Research
Open Access
Positive periodic solutions for p-Laplacian neutral differential equations with a singularity
- Zhiyan Li1 and
- Fanchao Kong2Email author
Received: 2 October 2016
Accepted: 6 April 2017
Published: 18 April 2017
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.
Keywords
neutral differential equations positive periodic solutions continuation theorem p-Laplacian singularity1 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.
$$ \bigl(\varphi_{p} \bigl(x(t)-cx(t-\tau) \bigr)' \bigr)'=g \bigl(t,x \bigl(t-\tau(t) \bigr) \bigr)+e(t) $$
(1.1)
$$ \bigl(\varphi_{p} \bigl(x(t)-cx(t-\delta) \bigr)' \bigr)'=f \bigl(t,x(t) \bigr)x'(t)+ \sum ^{n}_{j=1}\beta_{j}g \bigl(t,x \bigl(t- \tau_{j}(t) \bigr) \bigr)+p(t), $$
(1.2)
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.
$$ \bigl(\varphi_{p} \bigl(x(t)-cx(t-\delta) \bigr)'' \bigr)''=f \bigl(x(t) \bigr)x'(t)+g(t,x \bigl(t-\tau\bigl(t, \vert x \vert _{\infty}\bigr) \bigr)+e(t), $$
(1.3)
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 \(x=0\).
$$ x''(t)+f \bigl(x(t) \bigr)x'(t)+g \bigl(t,x(t-\sigma) \bigr)=0, $$
(1.4)
As usual, we say that g has a singularity of repulsive type, if and g has a singularity of attractive type, if
$$\lim_{x\rightarrow0^{+}} g(t,x)=-\infty, \quad \text{uniformly for a.e. } t \in(- \infty, +\infty), $$
$$\lim_{x\rightarrow0^{+}} g(t,x)=+\infty, \quad \text{uniformly for a.e. } t \in(- \infty, +\infty). $$
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 \(\int_{0}^{T} e(t)\,dt=0\).
$$ \bigl[\varphi_{p} \bigl( \bigl(x(t)-cx(t-\gamma) \bigr)^{(m)} \bigr) \bigr]^{(m)}+f \bigl(x(t) \bigr) x'(t)+g \bigl(t,x(t- \tau) \bigr)=e(t), $$
(1.5)
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 \(\Vert x \Vert =\max\{ \vert x \vert _{0}, \vert x' \vert _{0} \}\).
$$C_{T}= \bigl\{ x\mid x \in C(\mathbb{R},\mathbb{R}), x(t+T)\equiv x(t) \bigr\} $$
$$C^{1}_{T}= \bigl\{ x\mid x \in C^{1}(\mathbb{R}, \mathbb{R}), x(t+T)\equiv x(t) \bigr\} $$
Define a linear operator
$$A:\quad C_{T}\rightarrow C_{T}, \quad\quad (Ax) (t)=x(t)-c x(t-\gamma). $$
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.
$$\bigl\vert x^{(i)} \bigr\vert _{0}\leq M_{i}(m) \int_{0}^{T} \bigl\vert x^{(m)}(t) \bigr\vert \,dt,\quad i=1,2,\ldots,m-1, $$
$$\begin{aligned} M_{i}(m)= \textstyle\begin{cases} M_{2s-1}(m)=T^{m-2s}\sqrt{-B_{2m-4s}/12(2m-4s)!}, &s=1,2,\ldots,\frac {m}{2}-1,\\ M_{2s}(m)=\frac{(-1)^{\frac{m-2s}{2}+1} T^{m-2s-1}B_{m-2s}}{(m-2s)!}, &s=1,\ldots,\frac{m}{2}-1,\\ M_{m-1}(m)=\frac{1}{2}, \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned} M_{i}(m)= \textstyle\begin{cases} M_{2s+1}(m)=\frac{(-1)^{\frac{m-2s-1}{2}+1} T^{m-2s-2}B_{m-2s-1}}{(m-2s-1)!}, &s=0,1,\ldots,\frac{m+1}{2}-2,\\ M_{2s}(m)=T^{m-2s-1}\sqrt{-B_{2m-4s-2}/12(2m-4s-2)!}, &s=1,2,\ldots ,\frac{m+1}{2}-2,\\ M_{m-1}(m)=\frac{1}{2}, \end{cases}\displaystyle \end{aligned}$$
$$B_{0}=1, \quad\quad B_{p}=\frac{-\sum^{p-1}_{i=0}C^{i}_{p+1}B_{i}}{p+1}, $$
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.
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).
$$ \textstyle\begin{cases}u^{(m)}(t)= [A^{-1}\varphi_{q}(v)](t),\\ v^{(m)}(t)=-f(u(t))u'(t)-g(t,u(t-\tau))+e(t), \end{cases} $$
(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.
$$X= \bigl\{ x=(u,v)^{\top}\in C^{1} \bigl(\mathbb{R}, \mathbb{R}^{2} \bigr):x(t+T)\equiv x(t) \bigr\} $$
$$Y= \bigl\{ x=(u,v)^{\top}\in C \bigl(\mathbb{R},\mathbb{R}^{2} \bigr):x(t+T)\equiv x(t) \bigr\} $$
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.
