Abstract
In this paper, we consider the existence of a positive periodic solution for the following kind of high-order p-Laplacian neutral singular Rayleigh equation with variable coefficient:
Our proof is based on Mawhin’s continuation theory.
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1 Introduction
In this paper, we consider the following high-order p-Laplacian neutral singular Rayleigh equation with variable coefficient:
where \(p>1\), \(\varphi_{p}(x)= \vert x \vert ^{p-2}x\) for \(x\neq 0\) and \(\varphi_{p}(0)=0\), \(c\in C^{n} (\mathbb{R},\mathbb{R})\) and \(c(t+T)\equiv c(t)\), f is a continuous function defined in \(\mathbb{R}^{2}\) and periodic in t with \(f(t,\cdot)=f(t+T,\cdot)\) and \(f(t,0)=0\), \(g(t,x)=g_{0}(x)+g_{1}(t,x)\), where \(g_{1}:\mathbb{R}\times(0,+\infty)\to \mathbb{R}\) is an \(L^{2}\)-Carathéodory function, \(g_{1}(t,\cdot)=g_{1}(t+T,\cdot)\), \(g_{0}\in C((0,\infty);\mathbb{R})\) has a singularity at \(x=0\), \(e:\mathbb{R}\rightarrow\mathbb{R}\) is a continuous periodic function with \(e(t+T)\equiv e(t)\) and \(\int^{T}_{0}e(t)\,dt=0\), T is a positive constant, and n and m are positive integers.
In recent years, there are many works on periodic solutions for high-order neutral differential equations (see [1–11] and the references therein). Wang and Lu [5] in 2007 investigated the existence of periodic solution for the following high-order neutral functional differential equation with distributed delay:
Using the continuation theorem of coincidence degree theory, they obtained the existence of periodic solutions for (1.2). Afterwards, Ren et al. considered the following high-order p-Laplacian neutral differential equation
They obtained the existence of periodic solutions for (1.3) in the general case (\(\vert c \vert \neq1\)) in [10] and in the critical case (\(\vert c \vert =1\)) in [9], respectively.
At the same time, some authors began to consider high-order neutral differential equation with singularity. Recently, applying the coincidence degree theory and some analysis skills, Xin et al. [11] discussed the existence of a positive periodic solution for the following neutral Liénard equation with singularity:
Inspired by these results in [5, 9–11], in this paper, we consider the existence of a positive periodic solution for (1.1) with singularity by applications of Mawhin’s continuation theory. The obvious difficulty lies in the following two respects. Firstly, \((x(t)-c(t)x(t-\sigma))^{(n)}\neq x^{(n)}(t)-c(t)x^{(n)}(t-\sigma)\), and the calculation of \((x(t)-c(t)x(t-\sigma))^{(n)}\) is very complicated. Secondly, a priori bounds of periodic solutions are not easy to estimate.
2 Preparation
Firstly, we give qualitative properties of the neutral operator \((Ax)(t):=x(t)-c(t)x(t-\sigma)\).
Lemma 2.1
(see [12])
If \(\vert c(t) \vert \neq1\), then the operator A has a continuous inverse \(A^{-1}\) on \(C_{T}:=\{\phi\in C(\mathbb{R},\mathbb{R}):\phi(t+T)\equiv\phi(t)\}\), satisfying
where \(\Gamma:= \bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} 1- \vert c \vert _{\infty}&\textit{for } \vert c \vert _{\infty}:=\max_{t\in[0,T]} \vert c(t) \vert <1,\\ \vert c \vert _{0}-1&\textit{for } \vert c \vert _{0}:=\min_{t\in[0,T]} \vert c(t) \vert >1. \end{array} } \)
Lemma 2.2
(Gaines and Mawhin [13])
Let X and Y be two Banach spaces, and let \(L:D(L)\subset X\rightarrow Y\) be a Fredholm operator with index zero. Let \(\Omega\subset X\) be an open bounded set, and let \(N:\overline{\Omega}\rightarrow Y \) be L-compact on Ω̅. Assume that the following conditions hold:
-
(1)
\(Lx\neq\lambda Nx\), \(\forall x\in\partial\Omega\cap D(L)\), \(\lambda\in(0,1)\);
-
(2)
\(Nx\notin\operatorname{Im} L\), \(\forall x\in\partial \Omega\cap\operatorname{Ker} L\);
-
(3)
\(\deg\{JQN,\Omega\cap\operatorname{Ker} L,0\}\neq0\), where \(J:\operatorname{Im} Q\rightarrow\operatorname{Ker} L\) is an isomorphism.
