3 Proofs of Theorems 1.1 and 1.5
Proof of Theorem 1.1
Consider the following operator equation:
where L and N are defined by equations (2.2) and (2.3). Set
If \(x(t)=(x_{1}(t),x_{2}(t))^{\top }\in \varOmega _{1}\), then
since \((Ax_{1}^{(n)})(t)=(Ax_{1})^{(n)}(t)\). Substituting \(x_{2}(t)=\frac{1}{ \lambda ^{p-1}}(\phi _{p}(Ax_{1})^{(n)}(t))\) into the second equation of equation (3.1), we get
Integrating both sides of equation (3.2) over \([0,T]\), we have
since \(\int ^{T}_{0}(\phi _{p}(Ax_{1})''(t))''=0\). From equation (3.3) and condition \((H_{1})\), we deduce
Then, by condition \((H_{2})\), we know that there are two points \(\xi , \eta \in (0,T)\) such that
Then, from (3.4), we have
Multiplying both sides of equation (3.2) by \((Ax_{1})(t)\) and integrating over the interval \([0,T]\), we get
Substituting \(\int ^{T}_{0}\phi _{p}((Ax_{1})^{(n)}(t))^{(n)}(Ax_{1})(t)\,dt=(-1)^{n} \int ^{T}_{0} \vert (Ax_{1})^{(n)}(t) \vert ^{p}\,dt\) into equation (3.6), we see that
Furthermore, we obtain
Therefore, from condition \((H_{1})\), it is clear that
where \(\Vert e \Vert :=\max_{t\in [0,T]} \vert e(t) \vert \). From conditions \((H_{1})\), \((H_{2})\) and equation (3.3), we obtain
from \(\frac{a(1+ \vert c \vert )T}{(1-\sigma ') \vert 1- \vert c \vert \vert ^{p}} (\frac{T}{\pi _{p}} )^{2p-1}>0\), we obtain \(\sigma '<1\). Substituting equations (3.5) and (3.8) into (3.7), and we have
Using the Hölder inequality, we deduce
where \(P_{1}:=(1+ \vert c \vert )(N+b+ \Vert e \Vert )T\). Let \(\nu (t)=x_{1}(t+\eta )-x_{1}( \eta )\), here \(x_{1}(\eta )\leq D_{2}\) is as in equation (3.4), and then \(\nu (0)=\nu (T)=0\). By Lemma 2.3 and Minkowski’s inequality [16], we have
since \(\int ^{T}_{0} \vert \nu '(t) \vert ^{p}\,dt=\int ^{T}_{0} \vert x_{1}'(t) \vert ^{p}\,dt\). On the other hand, in view of \(x_{1}(0)=x_{1}(T)\), there exists a point \(t_{1}\in (0,T)\) such that \(x_{1}'(t_{1})=0\). Let \(\nu _{1}(t)=x_{1}'(t+t _{1})\), it is easy to see that \(\nu _{1}(0)=\nu _{1}(T)=0\). Applying inductive method, from \(x_{1}^{(n-2)}(0)=x^{(n-2)}(T)\), there exists a point \(t_{n-1}\in (0,T)\) such that \(x_{1}^{(n-1)}(t_{n-1})=0\). Let \(\nu _{n-1}(t)=x_{1}^{(n-1)}(t+t_{n-1})\), then we get \(\nu _{n-1}(0)= \nu _{n-1}(T)=0\). By Lemma 2.3,
Substituting equations (3.11) into (3.10), we arrive at
Furthermore, substituting equations (3.11) and (3.12) into (3.9), it is easy to verify that
Next, we introduce a classical inequality, there exists a positive constant \(k(p)>0\) (dependent on p),
Then, we consider the following two cases.
