Open Access

Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument

Advances in Difference Equations20162016:41

https://doi.org/10.1186/s13662-015-0721-2

Received: 22 May 2015

Accepted: 8 December 2015

Published: 3 February 2016

Abstract

In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument:
$$ \bigl(\varphi_{p}\bigl(x'(t)\bigr)\bigr)'+f \bigl(x(t)\bigr)x'(t)+g\bigl(t,x(t-\sigma)\bigr)=e(t). $$
By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established.

Keywords

positive solution p-Laplacian Liénard equation singularity deviating argument

MSC

34C25 34K13 34K40

1 Introduction

In this paper, we consider the following p-Laplacian Liénard type differential equation with singularity and deviating argument:
$$ \bigl(\varphi_{p}\bigl(x'(t)\bigr) \bigr)'+f\bigl(x(t)\bigr)x'(t)+g\bigl(t,x(t-\sigma) \bigr)=e(t), $$
(1.1)
where \(\varphi_{p}:\mathbb{R}\rightarrow\mathbb{R}\) is given by \(\varphi _{p}(s)=\vert s\vert ^{p-2}s\), here \(p>1\) is a constant, f is continuous function; g is a continuous function defined on \(\mathbb{R}^{2}\) and periodic in t with \(g(t,\cdot)=g(t+T,\cdot)\), g has a singularity at \(x=0\); σ is a constant and \(0\leq \sigma< T\); \(e:\mathbb{R}\rightarrow\mathbb{R}\) are continuous periodic functions with \(e(t+T)\equiv e(t)\) and \(\int^{T}_{0}e(t)\,dt=0\).
As is well known, the existence of periodic solutions for Liénard type differential equations was extensively studied (see [110] and the references therein). In recent years, there also appeared some results on a Liénard type differential equation with singularity; see [11, 12]. In 1996, using coincidence degree theory, Zhang considered the existence of T-periodic solutions for the scalar Liénard equation
$$ x''(t)+f\bigl(x(t)\bigr)x'(t)+g \bigl(t,x(t)\bigr)=0, $$
when g becomes unbounded as \(x\rightarrow0^{+}\). The main emphasis was on the repulsive case, i.e. when \(g(t,x)\rightarrow+\infty \), as \(x\rightarrow0^{+}\). Afterwards, Wang [12] studied the existence of periodic solutions of the Liénard equation with a singularity and a deviating argument,
$$x''(t)+ f\bigl(x(t)\bigr)x'(t)+ g \bigl(t,x(t-\sigma)\bigr)=0, $$
where σ is a constant. When g has a strong singularity at \(x = 0\) and satisfies a new small force condition at \(x =\infty\), the author proved that the given equation has at least one positive T-periodic solution.

However, the Liénard type differential equation (1.1), in which there is a p-Laplacian Liénard type differential equation, has not attracted much attention in the literature. There are not so many existence results for (1.1) even as regards the p-Laplacian Liénard type differential equation with singularity and deviating argument. In this paper, we try to fill this gap and establish the existence of a positive periodic solution of (1.1) using coincidence degree theory. Our new results generalize in several aspects some recent results contained in [11, 12].

2 Preparation

Let X and Y be real Banach spaces and \(L:D(L)\subset X\rightarrow Y\) be a Fredholm operator with index zero, here \(D(L)\) denotes the domain of L. This means that ImL is closed in Y and \(\dim \operatorname {Ker}L=\dim(Y/\operatorname {Im}L)<+\infty\). Consider supplementary subspaces \(X_{1}\), \(Y_{1}\) of X, Y, respectively, such that \(X=\operatorname {Ker}L \oplus X_{1}\), \(Y=\operatorname {Im}L\oplus Y_{1}\). Let \(P:X\rightarrow \operatorname {Ker}L\) and \(Q:Y\rightarrow Y_{1}\) denote the natural projections. Clearly, \(\operatorname {Ker}L\cap(D(L)\cap X_{1})=\{0\}\) and so the restriction \(L_{P}:=L|_{D(L)\cap X_{1}}\) is invertible. Let K denote the inverse of \(L_{P}\).

Let Ω be an open bounded subset of X with \(D(L)\cap\Omega\neq\emptyset\). A map \(N:\overline{\Omega}\rightarrow Y\) is said to be L-compact in Ω̅ if \(QN(\overline{\Omega})\) is bounded and the operator \(K(I-Q)N:\overline{\Omega}\rightarrow X\) is compact.

