Abstract
In this paper, we study the positive periodic solutions of a kind of p-Laplacian neutral differential equation with a singularity. By applying the continuation theorem and some analytic techniques, we shall establish several new criteria for the existence of positive periodic solutions for the considered problem. Some recent results in the literature are generalized and improved. Three examples are given to illustrate the effectiveness of our results.
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1 Introduction
In the past years, the study of periodic solutions for some types of p-Laplacian neutral differential equations has attracted much attention from researchers; see [1–12] and the references cited therein. For example, in [11] and [12], Zhu and Lu studied the periodic solutions for p-Laplacian neutral functional differential equations as follows:
and
respectively.
On the basis of work of [11] and [12], Wang and Zhu [7] further discussed existence of periodic solutions for a fourth-order p-Laplacian neutral functional differential equation of the form:
where \(p>1\), \(\varphi_{p} : \mathbb {R}\rightarrow \mathbb {R}\), \(\varphi_{p}(u)= \vert u \vert ^{p-2}u\) for \(u\neq0\) and \(\varphi_{p}(0)=0\); \(f\in C(\mathbb {R},\mathbb {R})\); \(g, \tau\in C(\mathbb {R}^{2}, \mathbb {R})\) with \(g(t+T,\cdot)=g(t,\cdot)\); \(\tau (t+T)\equiv\tau(t)\); \(e\in C(\mathbb {R},\mathbb {R})\) with \(e(t+T)\equiv e(t)\); \(T>0\), \(\vert c \vert \neq1\) and δ are given constants.
In recent years, by applying the method of coincidence degree, many good results of the existence of positive periodic solutions for some types of differential equations with a singularity have been obtained; see [13–22] and the references cited therein. For example, Wang in [21] studied the periodic solutions for the Liénard equation with a singularity and a deviating argument, which extends the results of Zhang in [22]
where \(g: \mathbb{R}\times(0,+\infty)\rightarrow\mathbb{R}\) is an \(L^{2}\)-Carathéodory funtion, \(g(t,x)\) is a T-periodic function in the first argument and can be singular at \(x=0\).
As usual, we say that g has a singularity of repulsive type, if
and g has a singularity of attractive type, if
Compared with the classical p-Laplacian neutral differential problems or the singular problems, p-Laplacian neutral differential problems with singular effects have been scarcely studied, not to mention the high-order p-Laplacian neutral differential equations with a singularity. This motivated us to carry out a study. In this paper, we consider the following high-order p-Laplacian neutral differential equation with a singularity and a deviating argument:
where \(\varphi_{p}:\mathbb{R}\rightarrow\mathbb{R}\), \(\varphi_{p}(u)= \vert u \vert ^{p-2}u\), \(p\geq2\); m is a positive integer; c is a constant with \(\vert c \vert <1\); \(0\leq\gamma, \tau< T\); \(f:\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(g: [0,T]\times(0,+\infty)\rightarrow\mathbb{R}\) is a continuous function and can be singular at \(u=0\), \(e(t)\) is T-periodic with \(\int_{0}^{T} e(t)\,dt=0\).
Remark 1.1
Obviously, if there is no singularity, (1.1) is the specially case of (1.5). The equations in [1, 3–6, 10] are also the specially cases of (1.5) if \(m\equiv1\) and there are no singularities. So we extend the result in [1, 3–7, 9–11]. Therefore, we extend the corresponding results in literature to the high-order case and the singular case.
The paper is organized as follows. Section 2 is devoted to introducing some definitions and recalling some preliminary results that will be extensively used. The existence results will be obtained in Section 3. Finally, two examples are given to illustrate the effectiveness of our result in Section 4.
2 Preliminaries
Let
with the norm \(\vert x \vert _{0}=\max_{t\in[0,T]} \vert x(t) \vert \), and
with the norm \(\Vert x \Vert =\max\{ \vert x \vert _{0}, \vert x' \vert _{0} \}\).
Define a linear operator
Lemma 2.1
[23]
If \(\vert c \vert < 1\), then A has continuous bounded inverse on X, and
-
(1)
\(\Vert A^{-1} x \Vert \leq\frac{ \vert x \vert _{0}}{ \vert 1- \vert c \vert \vert }\), \(\forall x\in X\);
-
(2)
\(\int_{0}^{T} \vert (A^{-1}f)(t) \vert \,dt \leq\frac {1}{1- \vert c \vert }\int_{0}^{T} \vert f(t) \vert \,dt\), \(\forall f\in X\).