$$ L:\quad D(L)\subset X \rightarrow Y, \quad\quad Lx= \bigl(u^{(m)}, v^{(m)} \bigr)^{\top}, $$
(2.2)
$$ \begin{aligned} &N: \quad D(N)\subset X \rightarrow Y, \\ &(Nx) (t)= \begin{pmatrix} [A^{-1}\varphi_{q}(v)](t) \\ -f(u(t))u'(t)-g(t,u(t-\tau))+e(t) \end{pmatrix} , \quad \forall t\in \mathbb{R}, \end{aligned} $$
(2.3)
$$\begin{aligned}& D(L)= \bigl\{ x\in C^{m} \bigl(\mathbb{R},\mathbb{R}^{2} \bigr), x(t+T)\equiv x(t) \bigr\} , \\& D(N)= \bigl\{ x=(u,v)^{\top}\in X:u(t)>0,t\in[0, T] \bigr\} . \end{aligned}$$
$$\ker L= \mathbb{R}^{2}, \quad\text{and} \quad \operatorname {Im}L= \biggl\{ y\in Y, \int_{0}^{T}y(s)\,ds=0 \biggr\} . $$
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 \(c_{j}=\frac{T^{j}}{(j+1)!}\), \(j=1,2,\ldots,m-2\).
$$Px=x(0); \quad\quad Qy=\frac{1}{T} \int_{0}^{T}y(s)\,ds, $$
$$ (Ky) (t)=\sum^{m-1}_{i=1} \frac{1}{i!}x^{(i)}(0)t^{i}+\frac{1}{(m-1)!} \int_{0}^{t}(t-s)^{m-1}y(s) \,ds, $$
(2.4)
$$\mathbf{A}\mathbf{X}=\mathbf{D}, $$
$$\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 & \ldots& 0 & 0\\ c_{1} & 1 & 0 & \ldots&0 &0 \\ c_{2} & c_{1} & 1& \ldots&0 &0\\ \vdots& \vdots& \vdots& \ddots& \vdots&\vdots\\ c_{m-3} & c_{m-4} & c_{m-5} & \ldots&1 & 0\\ c_{m-2} & c_{m-3} & c_{m-4} & \ldots&c_{1} & 1 \end{pmatrix} , $$
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
\(Lx = \lambda Nx\), \(\lambda\in(0,1)\).
$$\frac{M^{p}_{1}(m)T^{2p-1}(1+ \vert c \vert )(2m_{0}T+\alpha)}{(1- \vert c \vert )^{p}} < 1, $$
$$A_{1}\leq u(t)\leq A_{2}, \quad\quad \vert v \vert _{0}\leq A_{3}, $$
Proof
Consider the following operator equation: where L and N is defined by (2.2) and (2.3), respectively.
$$Lx=\lambda Nx, \quad \lambda\in(0,1), $$
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
$$\Omega_{1}= \Bigl\{ (u,v)^{\top}\in X: \min _{t\in[0,T]}u(t)>0, Lx=\lambda Nx ,\lambda\in(0,1) \Bigr\} . $$
$$ \textstyle\begin{cases}u^{(m)}(t)=\lambda[A^{-1}\varphi_{q}(v)](t),\\ v^{(m)}(t)=-\lambda f(u(t))u'(t)-\lambda g(t,u(t-\tau))+ \lambda e(t). \end{cases} $$
(3.1)
$$ \bigl(\varphi_{p} \bigl((Au)^{(m)}(t) \bigr) \bigr)^{(m)}+\lambda^{p }f \bigl(u(t) \bigr)u'(t)+ \lambda^{p}g \bigl(t,u(t-\tau) \bigr)=\lambda^{p}e(t). $$
(3.2)
$$ \int_{0}^{T}g \bigl(t, u(t-\tau) \bigr)\,dt=0. $$
(3.3)
$$ D_{1}\leq u(\sigma)\leq D_{2}, $$
(3.4)
$$ \begin{aligned}[b] \vert u \vert _{0} &=\max _{t\in[0,T]} \bigl\vert u(t) \bigr\vert \leq\max _{t\in[0,T]} \biggl\vert u( \sigma)+ \int_{\sigma}^{t}u'(s)\,ds \biggr\vert \\ &\leq D_{2}+ \int_{0}^{T} \bigl\vert u'(s) \bigr\vert \,ds. \end{aligned} $$
(3.5)
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:
$$\begin{aligned} \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p}\,dt &\leq \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \int_{0}^{T} \bigl\vert f \bigl(u(t) \bigr) \bigr\vert \bigl\vert u' \bigr\vert \,dt \\ &\quad{} + \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \int_{0}^{T} \bigl\vert g \bigl(t,u(t-\tau) \bigr) \bigr\vert \,dt \\ &\quad{} + \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \int_{0}^{T} \bigl\vert e(t) \bigr\vert \,dt, \end{aligned} $$
