Abstract
In this paper, we first discuss some properties of the neutral operator with multiple delays and variable coefficients \((Ax)(t):=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t-\delta _{i})\). Afterwards, by using an extension of Mawhin’s continuation theorem, a second order p-Laplacian neutral differential equation
is studied. Some new results on the existence of a periodic solution are obtained. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from those known in the literature.
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1 Introduction
In this paper, we consider a second order p-Laplacian neutral differential equation
where \(\phi _{p}:\mathbb{R}\to \mathbb{R}\) is given by \(\phi _{p}(s)=|s|^{p-2}s\), here \(p>1\) is a constant, \(c_{i}(t)\in C ^{1}(\mathbb{R},\mathbb{R})\) and \(c_{i}(t+T)=c_{i}(t)\) and \(\delta _{i}\) are constants in \([0,T)\) for \(i=1,2,\dots ,n\); f̃: \([0,T]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is an \(L^{2}\)-Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for every \(0< r< s\) there exists \(h_{r,s}\in L^{2}[0,T]\) such that \(|\tilde{f}(t,x(t),x'(t))| \leq h_{r,s}\) for all \(x\in [r,s]\) and a.e. \(t\in [0,T]\).
The study of the properties of the neutral operator \((A_{1}x)(t):=x(t)-cx(t- \delta )\) began with the paper of Zhang [2]. In 2004, Lu and Ge [14] investigated an extension of \(A_{1}\), namely the neutral operator \((A_{2}x)(t):=x(t)-\sum_{i=1}^{n}c_{i}x(t- \delta _{i})\). Afterwards, Du [6] discussed the neutral operator \((A_{3}x)(t):=x(t)-c(t)x(t-\delta )\), here \(c(t)\) is a T-periodic function. And by using Mawhin’s continuation theorem and the properties of \(A_{3}\), they obtained sufficient conditions for the existence of periodic solutions to the following Liénard neutral differential equation:
In recent years, many works have been published on the existence of periodic solutions of second-order neutral differential equations (see [1, 3,4,5, 7, 9, 11,12,13, 16,17,18,19]). In 2007, Zhu and Lu [19] discussed the existence of periodic solutions for a p-Laplacian neutral differential equation
Since \((\phi _{p}(x'(t)))'\) is nonlinear (i.e., quasilinear), Mawhin’s continuation theorem [8] cannot be applied 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 Mawhin’s continuation theorem can be applied. Afterwards, Du [5] discussed the existence of a periodic solution for a p-Laplacian neutral differential equation
by applying Mawhin’s continuation theorem.
However, the existence of a periodic solution for p-Laplacian neutral differential equation (1.1) has not been studied until now. The obvious difficulty lies in the following two respects. First, although \((Ax)(t)=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t-\delta _{i})\) is a natural generalization of the operators \(A_{1}\), \(A_{2}\) and \(A_{3}\), the class of neutral differential equations with A typically possesses a more complicated nonlinearity than neutral differential equations with \(A_{1}\), \(A_{2}\) and \(A_{3}\). Second, we do not get \((Ax)'(t)=(Ax')(t)\), meanwhile a priori bounds of periodic solutions are not easy to estimate.
The remaining part of the paper is organized as follows. In Sect. 2, we analyze qualitative properties of the generalized neutral operator A. In Sect. 3, by employing an extension of Mawhin’s continuation theorem, we state and prove the existence of periodic solutions for Eq. (1.1). In Sect. 4, we investigate the existence of periodic solutions for a p-Laplacian neutral differential equation by applying Theorem 3.2. In comparison to [5] and [19], we avoid translating the equation into a two-dimensional system. In Sect. 5, we discuss the existence of periodic solutions for a p-Laplacian neutral differential equation with singularity by applying Theorem 3.2. In Sect. 6, we give four examples to demonstrate the validity of the methods.