Then the equation \(Lx=Nx\) has a solution in \(\overline{\Omega}\cap D(L)\).
Lemma 2.3
(see [11])
If \(x\in C^{1}_{T}:=\{x\in C^{1}(\mathbb{R},\mathbb{R}):x(t+T)\equiv x(t)\}\) and there exists a point \(t_{0}\in(0,T)\) such that \(\vert x(t_{0}) \vert < d\), then
where \(\vert x \vert _{\infty}:=\max_{t\in\mathbb{R}} \vert x(t) \vert \).
To use the continuation degree theorem, we rewrite (1.1) in the form
where \(\frac{1}{p}+\frac{1}{q}=1\). Clearly, if a periodic solution of (2.1) is \(x(t):= \bigl ({\scriptsize\begin{matrix}{} x_{1}\cr x_{2} \end{matrix}} \bigr ) \), then \(x_{1}(t)\) must be a periodic solution of (1.1). Thus, the problem of finding a periodic solution for (1.1) reduces to finding a periodic solution for (2.1).
Now, set
with the norm \(\vert x \vert _{\infty}=\max\{ \vert x_{1} \vert _{\infty}, \vert x_{2} \vert _{\infty}\}\) and
with the norm \(\Vert x \Vert =\max\{ \vert x \vert _{\infty}, \vert x' \vert _{\infty}\}\). Clearly, both X and Y are Banach spaces. Meanwhile, define
by
and \(N: X\rightarrow Y\) by
Then (2.1) can be converted into the abstract equation \(Lx=Nx\).
If \(x= \bigl ({\scriptsize\begin{matrix}{} x_{1}\cr x_{2} \end{matrix}} \bigr ) \in \operatorname{Ker} L\), that is, \(\bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} (x_{1}(t)-c(t)x_{1}(t-\sigma))^{(n)}=0,\\ x_{2}^{(m)}(t)=0, \end{array} } \) then we have
where \(a_{0},\ldots,a_{n-1}, b_{0},\ldots,b_{m-1}\in\mathbb{R}\) are constant. From \(x_{1}(t)-c(t)x_{1}(t-\sigma)\in C_{T}\) and \(x_{2}(t)\in C_{T}\) we have \(a_{1}=\cdots=a_{n-1}=0\) and \(b_{1}=b_{2}=\cdots=b_{m-1}=0\). Let \(\phi(t)\neq0\) be a solution of \(x(t)-c(t)x(t-\sigma)=1\). Then \(\operatorname{Ker} L=x= \bigl ({\scriptsize\begin{matrix}{} a\phi(t),a\in\mathbb{R}\cr b,b\in\mathbb{R} \end{matrix}} \bigr ) \). From the definition of L we can easily see that
So L is a Fredholm operator with index zero.
Next, we will consider L-compact N. Let \(P:X\rightarrow\operatorname{Ker} L\) and \(Q:Y\rightarrow \operatorname{Im} Q\subset\mathbb {R}^{2}\) be defined by
Then \(\operatorname{Im} P=\operatorname{Ker} L\) and \(\operatorname{Ker} Q=\operatorname{Im} L\). Denote \(L_{P}=L\vert_{D(L)\cap\operatorname {Ker} P}\) and let \(L_{P}^{-1}: \operatorname{Im} L\rightarrow D(L)\) be the inverse of \(L_{P}\). Then
where \(a_{i}:=(Ax_{1})^{(i)}(0)\) are defined as follows:
\(Z=(a_{n-1},a_{n-2}\cdots,a_{1})^{\top}\), \(C=(c_{1},c_{2},\ldots,c_{n-1})^{\top }\), \(c_{i}=-\frac{1}{i!T}\int^{T}_{0}(T-s)^{i}y_{1}(s)\,ds\), and \(e_{j}=\frac{T^{j}}{(j+1)!}\), \(j=1,2,\ldots,n-2\). Similarly, we can get \(b_{1}:=x_{2}^{(i)}(0)\), \(i=1,2,\ldots,m-1\). Therefore, from (2.2) and (2.3) we get that N is L-compact on Ω̄.