$$ \begin{aligned}[b] \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p}\,dt &\leq \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \int_{0}^{T} \bigl(\alpha \bigl\vert u(t) \bigr\vert ^{p-2}+\beta \bigr) \bigl\vert u'(t) \bigr\vert \,dt \\ &\quad{} + \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \int_{0}^{T} \bigl\vert g \bigl(t,u(t-\tau) \bigr) \bigr\vert \,dt+ \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \vert e \vert _{0}T \\ &\leq \bigl(1+ \vert c \vert \bigr)\alpha \vert u \vert ^{p-1}_{0} \int_{0}^{T} \bigl\vert u'(t) \bigr\vert \,dt+ \bigl(1+ \vert c \vert \bigr)\beta \vert u \vert _{0} \int_{0}^{T} \bigl\vert u'(t) \bigr\vert \,dt \\ &\quad{} + \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \int_{0}^{T} \bigl\vert g \bigl(t,u(t-\tau) \bigr) \bigr\vert \,dt+ \bigl(1+ \vert c \vert \bigr) \vert u \vert _{0} \vert e \vert _{0}T. \end{aligned} $$
(3.6)
$$\begin{aligned}& I_{+}= \bigl\{ t\in[0,T]:g \bigl(t, u(t-\tau) \bigr)\geq0 \bigr\} ; \\& I_{-}= \bigl\{ t\in[0,T]:g \bigl(t, u(t-\tau) \bigr)\leq0 \bigr\} . \end{aligned}$$
$$ \begin{aligned}[b] \int_{0}^{T} \bigl\vert g \bigl(t,u(t-\tau) \bigr) \bigr\vert \,dt &= \int_{I_{+}}g \bigl(t,u(t-\tau) \bigr)\,dt- \int_{I_{-}}g \bigl(t,u(t-\tau) \bigr)\,dt \\ &=2 \int_{I_{+}}g \bigl(t,u(t-\tau) \bigr)\,dt \\ &\leq 2m_{0} \int_{0}^{T} u^{p-1}(t-\tau)\,dt+2 \int_{0}^{T}m_{1}\,dt \\ &\leq 2m_{0}T \vert u \vert ^{p-1}_{0}+2Tm_{1}. \end{aligned} $$
(3.7)
$$\begin{aligned}& \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p}\,dt \\& \quad \leq 2 \bigl(1+ \vert c \vert \bigr)m_{0}T \vert u \vert ^{p}_{0}+ \bigl(1+ \vert c \vert \bigr)\alpha \vert u \vert ^{p-1}_{0} \int_{0}^{T} \bigl\vert u'(t) \bigr\vert \,dt \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr)\beta \vert u \vert _{0} \int_{0}^{T} \bigl\vert u'(t) \bigr\vert \,dt +2 \bigl(1+ \vert c \vert \bigr)m_{1}T \vert u \vert _{0}+ \bigl(1+ \vert c \vert \bigr) \vert e \vert _{0}T \vert u \vert _{0} \\& \quad \leq 2 \bigl(1+ \vert c \vert \bigr)m_{0}T^{p+1}M^{p}_{1}(m) \biggl(\frac{D_{2}}{TM_{1}(m)}+ \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \biggr)^{p} \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr)\alpha T^{p}M^{p}_{1}(m) \biggl(\frac{D_{2}}{TM_{1}(m)}+ \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \biggr)^{p-1} \cdot \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr)\beta T^{2}M^{2}_{1}(m) \biggl( \frac{D_{2}}{TM_{1}(m)}+ \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \biggr)\cdot \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \\& \quad\quad{} +2 \bigl(1+ \vert c \vert \bigr)m_{1}T^{2}M_{1}(m) \biggl( \frac{D_{2}}{TM_{1}(m)}+ \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \biggr) \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr) \vert e \vert _{0}T^{2}M_{1}(m) \biggl(\frac{D_{2}}{TM_{1}(m)}+ \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \biggr). \end{aligned}$$
(3.8)
$$ \begin{aligned}[b] \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt & = \int_{0}^{T} \bigl\vert \bigl(A^{-1}(Au)^{(m)} \bigr) (t) \bigr\vert \,dt \\ &\leq \frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{ 1- \vert c \vert } . \end{aligned} $$
(3.9)
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
$$ \begin{aligned}[b] \vert u \vert _{0} &\leq D_{2}+ \int_{0}^{T} \bigl\vert u'(s) \bigr\vert \,ds\leq D_{2}+T \bigl\vert u' \bigr\vert _{0} \\ &\leq D_{2}+TM_{1}(m) \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \\ &=D_{2}. \end{aligned} $$
(3.10)
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