Lemma 2.1

(Gaines and Mawhin [13])

Suppose that X and Y are two Banach spaces, and \(L:D(L)\subset X\rightarrow Y\) is a Fredholm operator with index zero. Let \(\Omega\subset X\) be an open bounded set and \(N:\overline{\Omega}\rightarrow Y \) be L-compact on Ω̅. Assume that the following conditions hold:
  1. (1)

    \(Lx\neq\lambda Nx\), \(\forall x\in\partial\Omega\cap D(L)\), \(\lambda\in(0,1)\);

     
  2. (2)

    \(Nx\notin \operatorname {Im}L\), \(\forall x\in\partial\Omega\cap \operatorname {Ker}L\);

     
  3. (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 \(\overline{\Omega}\cap D(L)\).
For the sake of convenience, throughout this paper we will adopt the following notation:
$$\begin{aligned}& \vert u\vert _{\infty}=\max_{t\in[0,T]}\bigl\vert u(t) \bigr\vert ,\qquad \vert u\vert _{0}=\min_{t\in[0,T]}\bigl\vert u(t)\bigr\vert , \\& \vert u\vert _{p}= \biggl(\int^{T}_{0}\vert u\vert ^{p}\,dt \biggr)^{\frac{1}{p}}, \qquad \bar{h}=\frac{1}{T} \int^{T}_{0}h(t)\,dt. \end{aligned}$$

Lemma 2.2

([14])

If \(\omega\in C^{1}(\mathbb{R},\mathbb{R})\) and \(\omega(0)=\omega(T)=0\), then
$$\int^{T}_{0}\bigl\vert \omega(t)\bigr\vert ^{p}\,dt\leq \biggl(\frac{T}{\pi_{p}} \biggr)^{p} \int^{T}_{0}\bigl\vert \omega'(t) \bigr\vert ^{p}\,dt, $$
where \(1\leq p<\infty\), \(\pi_{p}=2\int^{(p-1)/p}_{0}\frac{ds}{(1-\frac{s^{p}}{p-1})^{1/p}}=\frac {2\pi(p-1)^{1/p}}{p\sin(\pi/p)}\).

Lemma 2.3

If \(x\in C^{1}(\mathbb{R},\mathbb{R})\) with \(x(t+T)=x(t)\), and \(t_{0}\in[0,T]\) such that \(\vert x(t_{0})\vert < d\), then
$$\biggl( \int^{T}_{0}\bigl\vert x(t)\bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}\leq \biggl(\frac{T}{\pi _{p}} \biggr) \biggl( \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}+dT^{\frac{1}{p}}. $$