2 Analysis of the generalized neutral operator
Let
Set \(C_{T}:=\{x\in C(\mathbb{R},\mathbb{R}):x(t+T)=x(t),t\in \mathbb{R}\}\), then \((C_{T},\|\cdot \|)\) is a Banach space. Define operators \(A,B:C_{T}\rightarrow C_{T}\), by
Lemma 2.1
If \(\sum_{i=1}^{n}\|c_{i}\|\neq 1\), then operator A has a continuous inverse \(A^{-1}\) on \(C_{T}\), satisfying
-
(1)
$$ \bigl\vert \bigl(A^{-1}x\bigr) (t) \bigr\vert \leq \textstyle\begin{cases} \frac{ \Vert x \Vert }{1-\sum_{i=1}^{n} \Vert c_{i} \Vert }, \quad \textit{for }\sum_{i=1}^{n} \Vert c_{i} \Vert < 1; \\ \frac{\frac{1}{ \Vert c_{k} \Vert } \Vert x \Vert }{1-\frac{1}{ \Vert c_{k} \Vert }-\sum_{i=1,i\neq k}^{n} \Vert \frac{c_{i}}{c_{k}} \Vert }, \quad \textit{for }\sum_{i=1}^{n} \Vert c_{i} \Vert >1; \end{cases} $$
-
(2)
$$ \int ^{T}_{0} \bigl\vert \bigl(A^{-1}x \bigr) (t) \bigr\vert \,dt\leq \textstyle\begin{cases} \frac{1}{1-\sum_{i=1}^{n} \Vert c_{i} \Vert }\int ^{T}_{0} \vert x(t) \vert \,dt, \quad \textit{for }\sum_{i=1}^{n} \Vert c_{i} \Vert < 1; \\ \frac{\frac{1}{ \Vert c_{k} \Vert }}{1-\frac{1}{ \Vert c_{k} \Vert }-\sum_{i=1,i \neq k}^{n} \Vert \frac{c_{i}}{c_{k}} \Vert }\int ^{T}_{0} \vert x(t) \vert \,dt, \quad \textit{for }\sum_{i=1}^{n} \Vert c_{i} \Vert >1. \end{cases} $$
Proof
Case 1:
Therefore, we have
and
Since \(A=I-B\) and \(\|B\|<1\), we get that A has a continuous inverse \(A^{-1}\): \(C_{T}\rightarrow C_{T}\) with
where \(B^{0}=I\). Then
Moreover,
Case 2: \(\sum_{i=1}^{n}\|c_{i}\|>1\).
The operator \((Ax)(t)=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t-\delta _{i})\) can be converted to
Let \(t_{1}=t-\delta _{k}\), it is clear that
Define
Therefore, \((Ex)(t_{1}+\delta _{k})=x(t_{1}+\delta _{k})-\sum_{i=1}^{n}e_{i}(t_{1}+\delta _{k})x(t_{1}-\varepsilon _{i})\) and, from Case 1, we get
Moreover, since \((A^{-1}x)(t)=-\frac{1}{c_{k}(t)}(E^{-1}x)(t)\), we have
Meanwhile, we obtain
□
3 Periodic solutions for equation (1.1)
In order to use an extension of Mawhin’s continuation theorem [10], we recall it firstly.
Let X and Z be Banach spaces with norms \(\|\cdot \|_{X}\) and \(\|\cdot \|_{Z}\), respectively. A continuous operator \(M:X\cap \operatorname{dom} M\to Z\) is said to be quasilinear if
-
(1)
\(\operatorname{Im}M:=M(X\cap \operatorname{dom}M)\) is a closed subset of Z;
-
(2)
\(\ker M:=\{x\in X\cap \operatorname{dom}M: Mx=0\} \) is a subspace of X with \(\dim \ker M<+\infty \).
Let \(X_{1}=\ker M\) and \(X_{2}\) be the complement space of \(X_{1}\) in X, then \(X=X_{1}\oplus X_{2}\). Furthermore, \(Z_{1}\) is a subspace of Z and \(Z_{2}\) is the complement space of \(Z_{1}\) in Z, so \(Z=Z_{1}\oplus Z_{2}\). Suppose that \(P:X\rightarrow X_{1}\) and \(Q: Z\rightarrow Z_{1}\) are two projections and \(\varOmega \subset X\) is an open and bounded set with the origin \(\theta \in \varOmega \).
Let \(N_{\lambda }:\bar{\varOmega }\to Z\), \(\lambda \in [0,1]\) be a continuous operator. Denote \(N_{1}\) by N, and let \(\sum_{\lambda }=\{x\in \bar{ \varOmega }:Mx=N_{\lambda }x\}\). Then \(N_{\lambda }\) is said to be M-compact in Ω̄ if
-
(3)
there is a vector subspace \(Z_{1}\) of Z with \(\dim Z_{1}=\dim X _{1}\) and an operator \(R:\bar{\varOmega }\times X_{2}\) being continuous and compact such that for \(\lambda \in [0,1]\),
$$\begin{aligned}& (I-Q)N_{\lambda }(\bar{\varOmega })\subset \operatorname{Im}M\subset (I-Q)Z, \end{aligned}$$(3.1)$$\begin{aligned}& QN_{\lambda }x=0, \quad \lambda \in (0,1) \Leftrightarrow QNx=0, \end{aligned}$$(3.2)$$\begin{aligned}& R(\cdot ,0) \mbox{ is the zero operator and } R(\cdot ,\lambda )|_{ \sum _{\lambda }}=(I-P)|_{\sum _{\lambda }}, \end{aligned}$$(3.3)and
$$ M\bigl[P+R(\cdot ,\lambda )\bigr]=(I-Q)N_{\lambda }. $$(3.4)Let \(J:Z_{1}\to X_{1}\) be a homeomorphism with \(J(\theta )=\theta \).