$$\begin{aligned}& \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p}\,dt \\& \quad \leq 2 \bigl(1+ \vert c \vert \bigr)m_{0}T^{p+1}M^{p}_{1}(m) \biggl(\frac{D_{2}}{TM_{1}(m)}+\frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \biggr)^{p} \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr)\alpha T^{p}M^{p}_{1}(m) \biggl(\frac{D_{2}}{TM_{1}(m)}+ \frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \biggr)^{p-1}\cdot \frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr)\beta T^{2}M^{2}_{1}(m) \biggl( \frac{D_{2}}{TM_{1}(m)}+\frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \biggr)\cdot\frac {\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \\& \quad\quad{} +2 \bigl(1+ \vert c \vert \bigr)m_{1}T^{2}M_{1}(m) \biggl( \frac{D_{2}}{TM_{1}(m)}+\frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \biggr) \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr) \vert e \vert _{0}T^{2}M_{1}(m) \biggl(\frac{D_{2}}{TM_{1}(m)}+ \frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \biggr). \end{aligned}$$
(3.11)
$$ \begin{aligned}[b] & \biggl(\frac{D_{2}}{TM_{1}(m)}+\frac{\int _{0}^{T} \vert (Au )^{(m)}(t) \vert \,dt}{1- \vert c \vert } \biggr)^{p} \\ &\quad =\frac{1}{(1- \vert c \vert )^{p}} \biggl( \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert \,dt \biggr)^{p} \biggl(1+\frac{\frac{D_{2}}{TM_{1}(m)}(1- \vert c \vert )}{\int_{0}^{T} \vert (Au)^{(m)}(t) \vert \,dt} \biggr)^{p}. \end{aligned} $$
(3.12)
$$ (1+x)^{p}< 1+(1+p)x, \quad x\in\bigl(0,l(p) \bigr]. $$
(3.13)
$$ \vert u \vert _{0} \leq D_{2}+TM_{1}(m) \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \leq D_{2}+\frac{ D_{2}}{l(p)}:=M_{1}. $$
(3.14)
$$ \begin{aligned}[b] & \biggl(\frac{D_{2}}{TM_{1}(m)}+\frac{\int _{0}^{T} \vert (Au)^{(m)}(t) \vert \,dt}{1- \vert c \vert } \biggr)^{p} \\ &\quad \leq\frac{1}{(1- \vert c \vert )^{p}} \biggl( \int_{0}^{T} \bigl\vert (Au)^{(m)} (t) \bigr\vert \,dt \biggr)^{p} \biggl(1+\frac{(p+1)\frac{D_{2}}{TM_{1}(m)}(1- \vert c \vert )}{\int_{0}^{T} \vert (Au)^{(m)}(t) \vert \,dt} \biggr) \\ &\quad = \frac{ ( \int_{0}^{T} \vert (Au)^{(m)}(t) \vert \,dt )^{p}}{(1- \vert c \vert )^{p}}+\frac{(p+1)\frac {D_{2}}{TM_{1}(m)}}{(1- \vert c \vert )^{p-1}}\cdot \biggl( \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert \,dt \biggr)^{p-1}. \end{aligned} $$
(3.15)
$$\begin{aligned}& \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p}\,dt \\& \quad \leq \frac{M^{p}_{1}(m)T^{2p-1}(1+ \vert c \vert )(2m_{0}T+\alpha)}{(1- \vert c \vert )^{p}} \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p}\,dt \\& \quad\quad{} +\frac{ T^{\frac{2p^{2}-3p+1}{p}} M^{p-1}_{1}(m)D_{2}(1+ \vert c \vert ) [2m_{0}T(p+1)+\alpha p] }{(1- \vert c \vert )^{p-1}} \cdot \biggl( \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p} \,dt \biggr)^{\frac{p-1}{p}} \\& \quad\quad{} + \frac{(1+ \vert c \vert )\beta T^{4p-2} M^{2}_{1}(m)}{(1- \vert c \vert )^{2}} \biggl( \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p} \,dt \biggr)^{\frac{2}{p}} \\& \quad\quad{} + \frac{(1+ \vert c \vert )T^{\frac{2p-1}{p}}M_{1}(m) [\beta D_{2}+2m_{1}T+ \vert e \vert _{0} T] }{1- \vert c \vert }\cdot \biggl( \int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p} \,dt \biggr)^{\frac{1}{p}} \\& \quad\quad{} + \bigl(1+ \vert c \vert \bigr)TD_{2} \bigl(2m_{1}+ \vert e \vert _{0} \bigr). \end{aligned}$$
(3.16)
$$\int_{0}^{T} \bigl\vert (Au)^{(m)}(t) \bigr\vert ^{p}\,dt\leq M_{2}. $$
$$ \begin{aligned}[b] \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt &\leq \frac{\int_{0}^{T} \vert (Au )^{(m)}(t) \vert \, dt}{1- \vert c \vert } \\ &\leq \frac{T^{\frac{p-1}{p}} (\int_{0}^{T} \vert (Au )^{(m)}(t) \vert ^{p}\,dt )^{\frac{1}{p}}}{1- \vert c \vert } \\ & \leq\frac{T^{\frac{p-1}{p}} M_{2} ^{\frac{1}{p}}}{1- \vert c \vert } :=M_{3}, \end{aligned} $$
(3.17)