Proof

Let \(\omega(t)=x(t+t_{0})-x(t_{0})\), and then \(\omega(0)=\omega(T)=0\). By Lemma 2.2 and Minkowski’s inequality, we have
$$\begin{aligned} \biggl( \int^{T}_{0}\bigl\vert x(t)\bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}&= \biggl( \int^{T}_{0}\bigl\vert \omega (t)+x(t_{0}) \bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \\ &\leq \biggl( \int^{T}_{0}\bigl\vert \omega(t)\bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}+ \biggl( \int ^{T}_{0}\bigl\vert x(t_{0})\bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \\ &\leq \biggl(\frac{T}{\pi_{p}} \biggr) \biggl( \int^{T}_{0}\bigl\vert \omega'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}+dT^{\frac{1}{p}} \\ &= \biggl(\frac{T}{\pi_{p}} \biggr) \biggl( \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{p}\,dt \biggr)^{\frac {1}{p}}+dT^{\frac{1}{p}}. \end{aligned}$$
This completes the proof of Lemma 2.3. □
In order to apply the topological degree theorem to study the existence of a positive periodic solution for (1.1), we rewrite (1.1) in the form
$$ \textstyle\begin{cases} x_{1}'(t)=\varphi_{q}(x_{2}(t)),\\ x_{2}'(t)=-f(x_{1}(t))x_{1}'(t)-g(t,x_{1}(t-\sigma))+e(t), \end{cases} $$
(2.1)
where \(\frac{1}{p}+\frac{1}{q}=1\). Clearly, if \(x(t)=(x_{1}(t),x_{2}(t))^{\top}\) is an T-periodic solution to (2.1), then \(x_{1}(t)\) must be an T-periodic solution to (1.1). Thus, the problem of finding an T-periodic solution for (1.1) reduces to finding one for (2.1).
Now, set \(X=Y=\{x=(x_{1}(t),x_{2}(t))\in C^{1}(\mathbb{R},\mathbb{R}^{2}): x(t+T)\equiv x(t)\}\) with the norm \(\Vert x\Vert =\max\{\vert x_{1}\vert _{\infty}, \vert x_{2}\vert _{\infty}\}\). Clearly, X and Y are both Banach spaces. Meanwhile, define
$$L:D(L)=\bigl\{ x\in C^{1}\bigl(\mathbb{R},\mathbb{R}^{2} \bigr): x(t+T) = x(t), t \in\mathbb {R}\bigr\} \subset X\rightarrow Y $$
by
$$ (Lx) (t)=\begin{pmatrix} x_{1}'(t)\\x_{2}'(t) \end{pmatrix} $$
and \(N: X\rightarrow Y\) by
$$ (Nx) (t)=\begin{pmatrix} \varphi_{q}(x_{2}(t))\\-f(x_{1}(t))x_{1}'(t)-g(t,x_{1}(t-\sigma))+e(t) \end{pmatrix}. $$
(2.2)
Then (2.1) can be converted to the abstract equation \(Lx=Nx\). From the definition of L, one can easily see that
$$\operatorname {Ker}L \cong\mathbb {R}^{2},\qquad \operatorname {Im}L= \Biggl\{ y\in Y: \int_{0}^{T} \begin{pmatrix} y_{1}(s)\\ y_{2}(s) \end{pmatrix} ds= \begin{pmatrix} 0\\ 0 \end{pmatrix} \Biggr\} . $$
So L is a Fredholm operator with index zero. Let \(P:X\rightarrow \operatorname {Ker}L\) and \(Q:Y\rightarrow \operatorname {Im}Q\subset\mathbb {R}^{2}\) be defined by
$$Px= \begin{pmatrix} (Ax_{1})(0)\\x_{2}(0) \end{pmatrix} ;\qquad Qy=\frac{1}{T} \int_{0}^{T} \begin{pmatrix} y_{1}(s)\\ y_{2}(s) \end{pmatrix} ds, $$
then \(\operatorname {Im}P=\operatorname {Ker}L\), \(\operatorname {Ker}Q=\operatorname {Im}L\). Let K denote the inverse of \(L|_{\operatorname {Ker}p\cap D(L)}\). It is easy to see that \(\operatorname {Ker}L=\operatorname {Im}Q=\mathbb{R}^{2}\) and
$$[Ky](t)= \int^{T}_{0}G(t,s)y(s)\,ds, $$
where
$$ G(t,s)= \textstyle\begin{cases} \frac{s}{T}, &0\leq s < t\leq T;\\ \frac{s-t}{T}, &0\leq t\leq s \leq T. \end{cases} $$
(2.3)
From (2.2) and (2.3), it is clear that QN and \(K(I-Q)N\) are continuous, \(QN(\overline{\Omega})\) is bounded and then \(K(I-Q)N(\overline{\Omega})\) is compact for any open bounded \(\Omega\subset X\), which means N is L-compact on Ω̅.

3 Main results

Assume that
$$ \psi(t)=\lim_{x\rightarrow+\infty}\sup\frac{g(t,x)}{x^{p-1}} $$
(3.1)
exists uniformly a.e. \(t\in[0,T]\), i.e., for any \(\varepsilon>0\) there is \(g_{\varepsilon}\in L^{2}(0,T)\) such that
$$ g(t,x)\leq\bigl(\psi(t)+\varepsilon\bigr)x+g_{\varepsilon}(t), $$
(3.2)
for all \(x>0\) and a.e. \(t\in[0,T]\). Moreover, \(\psi\in C(\mathbb{R},\mathbb{R})\) and \(\psi(t+T)=\psi(t)\).

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:

(H1) (Balance condition) There exist constants \(0< D_{1}< D_{2}\) such that if x is a positive continuous T-periodic function satisfying
$$\int^{T}_{0}g\bigl(t,x(t)\bigr)\,dt=0, $$
then
$$D_{1}\leq x(\tau)\leq D_{2}, $$
for some \(\tau\in[0,T]\).

(H2) (Degree condition) \(\bar{g}(x)<0\) for all \(x \in(0,D_{1})\), and \(\bar{g}(x)>0\) for all \(x>D_{2}\).

(H3) (Decomposition condition) \(g(t,x)=g_{0}(x)+g_{1}(t,x)\), where \(g_{0}\in C((0,\infty);\mathbb{R}) \) and \(g_{1}:[0,T]\times[0,\infty)\rightarrow\mathbb{R}\) is an \(L^{2}\)-Carathéodory function, i.e. it is measurable in the first variable and continuous in the second variable, and for any \(b>0\) there is \(h_{b}\in L^{2}(0,T;\mathbb{R}_{+})\) such that
$$\bigl\vert g_{1}(t,x)\bigr\vert \leq h_{b}(t),\quad \mbox{a.e. } t \in[0,T], \forall 0\leq x\leq b. $$

(H4) (Strong force condition at \(x=0\)) \(\int^{1}_{0}g_{0}(x)\,dx=-\infty\).