Next, we investigate existence of periodic solutions for Eq. (1.1) by applying the extension of Mawhin’s continuation theorem.
Lemma 3.1
([10])
Let X and Z be Banach spaces with norm \(\|\cdot \|_{X}\) and \(\|\cdot \|_{Z}\), respectively, and \(\varOmega \subset X\) be an open and bounded set with \(\theta \in \varOmega \). Suppose that \(M:X\cap \operatorname{dom}M\to Z\) is a quasilinear operator and
is an M-compact mapping. In addition, if
-
(a)
\(Mx\neq N_{\lambda }x\), \(\lambda \in (0,1)\), \(x\in \partial \varOmega \),
-
(b)
\(\deg \{JQN,\varOmega \cap \ker M,0\}\neq 0\),
where \(N=N_{1}\), then the abstract equation \(Mx=Nx\) has at least one solution in Ω̄.
Theorem 3.2
Assume \(\sum_{i=1}^{n}\|c_{i}\|\neq 1\), Ω is an open bounded set in \(C^{1}_{T}\). Suppose the following conditions hold:
-
(i)
For each \(\lambda \in (0,1)\), the equation
$$ \bigl(\phi _{p}(Ax)'(t) \bigr)'=\lambda \tilde{f}\bigl(t,x(t),x'(t)\bigr) $$(3.5)has no solution on ∂Ω.
-
(ii)
The equation
$$ F(a):=\frac{1}{T} \int ^{T}_{0}\tilde{f}(t,a,0)\,dt=0 $$has no solution on \(\partial \varOmega \cap \mathbb{R}\).
-
(iii)
The Brouwer degree
$$ \deg \{F,\varOmega \cap \mathbb{R},0\}\neq 0. $$Then Eq. (1.1) has at least one T-periodic solution on Ω̄.
Proof
In order to use Lemma 3.1 we study the existence of periodic solutions to Eq. (1.1). We set \(X:=\{x\in C[0,T]: x(0)=x(T)\}\) and \(Z:=C[0,T]\),
where \(\operatorname{dom}M:=\{u\in X:\phi _{p}(Au)'\in C^{1}(\mathbb{R},\mathbb{R}) \}\). Then \(\ker M=\mathbb{R}\). In fact,
where \(q>1\) is a constant with \(\frac{1}{p}+\frac{1}{q}=1\) and c, \(c_{1}\), \(c_{2}\) are constants in \(\mathbb{R}\). Since \((Ax)(0)=(Ax)(T)\), then we get \(\ker M=\{x\in X: (Ax)(t)\equiv c_{2} \}\). In addition,
So M is quasilinear. Let
Clearly, \(\operatorname{dim} X_{1}= \operatorname{dim} Z_{1}=1\), and \(X=X_{1}\oplus X_{2}\), \(P:X\to X_{1}\), \(Q:Z\to Z_{1}\), are defined by
For \(\forall \bar{\varOmega }\subset X\), define \(N_{\lambda }:\bar{ \varOmega }\to Z\) by
We claim that \((I-Q)N_{\lambda }(\bar{\varOmega })\subset \operatorname{Im}M=(I-Q)Z\) holds. In fact, for \(x\in \bar{\varOmega }\), we observe that
Therefore, we have \((I-Q)N_{\lambda }(\bar{\varOmega })\subset \operatorname{Im} M\).
Moreover, for any \(x\in Z\), it is obvious that
So, we have \((I-Q)Z\subset \operatorname{Im}M\). On the other hand, \(x\in \operatorname{Im}M\) and \(\int ^{T}_{0}x(t)\,dt=0\), so we have \(x(t)=x(t)- \int ^{T}_{0}x(t)\,dt\). Hence, we get \(x(t)\in (I-Q)Z\). Therefore, \(\operatorname{Im}M=(I-Q)Z\).
From \(QN_{\lambda }x=0\), we get \(\frac{\lambda }{T}\int ^{T}_{0} \tilde{f}(t,x(t),x'(t))\,dt=0\). Since \(\lambda \in (0,1)\), we have \(\frac{1}{T}\int ^{T}_{0}\tilde{f}(t,x(t), x'(t))\,dt=0\). Therefore, \(QNx=0\), and so Eq. (3.4) also holds.