$$\begin{aligned} \vert u \vert _{0} &\leq D_{2}+ \int_{0}^{T} \bigl\vert u'(s) \bigr\vert \,ds\leq D_{2}+T \bigl\vert u' \bigr\vert _{0} \\ &\leq D_{2}+TM_{1}(m) \int_{0}^{T} \bigl\vert u^{(m)}(t) \bigr\vert \,dt \\ &\leq D_{2}+ TM_{1}(m)M_{3}:=M_{4}. \end{aligned}$$
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
$$\begin{aligned} \int_{0}^{T} \bigl\vert v^{(m)}(t) \bigr\vert \,dt &\leq \lambda \biggl[ \int_{0}^{T} \bigl\vert f \bigl(u(t) \bigr) \bigr\vert \bigl\vert u'(t) \bigr\vert \,dt+ \int_{0}^{T} \bigl\vert g \bigl(t,u(t-\tau) \bigr) \bigr\vert \,dt \\ &\quad{} + \int_{0}^{T} \bigl\vert e(t) \bigr\vert \,dt \biggr], \end{aligned}$$
$$ \begin{aligned}[b] \int_{0}^{T} \bigl\vert v^{(m)}(t) \bigr\vert \,dt &\leq \lambda \bigl[ \alpha T \vert u \vert ^{p-2}_{0} \bigl\vert u' \bigr\vert _{0}+ \beta T \bigl\vert u' \bigr\vert _{0}+ 2 m_{0}T \vert u \vert ^{p-1}_{0} \\ &\quad{} + 2 Tm_{1} + \vert e \vert _{0}T \bigr]. \end{aligned} $$
(3.19)
$$\begin{aligned} \int_{0}^{T} \bigl\vert v^{(m)}(t) \bigr\vert \,dt &\leq \lambda \bigl[ \alpha T \vert u \vert ^{p-2}_{0} \bigl\vert u' \bigr\vert _{0}+ \beta T \bigl\vert u' \bigr\vert _{0}+ 2 m_{0}T \vert u \vert ^{p-1}_{0} \\ &\quad{} + 2 Tm_{1} + \vert e \vert _{0}T \bigr] \\ \leq& \lambda \bigl[ \alpha TM^{p-2}_{4} M_{1}(m)M_{3}+ \beta T M_{1}(m)M_{3}+ 2m_{0}T M^{p-1}_{4} \\ &\quad{} + 2Tm_{1} + \vert e \vert _{0}T \bigr] . \end{aligned}$$
(3.20)
$$\int_{0}^{T} \bigl\vert v(t) \bigr\vert ^{q-2}v(t)\,dt=0, $$
$$\begin{aligned} \bigl\vert v(t) \bigr\vert &= \biggl\vert \int_{\eta}^{t} v'(s)\,ds+v(\eta) \biggr\vert \leq \int_{0}^{T} \bigl\vert v'(s) \bigr\vert \,ds\leq T \bigl\vert v' \bigr\vert _{0}, \end{aligned}$$
$$ \begin{aligned}[b] \vert v \vert _{0}&=\max _{t\in[0,T]} \bigl\vert v(t) \bigr\vert \\ &\leq T \bigl\vert v' \bigr\vert _{0}\leq TM_{1}(m) \int_{0}^{T} \bigl\vert v^{(m)}(t) \bigr\vert \,dt \\ &\leq \lambda TM_{1}(m) \bigl[ \alpha TM^{p-2}_{4} M_{1}(m)M_{3}+ \beta T M_{1}(m)M_{3}+ 2m_{0}T M^{p-1}_{4} \\ &\quad{} + 2Tm_{1} + \vert e \vert _{0}T \bigr] \\ &\leq TM_{1}(m) \bigl[ \alpha TM^{p-2}_{4} M_{1}(m)M_{3}+ \beta T M_{1}(m)M_{3}+ 2m_{0}T M^{p-1}_{4} \\ &\quad{} + 2Tm_{1} + \vert e \vert _{0}T \bigr] \\ &:=A_{3} . \end{aligned} $$
(3.21)
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 \(t\in[0,\sigma]\).
$$ \begin{aligned}[b] v^{(m)}(t+\tau)&=- \lambda f \bigl(u(t+ \tau) \bigr)u'(t+\tau) \\ &\quad{} -\lambda \bigl[g_{1} \bigl(t+\tau, u(t) \bigr)+g_{0} \bigl(u(t) \bigr) \bigr] \\ &\quad{} +\lambda e(t+\tau). \end{aligned} $$
(3.22)
$$ \begin{aligned}[b] v^{(m)}(t+\tau)u'(t)&=- \lambda f \bigl(u(t+\tau) \bigr)u'(t+\tau)u'(t) \\ &\quad{} -\lambda \bigl[g_{1} \bigl(t+\tau, u(t) \bigr)+g_{0} \bigl(u(t) \bigr) \bigr] u'(t) \\ & \quad{} +\lambda e(t+\tau)u'(t). \end{aligned} $$
(3.23)
$$\begin{aligned} \lambda \int_{u(\sigma)}^{u(t)}g_{0}(u)\,du&=\lambda \int_{\sigma}^{t}g_{0} \bigl(u(t) \bigr)u'(t)\,dt \\ &=- \int_{\sigma}^{t}v^{(m)}(t+ \tau)u'(t)\,dt - \lambda \int_{\sigma}^{t}f \bigl(u(t+\tau) \bigr)u'(t+ \tau)u'(t)\,dt \\ &\quad{} -\lambda \int_{\sigma}^{t}g_{1} \bigl(t+\tau,u(t) \bigr)u'(t)\,dt+\lambda \int_{\sigma}^{t} e(t +\tau)u'(t)\,dt, \end{aligned}$$