3 Periodic solutions for (1.1) with repulsive singularity
For convenience, we list the following assumptions, which will further used repeatedly:
- (H1):
-
There exists a positive constant K such that \(\vert f(t,u) \vert \leq K\) for \((t,u)\in \mathbb{R}\times\mathbb{R}\).
- (H2):
-
There exist positive constants α and β such that \(\vert f(t,u) \vert \leq\alpha \vert u \vert ^{p-1}+\beta\) for \((t,u)\in\mathbb{R}\times\mathbb{R}\).
- (H3):
-
\(f(t,u)\geq0\) for \((t,u)\in\mathbb{R}\times\mathbb{R}\);
- (H4):
-
There exists a positive constant D such that \(g(t,x)>K\) for \(x>D\).
- (H5):
-
There exists a positive constant \(D_{1}\) such that \(g(t,x)> \vert e \vert _{\infty}\) for \(x>D_{1}\).
- (H6):
-
There exist positive constants γ, ζ such that
$$ g(t,x)\leq\gamma x^{p-1}+\zeta\quad \mbox{for all } x>0. $$ - (H7):
-
(Repulsive singularity) \(\int^{1}_{0}g_{0}(s)\,ds=-\infty\).
Theorem 3.1
Assume that (H1), (H4), and (H6)-(H7) hold. Then (1.1) has at least one T-periodic solution if
where \(c_{n-k}:=\max_{t\in[0,\omega]} \vert c^{(n-k)}(t) \vert \).
Proof
Consider the abstract equation
Set \(\Omega_{1}=\{x:Lx=\lambda Nx,\lambda\in (0,1)\}\). If \(x(t)=(x_{1}(t),x_{2}(t))^{\top}\in\Omega_{1}\), then
Substituting \(x_{2}(t)=\lambda^{1-p}\varphi_{p}[(Ax_{1})^{(n)}(t)]\) into the second equation of (3.1), we have
Integrating both sides of (3.2) from 0 to T, we have
From the mean value theorem, there exists a point \(\xi\in(0,T)\) such that
Then by (H1) we have
and in view of (H4), we get that \(x_{1}(\xi)\leq D\). Since \(x_{1}(t)\) is periodic with period T and \(x_{1}(t)>0\) for \(t\in[0,T]\). Then \(0< x_{1}(\xi)\leq D\). Therefore, from Lemma 2.3 we can get
From (3.4) and the Wirtinger inequality (see [14], Lemma 2.4) we get
Since \(x_{1}^{(i-1)}(0)=x_{1}^{(i-1)}(T)\), \(i=1,2\ldots,n-1\), there exists a point \(t_{i}^{*}\in[0,T]\) such that \(x_{1}^{(i)}(t_{i}^{*})=0\). From the Hölder and Wirtinger inequalities, we can easily get
On the other hand, since \((Ax_{1})(t)=x_{1}(t)-c(t)x_{1}(t-\sigma)\), we have
So, we can get
Applying Lemma 2.2, (3.5), and (3.6), we have
Since \(\Gamma-\frac{T}{2} (\sum_{k=0}^{n-1}C_{n}^{k}c_{n-k} (\frac{T}{2\pi } )^{n-1-k} )>0\), we have
In view of \(\int^{T}_{0}(\varphi_{q}(x_{2}(t)))\,dt=\int^{T}_{0}(Ax_{1}(t))^{(n)}(t)\,dt=0\), there exists a point \(t_{2}\in(0,T)\) such that \(x_{2}(t_{2})=0\). From the Wirtinger inequality and from (3.4) we easily get
Besides, from \(x_{2}^{(m-2)}(0)=x_{2}^{(m-2)}(T)\), there exists a point \(t_{3}\in(0,T)\) such that \(x_{2}^{(m-1)}(t_{3})=0\), which, together with the integration of the second equation of (3.1) on interval \([0,T]\), yield
since \(\vert f(t,u) \vert \leq K\) form (H1). From (H1) and (H6) we have