Let \(J:Z_{1}\to X_{1}\), \(J(x)=x\), then \(J(0)=0\). Define \(R:\bar{\varOmega }\times [0,1]\to X_{2}\),
where \(a\in R\) is a constant such that
From Lemma 2.3 of [15], we know that a is uniquely defined by
where \(\tilde{a}(x,\lambda )\) is continuous on \(\bar{\varOmega }\times [0,1]\) and maps bounded sets of \(\bar{\varOmega } \times [0,1]\) into bounded sets of \(\mathbb{R}\).
From Eq. (3.4), one can find that
Now, for any \(x\in \sum_{\lambda }=\{x\in \bar{\varOmega }: Mx=N_{\lambda }x\}=\{x\in \bar{\varOmega }: (\phi _{p}(Ax)'(t))'=\lambda \tilde{f}(t,x(t),x'(t)) \} \), we have \(\int ^{T}_{0}\tilde{f}(t,x(t),x'(t))\,dt=0\), which, together with Eq. (3.7), gives
Taking \(a=\phi _{p}(Ax)'(0)\), we then have
where a is unique, and we see that
Thus, we derive
which yields the second part of Eq. (3.3). Meanwhile, if \(\lambda =0\), then
where \(c_{3}\in \mathbb{R}\) is a constant, so by the continuity of \(\tilde{a}(x,\lambda )\) with respect to \((x,\lambda )\), \(a=\tilde{a}(x,0)= \phi _{p}(Ac)'(0)=0\). Hence,
which yields the first part of Eq. (3.3). Furthermore, we consider
and, in fact,
Integrating both sides of (3.9) over \([0,s]\), we have
Therefore, we arrive at
where \(a:=\phi _{p}(A(P+R))'(0)\). Then, we get
Integrating both sides of (3.10) over \([0,t]\), we derive
i.e.,
Since \(R(x,\lambda )(0)=0\), \(P(t)=P(0)\), we obtain
Hence, we have that \(N_{\lambda }\) is M-compact on Ω̄. Obviously, the equation
can be converted to
where M and \(N_{\lambda }\) are defined by Eqs. (3.6) and (3.7), respectively. As proved above,
is an M-compact mapping. From assumption (i), one finds
and assumptions (ii) and (iii) imply that \(\deg \{JQN,\varOmega \cap \ker M,\theta \}\) is valid and
So by applications of Lemma 3.1, we see that Eq. (1.1) has a T-periodic solution. □
4 Application of Theorem 3.2: p-Laplacian equation
As an application, we consider the following p-Laplacian neutral Liénard equation:
where \(\phi _{p}:\mathbb{R}\to \mathbb{R}\) is given by \(\phi _{p}(s)=|s|^{p-2}s\), here \(p>1\) is a constant, g is a continuous function defined on \(\mathbb{R}^{2}\) and periodic in t with \(g(t,\cdot )=g(t+T,\cdot )\), \(f\in C(\mathbb{R},\mathbb{R})\), e is a continuous periodic function defined on \(\mathbb{R}\) with period T and \(\int ^{T}_{0}e(t)\,\,dt=0\). Next, by applications of Theorem 3.2, we will investigate the existence of periodic solution for Eq. (4.1) in the case that \(\sum_{i=1}^{n}\|c_{i}\|\neq 1\).