Lemma 2.2
[24]
Let \(T>0\) be a constant, \(x\in C^{m}(\mathbb{R},\mathbb{R})\), \(m\geq2\), and \(x(t+T)\equiv x(t)\), \(\vert x^{(i)} \vert _{0}=\max_{t\in [0,T]} \vert x^{(i)}(t) \vert \), then there are \(M_{i}(m)>0\) independent of x such that
where if m is an even integer,
if m is an odd integer,
and \(B_{m-2s}\), \(B_{2m-4s}\), \(B_{m-2s-1}\), \(B_{2m-4s-2}\) are Bernoulli numbers, which can be calculated by the following recursion formula:
where \(C^{i}_{p+1}\) is the number of combinations.
Lemma 2.3
[25]
Let X and Y be two real Banach spaces, \(L: D(L)\subset X \rightarrow Y\) be a Fredholm operator with index zero, \(\Omega\subset X\) be an open bounded set, and \(N:\bar{\Omega}\subset X\rightarrow Y\) be L-compact on Ω̄. Suppose that all of the following conditions hold:
-
(1)
\(Lx\neq\lambda Nx\), \(\forall x \in\partial\Omega\cap D(L)\), \(\forall \lambda\in(0,1)\);
-
(2)
\(QNx\neq0\), \(\forall x\in\partial\Omega\cap\ker L\);
-
(3)
\(\deg\{JQN,\Omega\cap\ker L,0\}\neq0\), where \(J: \operatorname {Im}Q\rightarrow \ker L\) is an homeomorphism map.
Then the equation \(Lx=Nx\) has at least one solution on \(D(L)\cap\bar {\Omega}\).
In order to use the continuation theorem to study the positive T-periodic solutions for equation (1.5), we consider the following system:
where \(q>1\) is a constant with \(1/p+1/q=1\). Clearly, if \(x(t)=(u(t),v(t))^{\top}\) is a T-periodic solution of system (2.1), then \(u(t)\) must be a T-periodic solution of equation (1.5). Thus, the problem of finding a positive T-periodic solution for (1.5) reduces to finding one for (2.1).
Define
with the norm \(\Vert x \Vert _{X}=\max\{ \Vert u \Vert ; \Vert v \Vert \}\),
with the norm \(\Vert x \Vert _{Y}=\max\{ \vert u \vert _{0}; \vert v \vert _{0}\}\). Clearly, X and Y are two Banach spaces.
Define the linear operator
and
where
Then we can see that equation (2.1) can be converted to the abstract equation \(Lx= Nx\). Moreover, from the definition of L, we can have
Clearly, L is a Fredholm operator with index zero.
Let projectors \(P: X \rightarrow\ker L\) and \(Q: Y \rightarrow Y/\operatorname{Im}L\) be defined by
and let K represent the inverse of \(L_{\ker P\cap D(L)}\). Obviously, \(\ker Q =\operatorname {Im}L = \mathbb{R}^{2}\) and
where \(x^{(i)}(0)\) (\(i=1,2,\ldots,m-1\)) are defined by the equation
where
\(\mathbf{X}= (x^{(m-1)}(0), x^{(m-2)}(0),\ldots,x''(0),x'(0) )^{\top}\), \(\mathbf{D}= (d_{1},d_{2},\ldots,d_{m-2},d_{m-1} )^{\top}\), \(d_{i}=-\frac{1}{i!T}\int_{0}^{T}(T-s)^{i}y(s)\,ds\), \(i=1,2,\ldots,m-1\), and \(c_{j}=\frac{T^{j}}{(j+1)!}\), \(j=1,2,\ldots,m-2\).
For the sake of convenience, we list the following assumptions:
- [\(H_{1}\)]:
-
There exist positive constants \(D_{1}\) and \(D_{2}\) with \(D_{1}< D_{2}\) such that
-
(1)
for each positive continuous T-periodic function \(x(t)\) satisfying \(\int_{0}^{T} g(t, x(t))\,dt=0\), then there exists a positive point \(\sigma\in[0,T]\) such that
$$D_{1}\leq x(\sigma)\leq D_{2}; $$ -
(2)
\(\overline{g}(x)<0\) for all \(x\in(0,D_{1})\) and \(\overline {g}(x)>0\) for all \(x>D_{2}\), where \(\overline{g}(x)=\frac{1}{T}\int_{0}^{T} g(t,x)\,dt\), \(x>0\).
-
(1)
- [\(H_{2}\)]:
-
\(g(t,x)=g_{1}(t,x)+g_{0}(x)\) for all \(u \in(0, +\infty )\), where \(g_{0}: (0,+\infty)\rightarrow\mathbb{R}\) is a continuous function, \(g_{1}: [0,T]\times[0,+\infty)\rightarrow\mathbb{R}\) is a continuous function and
-
(1)
there exist positive constants \(m_{0}\) and \(m_{1}\) such that
$$g(t,x)\leq m_{0}x^{p-1}+m_{1}, \quad \text{for all } (t,x)\in[0,T]\times(0,+\infty); $$ -
(2)
\(\int_{0}^{1} g_{0}(x)\,dx=-\infty\).