Since \((1+x)^{k}\leq1+(1+k)x\) for \(x\in[0,\delta]\), where δ is a constant, which depends only on \(k>0\), substituting (3.10) into (3.9), we have
where \(N_{1}:=2\zeta T+2KT+T \vert e \vert _{\infty}\). Substituting (3.6) and (3.7) into (3.11), we have
Combining of (3.8) and (3.12) implies
So, we have
Since
there exists a positive constant \(M_{1}\) such that
Therefore, from (3.7) we have
From (3.6) we have
Hence, from (3.4) we have
From (3.6), (3.9), and (3.10) we have
From (3.8) we get
On the other hand, since \(g(t,x_{1})=g_{1}(t,x_{1}(t))+g_{0}(x_{1}(t))\), (3.2) can be rewritten as
Let \(\tau\in[0,T]\) for any \(\tau\leq t\leq T\). Multiplying both sides of (3.19) by \(x_{1}'(t)\) and integrating on \([\tau,t]\), we have
By (3.2), (3.12), (3.17), and (3.18) we have
Moreover, we have
where \(g_{M_{3}}:=\max_{0< x\leq M_{3}} \vert g_{1}(t,u) \vert \in L^{2}(0,T)\) and \(\Vert g_{M_{3}} \Vert _{2}:= (\int^{T}_{0} \vert g_{1}(t,x_{1}'(t)) \vert ^{2}\,dt )^{\frac{1}{2}}\). Substituting (3.21) and (3.22) into (3.20), we have
From repulsive singular condition (H7) we know that there exists a constant \(M_{5}>0\) such that
The case \(t\in[0,\tau]\) can be treated similarly.
Let
where \(0< E_{5}< M_{5}\), \(E_{1}>\max\{D,M_{3}\}\), \(E_{2}>M_{2}\), \(E_{3}>M_{1}\), and \(E_{4}>M_{4}\). Next, we shall prove that \(\Omega_{2}\) is a bounded set. In fact, for all \(x\in\Omega_{2}\), \(x_{2}=0\), \(x_{1}=a_{0}\phi(t)\), and \(a_{0}\in\mathbb{R}^{+}\), we have
From assumption (H1) we have \(0< a_{0}\phi(t)\leq D\). So \(\Omega_{2}\) is a bounded set.
Let \(\Omega=\{x\in(x_{1},x_{2})^{\top}: \Vert x \Vert \leq M\}\), where \(M=\max\{E_{1},E_{2},E_{3},E_{4}\}\). Then \(\Omega_{1}\cup\Omega_{2}\subset\Omega\), and, as it follows from the above proof, \(Lx\neq\lambda Nx\) for all \((x,\lambda)\in\partial \Omega\times(0,1)\), so that conditions (1) and (2) of Lemma 2.2 are both satisfied. Define the isomorphism \(J:\operatorname{Im} Q\rightarrow\operatorname{Ker} L\) as follows:
Let \(H(\mu,x)=-\mu x+(1-\mu)JQNx\), \((\mu,x)\in[0,1]\times\Omega\). Then, for all \((\mu,x)\in(0,1)\times(\partial\Omega\cap\operatorname {Ker} L)\),
since \(\int^{T}_{0}e(t)\,dt=0\) and \(f(t,0)=0\). From (H4) it is obvious that \(x^{\top}H(\mu,x)<0\) for all \((\mu,x)\in(0,1)\times (\partial\Omega\cap\operatorname{Ker} L)\). Hence
So condition (3) of Lemma 2.2 is satisfied. Applying Lemma 2.2, we conclude that equation \(Lx=Nx\) has a solution \(x=(x_{1},x_{2})^{\top}\) on \(\bar{\Omega}\cap D(L)\), that is, (1.1) has a T-periodic solution \(x_{1}(t)\). □
Theorem 3.2
Assume that (H2)-(H3) and (H5)-(H7) hold. Then (1.1) has at least a nonconstant T-periodic solution if
Proof
We follow the same strategy and notation as the proof of Theorem 3.1. Now, we consider \(\Vert x' \Vert \leq M_{2}\).