Define
Theorem 4.1
Suppose \(\sum_{i=1}^{n}\|c_{i}\|\neq 1\) holds. Assume the following conditions hold:
- \((H_{1})\) :
-
There exists a constant \(D>0\) such that
$$ xg(t,x)>0, \quad \forall (t,x)\in [0,T]\times \mathbb{R},\textit{ with } \vert x \vert >D. $$ - \((H_{2})\) :
-
There exist positive constants m, ñ such that
$$ \bigl\vert f(x) \bigr\vert \leq m \vert x \vert ^{p-2}+ \tilde{n}, \quad x\in \mathbb{R}. $$ - \((H_{3})\) :
-
There exist positive constants a, b, B such that
$$ \bigl\vert g(t,x) \bigr\vert \leq a \vert x \vert ^{p-1}+b, \quad \textit{for } \vert x \vert >B \textit{ and }t\in [0,T]. $$Then Eq. (4.1) has at least one T-periodic solution, if
$$ \sigma T^{\frac{1}{q}} \biggl(\frac{m\sum_{i=1}^{n} \Vert c_{i} \Vert }{2^{p-1}}+\frac{aT (1+\sum_{i=1}^{n} \Vert c_{i} \Vert )}{2^{p}} \biggr)^{ \frac{1}{p}} +\frac{\sigma T\sum_{i=1}^{n} \Vert c'_{i} \Vert }{2}< 1. $$
Proof
Consider the homotopic equation
Firstly, we claim that the set of all T-periodic solutions of Eq. (4.2) is bounded. Let \(x(t)\in C_{T}\) be an arbitrary T-periodic solution of Eq. (4.2). Integrating both sides of (4.2) over \([0,T]\), we have
From the mean-value theorem for integrals, there is a constant \(\xi \in [0,T]\) such that
In view of condition \((H_{1})\), we obtain
Then, we have
Multiplying both sides of Eq. (4.2) by \((Ax)(t)\) and integrating over the interval \([0,T]\), we get
Substituting \(\int ^{T}_{0}(\phi _{p}(Ax)'(t))'(Ax)(t)\,dt=-\int ^{T}_{0}|(Ax)'(t)|^{p}\,dt\), \(\int ^{T}_{0}f(x(t))x'(t)x(t)\,dt=0\) into Eq. (4.5), we see that
Thus, we have
Define
Using conditions \((H_{2})\) and \((H_{3})\), we arrive at
where \(\|e\|:=\max_{t\in [0,T]}|e(t)|\), \(\|g_{B}\|:=\max_{|x(t)|\leq B}|g(t,x(t))|\) and \(N_{1}:= (1+\sum_{i=1}^{n}\|c_{i}\| )T(\|g_{B}\|+b+\|e\|)\). Substituting Eq. (4.4) into Eq. (4.6), we get
Next, we introduce a classical inequality: there exists a \(\kappa (p)>0\), which is depends on p only, such that
Then, we consider the following two cases:
Case 1: If \(\frac{D}{\frac{1}{2}\int ^{T}_{0}|x'(t)|\,dt}> \kappa (p)\), we deduce
From Eq. (4.4), it is clear that
Case 2: If \(\frac{D}{\frac{1}{2}\int ^{T}_{0}|x'(t)|\,dt}< \kappa (p)\), then
Since \((Ax)(t)=x(t)-\sum_{i=1}^{n}x(t-\delta _{i})\), we have
and
By applying Lemma 2.1 and Hölder inequality, we get
where \(\|c'_{i}\|:=\max_{t\in [0,T]}|c'_{i}(t)|\), for \(i=1,2,\dots ,n\). Substituting Eq. (4.10) into Eq. (4.11), since \((\tilde{a}+\tilde{b})^{k}\leq \tilde{a}^{k}+\tilde{b}^{k}\), \(0< k \leq 1\), we have
Since \(\sigma T^{\frac{1}{q}} (\frac{m\sum_{i=1}^{n}\|c _{i}\|}{2^{p-1}}+ \frac{aT (1+\sum_{i=1}^{n}\|c_{i}\| )}{2^{p}} ) ^{\frac{1}{p}} +\frac{\sigma T\sum_{i=1}^{n}\|c'_{i}\|}{2}<1\), it is easily see that there exists a constant \(M_{1}'>0\) (independent of λ) such that
From Eq. (4.4), we obtain
Let \(M_{1}=\sqrt{M_{11}^{2}+M_{12}^{2}}+1\). As \((Ax)(0)=(Ax)(T)\), there exists a point \(t_{0}\in (0,T)\) such that \((Ax)'(t_{0})=0\). Moreover, since \(\phi _{p}(0)=0\), due to Eq. (4.14), it is obvious that
where \(\|f_{M_{1}}\|:=\max_{|x(t)|\leq M_{1}}|f(x(t))|\) and \(\|g_{M_{1}}\|:=\max_{|x(t)|\leq M_{1}}|g(t,x(t))|\). Next we claim that there exists a positive constant \(M_{2}^{*}>M_{2}'+1\), such that, for all \(t\in \mathbb{R}\),
In fact, if \((Ax)'\) is not bounded, there exists a positive constant \(M''_{2}\) such that \(\|(Ax)'\|>M''_{2}\) for some \((Ax)'\in \mathbb{R}\). Therefore, we have \(\|\phi _{p}(Ax)'\|=\|(Ax)^{\prime \,p-1}\|\geq M''_{2}\), which is a contradiction. Hence, Eq. (4.15) holds. From Lemma 2.1 and Eq. (4.15), we have
Setting \(M=\sqrt{M_{1}^{2}+M_{2}^{2}}+1\), we get
and we know that Eq. (4.1) has no solution on ∂Ω as \(\lambda \in (0,1)\). When \(x(t)\in \partial \varOmega \cap \mathbb{R}\), \(x(t)=M+1\) or \(x(t)=-M-1\), and from Eq. (4.4) we know that \(M+1>D\). Thus, from condition \((H_{1})\), we see that
since \(\int ^{T}_{0}e(t)\,dt=0\). So condition (ii) of Theorem 3.2 is also satisfied. Set
Obviously, from condition \((H_{1})\), we can get \(xH(x,\mu )>0\) and thus \(H(x,\mu )\) is a homotopic transformation, as well as
So condition (iii) of Theorem 3.2 is satisfied. In view of Theorem 3.2, there exists at least one T-periodic solution. □
5 Application of Theorem 3.2: p-Laplacian equation with singularity
In this section, we consider Eq. (4.1) with a singularity. Here \(g(t,x(t))=g_{0}(x)+g_{1}(t,x(t))\), \(g_{0}\in C((0,\infty );R)\) and \(g_{1}\) is an \(L^{2}\)-Carathéodory function, and \(g_{0}\) has a singularity at \(x=0\), i.e.,
Next, we consider the existence of periodic solutions for Eq. (4.1) with singularity by applying Theorem 3.2.