$$\begin{aligned} &\lambda \biggl\vert \int_{u(\tau)}^{u(t)}g_{0}(u)\,du \biggr\vert \\ &\quad = \lambda \biggl\vert \int_{\tau}^{t}g_{0} \bigl(u(t) \bigr)u'(t)\,dt \biggr\vert \\ &\quad \leq \int_{0}^{T} \bigl\vert v^{(m)}(t+\tau) \bigr\vert \bigl\vert u'(t) \bigr\vert \,dt + \lambda \int_{0}^{T} \bigl\vert f \bigl(u(t+\tau) \bigr) \bigr\vert \bigl\vert u'(t+\tau) \bigr\vert \bigl\vert u'(t) \bigr\vert \,dt \\ &\quad \quad{}+\lambda \int_{0}^{T} \bigl\vert g_{1} \bigl(t+ \tau,u(t) \bigr) \bigr\vert \bigl\vert u'(t) \bigr\vert \,dt+ \lambda \int_{0}^{T} \bigl\vert e(t +\tau) \bigr\vert \bigl\vert u'(t) \bigr\vert \,dt \\ &\quad \leq\lambda \bigl\vert u' \bigr\vert _{0} \bigl[ \alpha TM^{p-2}_{4} M_{1}(m)M_{3}+ \beta T M_{1}(m)M_{3}+ 2m_{0}T M^{p-1}_{4}+ 2Tm_{1} + \vert e \vert _{0}T \bigr] \\ &\quad \quad{}+\lambda \bigl\vert u' \bigr\vert ^{2}_{0} \int_{0}^{T} \bigl\vert f \bigl(u(t+\tau) \bigr) \bigr\vert \,dt+ \lambda \bigl\vert u' \bigr\vert _{0} \int_{0}^{T} \bigl\vert g_{1} \bigl(t+ \tau,u(t) \bigr) \bigr\vert \,dt \\ &\quad \quad{}+\lambda \bigl\vert u' \bigr\vert _{0} \int_{0}^{T} \bigl\vert e(t +\tau) \bigr\vert \,dt. \end{aligned}$$
$$F_{M_{4}}=\max_{ \vert u \vert \leq M_{4}} \bigl\vert f(u) \bigr\vert \quad\text{and} \quad G_{M_{4}}=\max_{t\in[0,T], \vert u \vert \leq M_{4}} \bigl\vert g_{1}(t,u) \bigr\vert , $$
$$\begin{aligned} \lambda \biggl\vert \int_{u(\tau)}^{u(t)}g_{0}(u)\,du \biggr\vert & \leq \lambda \bigl\vert u' \bigr\vert _{0} \bigl[ \alpha TM^{p-2}_{4} M_{1}(m)M_{3}+ \beta T M_{1}(m)M_{3} \bigr] \\ &\quad{} + \lambda \bigl\vert u' \bigr\vert _{0} \bigl[ 2m_{0}T M^{p-1}_{4}+ 2Tm_{1} + \vert e \vert _{0}T \bigr] \\ & \quad{} +\lambda \bigl\vert u' \bigr\vert ^{2}_{0} TF_{M_{4}}+\lambda \bigl\vert u' \bigr\vert _{0}TG_{M_{4}}+ \lambda \bigl\vert u' \bigr\vert _{0}T \vert e \vert _{0}, \end{aligned}$$
$$ \begin{aligned}[b] \biggl\vert \int_{u(\tau)}^{u(t)}g_{0}(u)\,du \biggr\vert & \leq M_{1}(m)M_{3} \bigl[ \alpha TM^{p-2}_{4} M_{1}(m)M_{3}+ \beta T M_{1}(m)M_{3} \bigr] \\ &\quad{} + M_{1}(m)M_{3} \bigl[2m_{0}T M^{p-1}_{4}+ 2Tm_{1} + \vert e \vert _{0}T \bigr] \\ & \quad{} + \bigl[M_{1}(m)M_{3} \bigr]^{2} TF_{M_{4}}+ M_{1}(m)M_{3}TG_{M_{4}} \\ &\quad{} + M_{1}(m)M_{3}T \vert e \vert _{0} \\ &< +\infty. \end{aligned} $$
(3.24)
$$ u(t)\geq M_{5}. $$
(3.25)
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. □
$$0< A_{1}=\min\{D_{1},M_{5}\},\quad\text{and}\quad A_{2}=\max\{D_{2},M_{4}\}. $$
$$ A_{1}\leq u(t)\leq A_{2}. $$
(3.26)
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
$$\Omega= \biggl\{ x=(u,v)^{\top}\in X:\frac{A_{1}}{2}< u(t)< A_{2}+1, \vert v \vert _{0}< A_{3}+1 \biggr\} . $$
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. □
$$\omega=Bx=B \begin{pmatrix}u \\v \end{pmatrix} = \begin{pmatrix}u-\frac{A_{1}+A_{2}}{2} \\v \end{pmatrix} . $$
$$J:\quad \operatorname {Im}Q\rightarrow\ker L,\quad\quad J (u,v)= \begin{pmatrix}v \\ -u \end{pmatrix} , $$
$$H(\mu,x)=\mu B x+(1-\mu) JQN x, \quad \forall(x,\mu)\in(\Omega\cap\ker L) \times[0,1]. $$
$$ \begin{aligned} K(\mu,x)= \begin{pmatrix}\mu u-\frac{\mu(A_{1}+A_{2})}{2} \\ \mu v \end{pmatrix} +\frac{1-\mu}{T} \begin{pmatrix}\int_{0}^{T}[f(u)u'+g(t, u)]\,dt \\ (1- \vert c \vert )^{-1}\int_{0}^{T}\varphi_{q}(v)\,dt \end{pmatrix} . \end{aligned} $$
(3.27)
$$ K(\mu_{0},x_{0})= \begin{pmatrix}\mu_{0} u_{0}-\frac{\mu_{0}(A_{1}+A_{2})}{2} +(1-\mu_{0}) \overline{g}(u_{0}) \\ \mu_{0} v_{0}+\frac{1-\mu_{0}}{1- \vert c \vert }\varphi_{q}(v_{0}) \end{pmatrix} . $$
(3.28)
$$\mu_{0} v_{0}+\frac{1-\mu_{0}}{1- \vert c \vert }\varphi_{q}(v_{0})=0, $$
$$ \mu_{0} u_{0}- \frac{\mu_{0}(A_{1}+A_{2})}{2} +(1- \mu_{0}) \overline{g} (u_{0})< 0 $$
(3.29)
$$ \mu_{0} u_{0}- \frac{\mu_{0}(A_{1}+A_{2})}{2} +(1- \mu_{0}) \overline{g} (u_{0})>0. $$
(3.30)