We first claim that there is a constant \(\xi^{*}\in[0,T]\) such that
Since \(\int^{T}_{0}(\varphi_{p}(Ax_{1})'(t))'dt=0\), there exist two points \(\xi^{*}, \xi_{*}\in[0,T]\) such that
From (H3) and (3.2) we have
since \(f(\xi^{*},x_{1}'(\xi^{*}))>0\). Therefore, we get
From (H5) we have
Since \(x(t)>0\), we get \(0< x_{1}(\xi^{*})\leq D_{1}\). This proves (3.24).
Similarly, from (3.4) we have
From (3.9) and (H2) we get
From (3.10), (H2), and (H7) we have
Substituting (3.27) into (3.26), from (3.11) we have
where \(N_{2}=2T(\zeta+\beta)+ \Vert e \Vert T\). From (3.12), (3.13), and (3.14) we get
Since \(\frac{ (\frac{T^{2p}\gamma}{2^{2p-1}}+\frac{T^{p+1} \alpha}{2^{p}} ) (\frac{T}{2\pi} )^{(n-2)(p-1)+(m-2)}}{ (\Gamma-\frac{T}{2} \sum_{k=0}^{n-1}C_{n}^{k}c_{n-k} (\frac{T}{2\pi} )^{n-1-k} )^{p-1}}<1\), it is easy to see that there exists a positive constant \(M_{2}\) such that
The rest of the proof is the same as in Theorem 3.1. □
We illustrate our results with an example.
Example 3.1
Consider the neutral functional differential
where \(p=4\), σ and μ are constants, and \(0<\sigma<T\). It is clear that \(T=\frac{\pi}{2}\), \(n=3\), \(m=3\), \(c(t)=\frac{1}{64}\sin4t\), \(e(t)=5\cos4t\), \(c_{1}=\max_{t\in[0,T]}\vert\frac{1}{16}\cos 4t\vert=\frac{1}{16}\), \(c_{2}=\max_{t\in[0,T]}\vert-\frac{1}{4}\sin 4t\vert=\frac{1}{4}\), and \(c_{3}=\max_{t\in[0,T]}\vert-\cos4t\vert=1\). In this case, \(f(t,u)=\cos^{2}(2t)\sin u\), \(f(t,0)=0\), \(\vert f(t,u) \vert =\vert\cos^{2}(2t)\sin u\vert\leq1\), \(K=1\); and \(g(t,x)=\frac{1}{4\pi}(\sin4t+3)x^{3}(t)-\frac{1}{x^{\mu}}\leq\frac {1}{\pi}x^{3}+1\), \(\gamma=\frac{1}{\pi}\), \(\zeta=1\); Obviously, conditions (H1) and (H6)-(H7) hold. Choose \(D=4\pi\) such that (H4) holds. Now we consider the following condition:
So, by Theorem 3.1, (3.29) has at least one nonconstant \(\frac{\pi}{2}\)-periodic solution.
4 Conclusions
In summary, a periodic solution of (1.1) with singularity is illustrated by Theorems 3.1 and 3.2. In Theorem 3.1, we consider the existence of a periodic solution for (1.1) in the case \(\vert f(t,u) \vert \leq K\). Furthermore, in Theorem 3.2, we give a condition on \(f(t,u)\) that is weaker than \(\vert f(,u) \vert \leq K\) in Theorem 3.1, that is, we obtain the existence of periodic solution for (1.1) in the case where \(\vert f(t,u) \vert \leq\alpha \vert u \vert ^{p-1}+\beta\). From the mathematical point of view, the results are valuable to understand the periodic solutions for high-order neutral differential equations.
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Acknowledgements
YX, SWY, and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by National Natural Science Foundation of China (No. 11501170), China Postdoctoral Science Foundation funded project (2016M590886), Education Department of Henan Province project (No. 16B110006), Henan Polytechnic University Outstanding Youth Fund (J2015-02), and Henan Polytechnic University Doctor Fund (B2013-055).
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YX, SWY, and ZBC worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.
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Xin, Y., Yao, S. & Cheng, Z. Positive periodic solution for higher-order p-Laplacian neutral singular Rayleigh equation with variable coefficient. Bound Value Probl 2017, 153 (2017). https://doi.org/10.1186/s13661-017-0883-9
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DOI: https://doi.org/10.1186/s13661-017-0883-9