Theorem 5.1
Suppose \(\sum_{i=1}^{n}\|c_{i}\|\neq 1\) and condition \((H_{2})\) hold. Assume that the following conditions hold:
- \((H_{4})\) :
-
There exist positive constants \(0< D_{1}< D_{2}\) such that x is a positive continuous T-periodic function satisfying \(\int ^{T}_{0}g(t,x(t))\,dt<0\), for some \(x\in (0,D_{1})\) and \(\int ^{T} _{0}g(t,x(t))\,dt>0\), for some \(x\in (D_{2},\infty )\).
- \((H_{5})\) :
-
There exist positive constants α and β such that
$$ g(t,x)\leq \alpha x^{p-1}+\beta , \quad \textit{for }t\in [0,T], \textit{ and }x>0. $$(5.2)
Then Eq. (4.1) has at least one T-periodic solution if
Proof
Consider the homotopic equation
We follow the same strategy and notation as in the proof of Theorem 4.1. From condition \((H_{4})\), we know that there exists a constant \(D_{2}>0\) such that
From Eq. (4.5), we have
From Eq. (4.3) and condition \((H_{5})\), we get
where \(g^{+}:=\max \{g(t,x), 0\}\). Using condition \((H_{2})\) and Eq. (5.5), we derive
Following the same strategy and notation as in the proof of Theorem 4.1, we can obtain, since \(\sigma T^{\frac{1}{q}} (\frac{m \sum_{i=1}^{n}\|c_{i}\|}{2^{p-1}}+\frac{\alpha T (1+ \sum_{i=1}^{n}\|c_{i}\| )}{2^{p-1}} )^{\frac{1}{p}}+\frac{ \sigma T\sum_{i=1}^{n}\|c'_{i}\|}{2}<1\), that there exists a constant \(M_{3}'>0\) (independent of λ) such that
From Eq. (5.7), we get
From Eqs. (4.15), (4.16) and (5.8), we get that there exists a constant \(M_{3}^{*}\), such that, for all \(t\in \mathbb{R}\),
On the other hand, multiplying both sides of (5.3) by \(x'(t)\), we get
since \(g(t,x(t))=g_{0}(x)+g_{1}(t,x(t))\). Letting \(\tau \in [0,T]\), for any \(\tau \leq t\leq T\), we integrate Eq. (5.10) on \([\tau ,t]\) and get
Furthermore,
From Eq. (5.3), we have
where \(\|f_{M_{3}}\|:=\max_{|x(t)|\leq M_{3}}|f(x(t))|\). From Eqs. (5.7) and (5.8), we obtain
where \(\|g_{1M_{3}}\|:=\max_{|x(t)|\leq M_{3}}|g_{1}(t,x)|\),
From these inequalities, we get
In view of Eq. (5.1), we know that there exists a constant \(M_{4}>0\) such that
The case \(t\in [0,\tau ]\) can be treated similarly.
From Eqs. (5.8), (5.9) and (5.11), we have
where \(0< E_{1}<\min (M_{4},D_{1})\), \(E_{2}>\max (M_{3},D_{2})\). This proves the claim, and the rest of the proof is identical to that of Theorem 4.1. □
6 Examples
Example 6.1
Consider the p-Laplacian Liénard equation in the case \(\sum_{i=1}^{n}\|c_{i}\|<1\):
where \(p=3\), \(\delta _{1}\), \(\delta _{2}\) are constants and \(0<\delta _{1}\), \(\delta _{2}< T\).