$$\begin{aligned} \deg(JQN ,\Omega\cap\ker L ,0)&=\deg \bigl(K(0,x),\Omega\cap\ker L ,0 \bigr) \\ &=\deg \bigl(K(1,x),\Omega\cap\ker L ,0 \bigr) \\ &=\deg(Bx,\Omega\cap\ker L ,0) \\ &=\sum_{x\in B^{-1}(0)} \operatorname {sgn}\bigl(\det B'(x) \bigr) \\ &=1\neq0. \end{aligned}$$
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:
$$ \begin{aligned}[b] & \biggl(\varphi_{4} \biggl( \biggl(x(t)- \frac{1}{3}x \biggl(t-\frac{\pi}{4} \biggr) \biggr)^{(3)} \biggr) \biggr)^{(3)}+ \biggl(\frac{u^{4}(t)}{6+u^{2}(t)} +3 \biggr)u'(t) \\ &\quad{} +\frac{1}{64}(1+\sin4t)u ^{3}(t-\tau)-\frac{1}{u^{3}(t-\tau)}= \frac{1}{16}\sin4t. \end{aligned} $$
(4.1)
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. □
$$\begin{aligned}& f \bigl(u(t) \bigr)=\frac{u^{4}}{6+u^{2}}+3,\quad u\in(0, +\infty), \quad\quad e(t)= \frac{1}{16}\sin4t, \\& g \bigl(t,u(t-\tau) \bigr)=\frac{1}{64}(1+\sin4t)u ^{3}(t- \tau)- \frac{1}{u^{3}(t-\tau)},\quad u\in(0, +\infty). \end{aligned}$$
$$\begin{aligned}& m=3,\quad\quad p=4, \quad\quad \gamma=\frac{\pi}{4},\quad\quad c=\frac{1}{3}, \quad\quad T=\frac{\pi}{2}, \\& \alpha=\frac{1}{6}, \quad\quad \beta=3, \quad\quad m_{0}=\frac{1}{32},\quad\quad D_{1}=1,\quad\quad D_{2}=9. \end{aligned}$$
$$\begin{aligned} \frac{M^{p}_{1}(m)T^{2p-1}(1+ \vert c \vert )(2m_{0}T+\alpha )}{(1- \vert c \vert )^{p}} \approx0.026 < 1. \end{aligned}$$
Example 4.2
Consider the following fourth-order p-Laplacian neutral functional differential equation:
$$ \begin{aligned}[b]& \biggl(\varphi_{4} \biggl( \biggl(x(t)- \frac{1}{10}x \biggl(t-\frac{\pi}{8} \biggr) \biggr)^{(4)} \biggr) \biggr)^{(4)}+ \biggl(\frac{u^{4}(t)}{8+u^{2}(t)} +9 \biggr)u'(t) \\ &\quad{} +\frac{1}{64}(1+\sin2t)u ^{3}(t-\tau)-\frac{1}{u^{3}(t-\tau)}= \frac{1}{32}\sin2t. \end{aligned} $$
(4.2)
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. □
$$\begin{aligned}& f \bigl(u(t) \bigr)=\frac{u^{4}}{8+u^{2}}+9,\quad u\in(0, +\infty), \quad\quad e(t)= \frac{1}{32}\sin2t, \\& g \bigl(t,u(t-\tau) \bigr)=\frac{1}{64}(1+\sin2t)u ^{3}(t- \tau)- \frac{1}{u^{3}(t-\tau)},\quad u\in(0, +\infty). \end{aligned}$$
$$\begin{aligned}& m=4, \quad\quad p=4, \quad\quad \gamma=\frac{\pi}{8}, \quad\quad c=\frac{1}{10}, \quad\quad T=\pi, \\& \alpha=\frac{1}{8}, \quad\quad \beta=9, \quad\quad m_{0}=\frac{1}{32},\quad\quad D_{1}=1, \quad\quad D_{2}=9. \end{aligned}$$
$$\begin{aligned} \frac{M^{p}_{1}(m)T^{2p-1}(1+ \vert c \vert )(2m_{0}T+\alpha )}{(1- \vert c \vert )^{p}} \approx0.207< 1. \end{aligned}$$
Example 4.3
Consider the following fourth-order p-Laplacian neutral functional differential equation:
$$ \begin{aligned}[b] & \biggl(\varphi_{2} \biggl( \biggl(x(t)- \frac{1}{10}x \biggl(t-\frac{\pi}{8} \biggr) \biggr)^{(4)} \biggr) \biggr)^{(4)}+ 9 u'(t) \\ &\quad{} +\frac{1}{64}(1+\sin4t)u(t-\tau)-\frac{1}{u(t-\tau)}=\frac{1}{32} \sin4t. \end{aligned} $$
(4.3)
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. □
$$\begin{aligned}& f \bigl(u(t) \bigr)= 9,\quad u\in(0, +\infty), \quad\quad e(t)=\frac{1}{32}\sin4t, \\& g \bigl(t,u(t-\tau) \bigr)=\frac{1}{64}(1+\sin4t)u(t-\tau)- \frac{1}{u(t-\tau)},\quad u\in(0, +\infty). \end{aligned}$$
$$\begin{aligned}& m=4,\quad\quad p=2, \quad\quad \gamma=\frac{\pi}{8}, \quad\quad c=\frac{1}{10}, \quad\quad T=\frac{\pi}{2}, \\& \alpha=0, \quad\quad \beta=9, \quad\quad m_{0}=\frac{1}{32}, \quad\quad D_{1}=1,\quad\quad D_{2}=9. \end{aligned}$$
$$\begin{aligned} \frac{M^{p}_{1}(m)T^{2p-1}(1+ \vert c \vert )(2m_{0}T+\alpha )}{(1- \vert c \vert )^{p}} \approx0.042< 1. \end{aligned}$$
Declarations
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).