Case 1. If \(\frac{D_{2}T^{\frac{1}{p}}}{ (\frac{T}{\pi _{p}} )^{n} (\int ^{T}_{0} \vert x_{1}^{(n)}(t) \vert ^{p}\,dt ) ^{\frac{1}{p}}}>k(p)\), then it is obvious that
From equations (3.5) and (3.11), by using the Hölder inequality, we deduce
Case 2. If \(\frac{D_{2}T^{\frac{1}{p}}}{ (\frac{T}{\pi _{p}} )^{n} (\int ^{T}_{0} \vert x_{1}^{(n)}(t) \vert ^{p}\,dt ) ^{\frac{1}{p}}}< k(p)\), from equations (3.13) and (3.14), we obtain
Since \((Ax_{1})^{(n)}(t)=(Ax_{1}^{(n)})(t)\), from Lemma 2.2 and (3.16), we see that
Since
obviously, there exists a positive constant \(M'_{1}\) such that
From equations (3.5), (3.11), and (3.17), applying the Hölder inequality, we deduce
From equations (3.5), (3.11), and (3.17), we get
since \(x_{1}'(t_{1})=0\). From \(x_{2}^{(n-2)}(0)=x_{2}^{(n-2)}(T)\), there exists a point \(t_{2}^{*}\in (0,T)\) such that \(x_{2}^{(n-1)}(t_{2} ^{*})=0\). From the second equation of (3.1), equations (3.8), (3.18), (3.19), and condition \((H_{1})\), we obtain
Integrating the first equation of (3.1) over \([0,T]\), we have \(\int ^{T}_{0}x_{2}(t)\,dt=\int ^{T}_{0}\phi _{p}((Ax_{1})''(t))\,dt=0\), which implies there is a point \(t_{3}^{*}\in (0,T)\) such that \(x_{2}(t_{3} ^{*})=0\). So, from equations (3.5) and (3.20), applying the Hölder inequality and equation (3.11), it is easy to see that
On the other hand, from equation (3.2) and condition \((H_{4})\), we see that
Let \(\xi \in [0,T]\) be as in equation (3.4) for any \(t\in [\xi ,T]\). Multiplying both sides of equation (3.22) by \(x_{1}'(t-\sigma (t))(1-\sigma '(t))\) and integrating over \([\xi , t]\), we get
Furthermore, by equations (3.16), (3.18), (3.19), and (3.20), we have
where \(g_{M_{1}}=\max_{0\leq x\leq M_{1}} \vert g_{1}(t,x) \vert \in L^{2}(0,T)\) is as in condition \((H_{4})\). According to singular condition \((H_{5})\), we know that there exists a positive constant \(M_{5}\) such that
The case \(t\in [0,\xi ]\) can be treated similarly.
From equations (3.16), (3.18), (3.19), (3.20), and (3.24), we let
where \(0< E_{1}<\min (M_{5}, D_{1})\), \(E_{2}>\max (M_{1}, D_{2}) \), \(E_{3}>M_{2}\), \(E_{4}>M_{4}\), and \(E_{5}>M_{3}\). \(\varOmega _{2}=\{x:x \in \partial \varOmega \cap \operatorname{Ker} L\}\), then \(\forall x\in \partial \varOmega \cap \operatorname{Ker} L\)
If \(QNx=0\), then \(x_{2}(t)=0\), \(x_{1}=E_{2}\) or \(E_{1}\). But if \(x_{1}(t)=E_{2}\), we know
From condition \((H_{2})\), we have \(x_{1}(t)\leq D_{2}\leq E_{2}\), which yields a contradiction. Similarly, if \(x_{1}=E_{1}\), we also have \(QNx\neq 0\), i.e., \(\forall x\in \partial \varOmega \cap \operatorname{Ker} L\), \(x \notin \operatorname{Im} L\), so assumptions (1) and (2) of Lemma 2.1 are both satisfied. Define the isomorphism \(J:\operatorname{Im} Q\rightarrow \operatorname{Ker} L\) as follows:
Let \(H(\mu ,x)=-\mu x+(1-\mu )JQNx\), \((\mu ,x)\in [0,1]\times \varOmega \), then \(\forall (\mu ,x)\in (0,1)\times (\partial \varOmega \cap \operatorname{Ker} L)\),
From condition \((H_{2})\), we get \(x^{\top }H(\mu ,x)\neq 0\), \(\forall (\mu ,x)\in (0,1)\times (\partial \varOmega \cap \operatorname{Ker} L)\). Hence
So assumption (3) of Lemma 2.1 is satisfied. By applying Lemma 2.1, we conclude that equation \(Lx=Nx\) has a solution \(x=(x_{1},x_{2})^{\top }\) on \(\bar{\varOmega }\cap D(L)\), i.e., (2.1) has a T-periodic solution \(x_{1}(t)\). □
Next, we study the existence of a positive T-periodic solution for equation (1.1) with singularity of attractive type.