Comparing Eq. (6.1) with Eq. (4.1), it is easy to see that \(c_{1}(t)=\frac{1}{40}\sin (4t)\), \(c_{2}(t)=\frac{1}{60}\cos (4t-\frac{ \pi }{3})\), \(f(x)=\frac{1}{20}x\), \(g(t,x)=\frac{1}{40}(2+\sin 4t)x ^{2}\), \(e(t)=\sin (4t)\), \(T=\frac{\pi }{2}\). It is easy to see that there exists a constant \(D=1\) such that condition \((H_{1})\) holds. Obviously, we get \(|f(x)|=|\frac{1}{20}x|\leq \frac{1}{20}|x|+3 \), here \(m=\frac{1}{20}\), \(\tilde{n}=3\), and condition \((H_{2})\) holds. Consider \(|g(t,x)|=|\frac{1}{40}(2+\sin (4t))x^{2}|\leq \frac{3}{40}|x|^{2}+1\), here \(a=\frac{3}{40}\), \(b=1\). So, condition \((H_{3})\) is satisfied. Moreover, \(\|c_{i}\|=\frac{1}{40}\), \(\|c_{2}\|=\frac{1}{60}\). So, we have \(\sum_{i=1}^{2}\|c_{i}\|=\|c_{1}\|+\|c_{2}\|=\frac{1}{24}<1\). Also \(\sigma =\frac{1}{1-\|c_{1}\|-\|c_{2}\|}=\frac{24}{23}\), \(\|c'_{1}\|= \frac{1}{10}\) and \(\|c'_{2}\|=\frac{1}{15}\). Next, we consider the condition
Therefore, by Theorem 4.1, we know that Eq. (6.1) has at least one positive \(\frac{\pi }{2}\)-periodic solution.
Example 6.2
Consider the p-Laplacian Liénard equation in the case \(\sum_{i=1}^{n}\|c_{i}\|>1\):
where \(p=5\), \(\delta _{3}\), \(\delta _{4}\) are constants and \(0<\delta _{3}\), \(\delta _{4}< T\).
Comparing Eq. (6.2) with Eq. (4.1), it is easy to see that \(c_{1}(t)=\frac{1}{8}\cos (8t)+\frac{15}{8}\), \(c_{2}(t)= \frac{1}{64}\sin (8t-\frac{\pi }{6})\), \(f(x)=\frac{1}{24}x^{2}+1\), \(g(t,x)=\frac{1}{16}(2+\sin 8t)x^{3}\), \(e(t)=\cos (8t-\frac{\pi }{4})\). \(T=\frac{\pi }{4}\). It is easy to see that there exists a constant \(D=1\) such that \((H_{1})\) holds. Obviously, we get \(|f(x)|=| \frac{1}{24}x^{2}+1|\leq \frac{1}{24}|x|^{2}+2 \), here \(m= \frac{1}{24}\), \(\tilde{n}=2\), and condition \((H_{2})\) holds. Consider \(|g(t,x)|=|\frac{1}{16}(2+\sin 8t)x^{3}|\leq \frac{3}{16}|x|^{3}+1\), here \(a=\frac{3}{16}\), \(b=1\). So, condition \((H_{3})\) is satisfied. Furthermore, \(\|c_{1}\|=\frac{1}{8}+\frac{15}{8}=2\), \(\|c_{2}\|= \frac{1}{64}\), so we have \(\sum_{i=1}^{2}\|c_{i}\|=\|c_{1}\|+ \|c_{2}\|=\frac{129}{64}>1\), \(\sigma =\frac{\frac{1}{\|c_{k}\|}}{1-\frac{1}{ \|c_{k}\|}-\sum_{i=1,i\neq k}^{n}\|\frac{c_{i}}{c_{k}}\|}=\frac{ \frac{1}{2}}{1-\frac{1}{2}-\frac{\frac{1}{2}}{\frac{1}{64}}}= \frac{64}{63}\), \(\|c'_{1}\|=1\) and \(\|c'_{2}\|=\frac{1}{8}\). Next, we consider the condition
Therefore, by Theorem 4.1, we know that Eq. (6.2) has at least one positive \(\frac{\pi }{4}\)-periodic solution.
Example 6.3
Consider the following p-Laplacian Liénard equation with singularity
where \(p=5\), \(\mu \geq 1\), \(\delta _{1}\), \(\delta _{2}\) and \(\delta _{3}\) are constants and \(0<\delta _{1}\), \(\delta _{2},\delta _{3}< T\).