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.
Authors’ Affiliations
(1)
Department of Mathematics and Physics, Hohai University
(2)
Department of Mathematics, Hunan Normal University
References
- Du, B: Periodic solutions to a p-Laplacian neutral Duffing equation with variable parameter. Electron. J. Qual. Theory Differ. Equ. 2011, 55 (2011) MathSciNetView ArticleMATHGoogle Scholar
- He, Z, Shen, J: Existence of periodic solutions for p-Laplacian neutral Rayleigh equation. Adv. Differ. Equ. 2014, 67 (2014) MathSciNetView ArticleGoogle Scholar
- Lu, S: Periodic solutions to a second order p-Laplacian neutral functional differential system. Nonlinear Anal. 69, 4215-4229 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Lu, S: Existence of periodic solutions of a p-Laplacian neutral functional differential equation. Nonlinear Anal. 70, 231-243 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Lu, S: On the existence of periodic solutions to a p-Laplacian neutral differential equation in the critical case. Nonlinear Anal., Real World Appl. 10, 2884-2893 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Liang, F, Guo, L, Lu, S: Existence of periodic solutions for a p-Laplacian neutral functional differential equation. Nonlinear Anal. 71, 427-436 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Wang, K, Zhu, Y: Periodic solutions for a fourth-order p-Laplacian neutral functional differential equation. J. Franklin Inst. 347, 1158-1170 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Wang, K, Zhu, Y: Periodic solutions for a higher order p-Laplacian neutral functional differential equation with a deviating argument. Nonlinear Anal., Theory Methods Appl. 71(9), 3906-3913 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Z, Song, C: Periodic solutions for the p-Laplacian neutral functional differential system. Adv. Differ. Equ. 2013, 367 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Yang, Q, Du, B: Periodic solutions to a generalized Liénard neutral functional differential system with p-Laplacian. J. Inequal. Appl. 2012, 270 (2012) View ArticleMATHGoogle Scholar
- Zhu, Y, Lu, S: Periodic solutions for p-Laplacian neutral functional differential equation with deviating arguments. J. Math. Anal. Appl. 325, 377-385 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Zhu, Y, Lu, S: Periodic solutions for p-Laplacian neutral functional differential equation with multiple deviating arguments. J. Math. Anal. Appl. 336, 1357-1367 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Fonda, A, Manásevich, R, Zanolin, F: Subharmonic solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 24, 1294-1311 (1993) MathSciNetView ArticleMATHGoogle Scholar
- Hakl, R, Torres, PJ, Zamora, M: Periodic solutions to singular second order differential equations: the repulsive case. Nonlinear Anal., Theory Methods Appl. 74, 7078-7093 (2011) View ArticleMATHGoogle Scholar
- Lazer, A, Solimini, S: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 88, 109-114 (1987) MathSciNetView ArticleMATHGoogle Scholar
- Lu, S, Kong, F: Periodic solutions for a kind of prescribed mean curvature Liénard equation with a singularity and a deviating argument. Adv. Differ. Equ. 2015, 151 (2015) View ArticleGoogle Scholar
- Lu, S, Zhong, T, Chen, L: Periodic solutions for p-Laplacian Rayleigh equations with singularities. Bound. Value Probl. 2016, 96 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Lu, S, Zhong, T, Gao, Y: Periodic solutions of p-Laplacian equations with singularities. Adv. Differ. Equ. 2016, 146 (2016) MathSciNetView ArticleGoogle Scholar
- Wang, Z: Periodic solutions of the second order differential equations with singularities. Nonlinear Anal., Theory Methods Appl. 58, 319-331 (2004) View ArticleMATHGoogle Scholar
- Wang, Z, Ma, T: Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities. Nonlinearity 25, 279-307 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Z: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal., Real World Appl. 16, 227-234 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, M: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203, 254-269 (1996) MathSciNetView ArticleMATHGoogle Scholar
- Lu, S, Ge, W: Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument. Nonlinear Anal., Theory Methods Appl. 56, 501-514 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Li, X, Lu, S: Periodic solutions for a kind of high-order p-Laplacian differential equation with sign-changing coefficient ahead of the nonlinear term. Nonlinear Anal. 70, 1011-1022 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Gaines, R, Mawhin, J: Coincidence Degree and Nonlinear Differential Equations. Lecture Notes in Math. Springer, Berlin (1977) View ArticleMATHGoogle Scholar
Copyright
© The Author(s) 2017