-
(1)
- [\(H_{3}\)]:
-
There exist positive constants α, β such that
$$f(x)\leq\alpha x^{p-2} +\beta,\quad \text{for all } x\in(0,+\infty). $$
3 Main results
Theorem 3.1
Suppose that conditions [\(H_{1}\)]-[\(H_{3}\)] hold and
then there exist positive constants \(A_{1}\), \(A_{2}\), \(A_{3}\) and ρ, which are independent of λ such that
where \(u(t)\) is any solution to the equation \(Lx = \lambda Nx\), \(\lambda\in(0,1)\).
Proof
Consider the following operator equation:
where L and N is defined by (2.2) and (2.3), respectively.
Define
If \(x=(u,v)^{\top}\in\Omega_{1}\), then \((u,v)\) satisfies
From the first equation of (3.1), we get \(v(t)=\varphi _{p}(\lambda^{-1}(Au)^{(m)}(t))\), and combining with the second equation of (3.1) yields
Integrating equation (3.2) on the interval \([0,T]\), we have
It follows from condition (1) in assumption \([H_{1}]\) that there exist positive constants \(D_{1}\), \(D_{2}\) and \(\sigma\in[0,T]\) such that
then we get
Multiplying both sides of (3.2) by \((Au)(t)\) and integrating on the interval \([0,T]\), we get
which combining with [\(H_{3}\)] yields
Write
Then it follows from (3.3) and [\(H_{2}\)](1) that
Substituting (3.7) into (3.6), combining with (3.5) and Lemma 2.2, we can have
It follows from conclusion (2) of Lemma 2.1 and by applying the Hölder inequality that
From the above inequality, we consider the following two cases:
Case 1
If \(\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt=0\), then \(\int_{0}^{T} \vert u^{(m)} (t) \vert \,dt=0\), it follows from (3.5) and Lemma 2.2 that
Case 2
If \(\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt>0\), then substituting (3.9) into (3.8), we can have
Furthermore,
By classical elementary inequalities, we see that there exists a \(l(p)>0\) which is dependent on p only, such that
If \(\frac{\frac{D_{2}}{TM_{1}(m)}(1- \vert c \vert )}{\int _{0}^{T} \vert (Au)^{(m)}(t) \vert \,dt}>l(p)\), then \(\int _{0}^{T} \vert (Au)^{(m)}(t) \vert \,dt<\frac{\frac {D_{2}}{TM_{1}(m)}(1- \vert c \vert )}{l(p)}\). It follows from (3.5), (3.9) and Lemma 2.2 that
If \(\frac{\frac{D_{2}}{TM_{1}(m)}(1- \vert c \vert )}{\int _{0}^{T} \vert (Au)^{(m)}(t) \vert \,dt}\leq l(p)\), then it follows from (3.12) and (3.13) that
Substituting (3.15) into (3.11) and by applying the Hölder inequality, we can see that
It follows from \(\frac{M^{p}_{1}(m)T^{2p-1}(1+ \vert c \vert )(2m_{0}T+\alpha)}{(1- \vert c \vert )^{p}} <1\) and \(p\geq2\) that there exists a positive constant \(M_{2}>0\) such that
Then by (3.9), we can see that
which together with (3.5) and Lemma 2.2 yields
Therefore, in both Case 1 and Case 2, we obtain
From the second equation of (3.1), we can get
from which by applying [\(H_{3}\)] and (3.7), we have
By Lemma 2.2, (3.17) and (3.18), we obtain
Moreover, integrating the first equation of (3.1) on the interval \([0,T]\), we have
which implies that there exists \(\eta\in[0,T]\) such that \(v(\eta )=0\). Thus,
by Lemma 2.2 and (3.20), we can obtain
On the other hand, from the second equation of (3.1) and [\(H_{2}\)], we can see that
Furthermore, multiplying both sides of equation (3.22) by \(u'(t)\), we have
Let \(\sigma\in[0,T]\) be as in (3.4). For any \(t\in[\sigma ,T]\), integrating equation (3.23) on the interval \([\sigma,t]\), we get
which together with (3.20) yields
Furthermore, set
then we have
by (3.17) and Lemma 2.2, we obtain
According to condition (2) in [\(H_{2}\)], we see that there exists a constant \(M_{5}>0\) such that, for \(t\in[\sigma,T]\),
In a similar way, we can handle the case of \(t\in[0,\sigma]\).