Comparing Eq. (6.3) with Eq. (4.1), it is easy to see that \(c_{1}(t)=\frac{1}{16}\cos (8t-\frac{\pi }{16}))\), \(c_{2}(t)= \frac{1}{24}\sin (8t)\), \(c_{3}(t)=\frac{1}{48}\cos (8t+\frac{\pi }{6})\), \(f(x)=\frac{1}{10}x^{3}\), \(g(t,x)=\frac{1}{32} (\frac{1}{2}+2 \sin (8t) )x^{4}-\frac{1}{x^{\mu }}\), \(e(t)=\cos (8t+\frac{ \pi }{4})\). \(T=\frac{\pi }{4}\). It is obvious that \((H_{2})\), \((H_{4})\) and \((H_{5})\) hold. \(\|c_{1}\|=\frac{1}{16}\), \(\|c_{2}\|= \frac{1}{24}\) and \(\|c_{3}\|=\frac{1}{48}\), so we have \(\sum_{i=1}^{3}\|c_{i}\|=\frac{1}{8}<1\), \(\sigma =\frac{1}{1-\sum_{i=1}^{3}\|c_{i}\|}=\frac{1}{1-\frac{1}{16}-\frac{1}{24}- \frac{1}{48}}=\frac{8}{7}\). Furthermore, \(\|c'_{1}\|=\frac{1}{2}\), \(\|c'_{2}\|=\frac{1}{4}\) and \(\|c'_{3}\|=\frac{1}{6}\). Next, we consider the condition
Therefore, by Theorem 5.1, we know that Eq. (6.3) has at least one positive \(\frac{\pi }{4}\)-periodic solution.
Example 6.4
Consider the following p-Laplacian Liénard equation with singularity
where \(p=3\), \(\mu \geq 1\), \(\delta _{1}\), \(\delta _{2}\), and \(\delta _{3}\) are constants and \(0<\delta _{1}\), \(\delta _{2},\delta _{3}< T\).
Comparing Eq. (6.4) with Eq. (4.1), it is easy to see that \(c_{1}(t)=\frac{1}{6}\cos (6t+\frac{\pi }{5})+\frac{11}{6}\), \(c_{2}(t)=\frac{1}{36}\cos (6t+\frac{\pi }{3})\), \(c_{3}(t)=- \frac{1}{24}\cos (6t+\frac{\pi }{6})\). \(f(x)=\frac{1}{149}x^{4}+2\), \(g(t,x)=\frac{1}{256}(1+\sin (6t))x^{5}-\frac{1}{x^{\mu }}\), \(e(t)=\sin (6t-\frac{\pi }{4})\), \(T=\frac{\pi }{3}\). It is obvious that \((H_{2})\), \((H_{4})\) and \((H_{5})\) hold. Furthermore, \(\|c_{1}\|= \frac{1}{6}+\frac{11}{6}=2\), \(\|c_{2}\|=\frac{1}{36}\) and \(\|c_{3}\|= \frac{1}{24}\), so we have \(\sum_{i=1}^{3}\|c_{i}\|= \frac{149}{72}>1\), \(\sigma =\frac{\frac{1}{\|c_{k}\|}}{1-\frac{1}{\|c _{k}\|}-\sum_{i=1,i\neq k}^{n}\|\frac{c_{i}}{c_{k}}\|}=\frac{ \frac{1}{2}}{1-\frac{1}{2}-\frac{\frac{1}{2}}{\frac{1}{36}}-\frac{ \frac{1}{2}}{\frac{1}{24}}}=\frac{576}{571}\), \(\|c'_{1}\|=1\), \(\|c'_{2}\|=\frac{1}{6}\) and \(\|c'_{3}\|=\frac{1}{4}\). Next, we consider the condition
Therefore, by Theorem 5.1, we know that Eq. (6.4) has at least one positive \(\frac{\pi }{3}\)-periodic solution.
7 Conclusions
In this paper, we first investigated some properties of the neutral operator with multiple delays and variable coefficients \((Ax)(t):=x(t)- \sum_{i=1}^{n}c_{i}(t)x(t-\delta _{i})\). Afterwards, by using an extension of Mawhin’s continuation theorem due to Ge and Ren, properties of the neutral operator A, we studied the existence of a periodic solution for equation (1.1). At last, by applying Theorem 3.2, we discussed the existence of a periodic solution for two p-Laplacian neutral differential equations. In comparison to [5] and [19], we avoided translating the equation into a two-dimensional system.
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Acknowledgements
ZHB, ZBC and SWY 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 National Natural Science Foundation of China (11501170, 71601072), China Postdoctoral Science Foundation funded project (2016M590886), Fundamental Research Funds for the Universities of Henan Provience (NSFRF170302).
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Bi, Z., Cheng, Z. & Yao, S. Periodic solutions for p-Laplacian neutral differential equation with multiple delay and variable coefficients. Adv Differ Equ 2019, 106 (2019). https://doi.org/10.1186/s13662-018-1942-y
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DOI: https://doi.org/10.1186/s13662-018-1942-y