Theorem 3.1

Assume that conditions (H1)-(H4) hold. Suppose the following condition is satisfied:

(H5) \((\frac{T}{\pi_{p}} )^{p}\vert \psi \vert _{\infty}<1\).

Then (1.1) has at least one positive T-periodic solution.

Proof

Consider the equation
$$Lx=\lambda Nx,\quad \lambda\in(0,1). $$
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
$$ \textstyle\begin{cases} x_{1}'(t)=\lambda\varphi_{q}(x_{2}(t)),\\ x_{2}'(t)=-\lambda f(x_{1}(t))x_{1}'(t) -\lambda g(t,x_{1}(t-\sigma))+\lambda e(t). \end{cases} $$
(3.3)
Substituting \(x_{2}(t)=\frac{1}{\lambda^{p-1}}\varphi_{p}(x_{1}'(t))\) into the second equation of (3.3)
$$ \bigl(\varphi_{p}\bigl(x_{1}'(t) \bigr)\bigr)'+\lambda^{p}f\bigl(x_{1}(t) \bigr)x_{1}'(t)+\lambda ^{p}g \bigl(t,x_{1}(t-\sigma)\bigr)=\lambda^{p}e(t). $$
(3.4)
Integrating both sides of (3.4) over \([0,T]\), we have
$$\int^{T}_{0}g\bigl(t,x_{1}(t-\sigma)\bigr) \,dt=0. $$
(3.5)
From (H1), there exist positive constants \(D_{1}\), \(D_{2}\), and \(\xi\in[0,T]\) such that
$$ D_{1}\leq x_{1}(\xi)\leq D_{2}. $$
(3.6)
Then we have
$$\bigl\vert x_{1}(t)\bigr\vert =\biggl\vert x_{1}(\xi)+ \int^{t}_{\xi}x_{1}'(s)\,ds \biggr\vert \leq D_{2}+ \int^{t}_{\xi}\bigl\vert x_{1}'(s) \bigr\vert \,ds,\quad t\in[\xi,\xi+T], $$
and
$$\bigl\vert x_{1}(t)\bigr\vert =\bigl\vert x_{1}(t-T) \bigr\vert =\biggl\vert x_{1}(\xi)- \int_{t-T}^{\xi}x_{1}'(s)\,ds \biggr\vert \leq D_{2} + \int_{t-T}^{\xi}\bigl\vert x_{1}'(s) \bigr\vert \,ds,\quad t\in[\xi,\xi+T]. $$
Combining the above two inequalities, we obtain
$$ \begin{aligned}[b] \vert x_{1}\vert _{\infty}&=\max_{t\in[0,T]}\bigl\vert x_{1}(t)\bigr\vert =\max_{t\in[\xi,\xi+T]}\bigl\vert x_{1}(t)\bigr\vert \\ &\leq\max_{t\in[\xi,\xi+T]} \biggl\{ D_{2}+\frac{1}{2} \biggl( \int^{t}_{\xi}\bigl\vert x_{1}'(s) \bigr\vert \,ds+ \int^{\xi}_{t-T}\bigl\vert x_{1}'(s) \bigr\vert \,ds \biggr) \biggr\} \\ &\leq D_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x_{1}'(s) \bigr\vert \,ds. \end{aligned} $$
(3.7)
Multiplying both sides of (3.4) by \(x_{1}(t)\) and integrating over the interval \([0,T]\), we get
$$\begin{aligned} &\int^{T}_{0}\bigl(\varphi_{p} \bigl(x_{1}'(t)\bigr)\bigr)'x_{1}(t) \,dt+\lambda^{p} \int ^{T}_{0}f\bigl(x_{1}(t) \bigr)x_{1}'(t)x_{1}(t)\,dt+\lambda^{p} \int^{T}_{0}g\bigl(t,x_{1}(t-\sigma ) \bigr)x_{1}(t)\,dt \\ &\quad =\lambda^{p} \int^{T}_{0}e(t)x_{1}(t)\,dt. \end{aligned}$$
(3.8)
Substituting \(\int^{T}_{0}(\varphi_{p}(x_{1}'(t)))'x_{1}(t)\,dt=-\int^{T}_{0}\vert x_{1}'(t)\vert ^{p}\,dt\), \(\int ^{T}_{0}f(x_{1}(t))x_{1}'(t)x_{1}(t)\,dt=0\) into (3.8), we have
$$ \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}d=\lambda^{p} \int^{T}_{0}g\bigl(t,x_{1}(t-\sigma) \bigr)x_{1}(t)\, dt-\lambda^{p} \int^{T}_{0}e(t)x_{1}(t)\,dt. $$
(3.9)
For any \(\varepsilon>0\), there exists a function \(g_{\varepsilon}\in L^{2}(0,T)\) such that (3.2) holds. Since \(x_{1}(t)>0\), \(t\in[0,T]\), it follows from (3.4) that