Let us define
Then by (3.4), (3.10), (3.18) and (3.25), we can obtain
Clearly, \(A_{1}\) and \(A_{2}\) are independent of λ. Therefore, the proof of Theorem 3.1 is complete. □
Theorem 3.2
Assume that all the conditions in Theorem 3.1 hold, then system (1.5) has at least one positive T-periodic solution.
Proof
Set
From (2.3) and (2.4), one can easily see that Ω is an open bounded subset of X and N is L-compact on Ω̅. Then the conditions (1) and (2) of Lemma 2.3 are satisfied.
In the following, we prove that condition (3) of Lemma 2.3 also holds.
Now, we let
Define a linear isomorphism
and define
Then we can get
In order to prove the condition (3) of Lemma 2.3 is also satisfied, firstly, we prove that \(K(\mu,x)\) is a homotopic mapping. By way of contradiction, i.e., suppose that there exist \(\mu_{0}\in [0,1]\) and \(x_{0}= \bigl({\scriptsize\begin{matrix}{}u_{0}\cr v_{0} \end{matrix}} \bigr) \in\partial(\Omega\cap\ker L)\) such that \(K(\mu_{0},x_{0})=0\). Then, substituting \(\mu_{0}\) and \(x_{0}\) into (3.27), we have
It follows from \(K(\mu_{0},x_{0})=0\) that
which together with \(\mu_{0}\in[0,1]\) gives \(v_{0}=0\). Thus, we can get \(u_{0}=\frac{A_{1}}{2}\) or \(A_{2}+1\). Furthermore, it follows from [\(H_{1}\)](2) that \(\overline{g} (\frac{A_{1}}{2})<0\) and \(\overline {g} (A_{2}+1)>0\), substituting \(u_{0}=\frac{A_{1}}{2}\) or \(A_{2}+1\) into (3.28), we can obtain
and
From (3.29) and (3.30), we have \(K(\mu_{0},x_{0})\neq0\), which is a contradiction. Therefore \(K(\mu,x)\) is a homotopic mapping and \(x^{\top}K(\mu,x)\neq0\). For all \((x,\mu)\in(\partial\Omega \cap\ker L )\times[0,1]\), we get
Thus, the condition (3) of Lemma 2.3 is also satisfied. Therefore, we can conclude that equation (1.5) has at least one positive T-periodic solution. □
4 Examples
In this section, we provide two examples to illustrate our main result.
Example 4.1
Consider the following third-order p-Laplacian neutral functional differential equation:
Conclusion
Problem (4.1) has at least one positive \(\pi /2\)-periodic solution.
Proof
Corresponding to (1.5), we have
Then we can have and choose
Moreover, by Lemma 2.2 we have \(M_{1}(3)=T/12\). Thus, the conditions [\(H_{1}\)]-[\(H_{3}\)] are satisfied. Meanwhile, we also have
Hence, by applying Theorem 3.1-3.2, we can see that equation (4.1) has at least one positive \(\pi/2\)-periodic solution. □
Example 4.2
Consider the following fourth-order p-Laplacian neutral functional differential equation:
Conclusion
Problem (4.2) has at least one positive π-periodic solution.
Proof
Corresponding to (1.5), we have
Then we can choose
Moreover, by Lemma 2.2, we get \(M_{1}(4)= \frac{T^{2}}{24\sqrt {15}}\). Thus, the conditions [\(H_{1}\)]-[\(H_{3}\)] are satisfied. Meanwhile, we also have
Hence, by applying Theorem 3.1-3.2, we can see that equation (4.2) has at least one positive π-periodic solution. □
Example 4.3
Consider the following fourth-order p-Laplacian neutral functional differential equation:
Conclusion
Problem (4.3) has at least one positive \(\frac{\pi}{2}\)-periodic solution.
Proof
Corresponding to (1.5), we have
Then we can choose
Moreover, by Lemma 2.2, we get \(M_{1}(4)= \frac{T^{2}}{24\sqrt {15}}\). Thus, the conditions [\(H_{1}\)]-[\(H_{3}\)] are satisfied. Meanwhile, we also have
Hence, by applying Theorem 3.1-3.2, we can see that equation (4.3) has at least one positive \(\frac{\pi }{2}\)-periodic solution. □
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Acknowledgements
This work was partially supported by the Fundamental Research Funds for the Central Universities (2016B07514) and the National Natural Science Foundation of China (Grant No. 11471109).
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Li, Z., Kong, F. Positive periodic solutions for p-Laplacian neutral differential equations with a singularity. Bound Value Probl 2017, 54 (2017). https://doi.org/10.1186/s13661-017-0790-0
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DOI: https://doi.org/10.1186/s13661-017-0790-0