Proof of Theorem 1.5
We follow the same strategy and notation as in the proof of Theorem 1.1. From equation (3.3) and condition \((H_{6})\), we know that there are points \(\mu , \nu \in (0,T)\) such that
Next, we consider \(\int ^{T}_{0} \vert g(t,x_{1}(t-\sigma (t))) \vert \,dt\). From equation (3.8) and conditions \((H_{1})\), \((H_{7})\), we obtain
The proof left is the same as that of Theorem 1.1. □
Finally, we present an example to illustrate our result.
Example 3.1
Consider the fourth-order neutral Rayleigh equation with singularity of repulsive type:
where \(\kappa \geq 1\) and \(p=4\), Ï„ is a constant and \(0\leq \tau < T\).
It is clear that \(T=\frac{\pi }{2}\), \(\sigma (t)=\frac{1}{8}\sin 4t\), \(\sigma '=\frac{1}{2}<1\), \(f(t,u)=(\sin ^{2}(2t)+5)\cos u\), \(g(t,x)=\frac{1}{4 \pi }(\cos 4t+3)x^{3}-\frac{1}{x^{\kappa }}\), \(a=\frac{1}{5\pi }\), \(\pi _{4}=\frac{2\pi (p-1)^{\frac{1}{p}}}{p\sin (\pi /p)}=\frac{2 \pi (4-1)^{\frac{1}{4}}}{4\cdot \frac{\sqrt{2}}{2}}= \pi \times (\frac{3}{4} )^{\frac{1}{4}}\). Take \(N=6\), \(a=\frac{1}{ \pi }\), \(b=1\). It is obvious that conditions \((H_{1})\)–\((H_{5})\) hold. Now we consider
Therefore, applying Theorem 1.1, we know that equation (3.27) has at least one positive \(\frac{\pi }{2}\)-periodic solution.
Example 3.2
Consider the following fourth-order neutral Rayleigh equation with singularity of attractive type:
where \(\kappa '\geq 1\) and \(p=2\), Ï„ is a constant and \(0\leq \tau < T\).
It is clear that \(T=\pi \), \(\sigma (t)=\frac{1}{4}\cos 2t\), \(\sigma '= \frac{1}{2}<1\), \(f(t,u)=-(\cos ^{2}t+100)\sin u\), \(g(t,x)=-\frac{1}{ \pi }(\sin 2t+5)x+\frac{5}{x^{\kappa '}}\), \(a=\frac{1}{5\pi }\), \(\pi _{2}=\frac{2\pi (p-1)^{\frac{1}{p}}}{p\sin (\pi /p)}=\frac{2\pi (2-1)^{ \frac{1}{2}}}{4\times \frac{1}{2}}= \pi \). Take \(N=6\), \(a=\frac{6}{ \pi }\), \(b=1\). It is easy to verify that conditions \((H_{1})\), \((H_{4})\), \((H _{6})\)–\((H_{8})\) hold. Now we consider
Therefore, applying Theorem 1.5, we know that equation (3.28) has at least one positive π-periodic solution.
4 Conclusions
In this article, we introduce the following existence of a positive T-periodic solution for 2n-order p-Laplacian neutral differential equation with singularities of attractive and repulsive type. Due to the friction term, \(f(t,x')\) may not satisfy \(\int ^{T}_{0}f(t,x'(t))\,dt=0\). This implies that the work on estimating a priori bounds of periodic solutions for equation (1.1) is more difficult than the corresponding work on equation (1.3) in [7]. In this paper, by using coincidence degree theory and conditions \((H_{1})\)–\((H_{5})\), we prove the existence of a positive T-periodic solution for equation (1.1) with singularity of repulsive type; applying conditions \((H_{1})\), \((H_{4})\), \((H _{6})\)–\((H_{8})\), we obtain that equation (1.1) with singularity of attractive type has at least one positive T-periodic solution.
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Acknowledgements
YX, SWY, and RCW are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.
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This work was supported by the National Natural Science Foundation of China (No. 71601072), Education Department of Henan Province project (No. 16B110006), Fundamental Research Funds for the Universities of Henan Province (NSFRF170302).
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YX, SWY, and RCW contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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YX, SWY, and RCW contributed to each part of this study equally and declare that they have no competing interests.
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Xin, Y., Yao, S. & Wang, R. Strong singularities of attractive and repulsive type to 2n-order neutral differential equation. J Inequal Appl 2019, 259 (2019). https://doi.org/10.1186/s13660-019-2210-8
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DOI: https://doi.org/10.1186/s13660-019-2210-8