$$ g\bigl(t,x_{1}(t-\sigma)\bigr)x_{1}(t)\leq \bigl(\psi(t)+\varepsilon\bigr)x_{1}^{p-1}(t-\sigma )x_{1}(t)+g_{\varepsilon}(t)x_{1}(t). $$
(3.10)
We infer from (3.9) and (3.10)
$$\begin{aligned} &\int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \\ &\quad \leq\lambda^{p} \int^{T}_{0}\bigl(\psi(t)+\varepsilon \bigr)x_{1}^{p-1}(t-\sigma)x_{1}(t)\,dt + \lambda^{p} \int^{T}_{0}\bigl(g_{\varepsilon}(t)+e(t) \bigr)x_{1}(t)\,dt \\ &\quad \leq \int^{T}_{0}\bigl(\bigl\vert \psi(t)\bigr\vert + \varepsilon\bigr)\bigl\vert x_{1}^{p-1}(t-\sigma)\bigr\vert \bigl\vert x_{1}(t)\bigr\vert \,dt + \int^{T}_{0}\bigl(\bigl\vert g_{\varepsilon}(t) \bigr\vert +\bigl\vert e(t)\bigr\vert \bigr)\bigl\vert x_{1}(t) \bigr\vert \,dt \\ &\quad \leq\bigl(\vert \psi \vert _{\infty}+\varepsilon\bigr) \biggl( \int^{T}_{0}\bigl\vert x_{1}(t-\sigma) \bigr\vert ^{p}\, dt \biggr)^{\frac{p-1}{p}} \biggl( \int^{T}_{0}\bigl\vert x_{1}(t)\bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \\ &\qquad {}+\vert x_{1}\vert _{\infty}\biggl( \int^{T}_{0}\bigl\vert g_{\varepsilon}(t)\bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t)\bigr\vert \, dt \biggr) \\ &\quad \leq\bigl(\vert \psi \vert _{\infty}+\varepsilon\bigr) \biggl( \int^{T}_{0}\bigl\vert x_{1}(t)\bigr\vert ^{p}\,dt \biggr) +\vert x_{1}\vert _{\infty}\biggl( \int^{T}_{0}\bigl\vert g_{\varepsilon}(t)\bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t)\bigr\vert \, dt \biggr). \end{aligned}$$
(3.11)
From Lemma 2.3 and (3.7), we have
$$ \biggl( \int^{T}_{0}\bigl\vert x_{1}(t)\bigr\vert ^{p} \biggr)^{\frac{1}{p}}\leq \biggl(\frac{T}{\pi _{p}} \biggr) \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}+D_{2}T^{\frac{1}{p}}. $$
(3.12)
Substituting (3.7), (3.12) into (3.11), we get
$$\begin{aligned} &\int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \\ &\quad \leq\bigl(\vert \psi \vert _{\infty}+ \varepsilon\bigr) \biggl( \biggl(\frac {T}{\pi_{p}} \biggr) \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}+D_{2}T^{\frac {1}{p}} \biggr)^{p} \\ &\qquad {}+ \biggl(D_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert \,dt \biggr) \biggl( \int^{T}_{0}\bigl\vert g_{\varepsilon}(t)\bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t)\bigr\vert \,dt \biggr) \\ &\quad \leq\bigl(\vert \psi \vert _{\infty}+\varepsilon\bigr) \biggl( \biggl( \frac{T}{\pi_{p}} \biggr)^{p} \int ^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \\ &\qquad {}+p \biggl(\frac{T}{\pi_{p}} \biggr)^{p-1} \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{p-1}{p}}D_{2}T^{\frac {1}{p}} +\cdots+D_{2}^{p}T \biggr) \\ &\qquad {}+ \biggl(D_{2}+ \frac{1}{2}T^{\frac{1}{q}} \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} \biggr) \bigl(T^{\frac{1}{2}}\bigl(\vert g_{\varepsilon} \vert _{2}+\vert e\vert _{2}\bigr) \bigr) \\ &\quad =\bigl(\vert \psi \vert _{\infty}+\varepsilon\bigr) \biggl( \frac{T}{\pi_{p}} \biggr)^{p} \int ^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \\ &\qquad {}+\bigl(\vert \psi \vert _{\infty}+ \varepsilon\bigr)p \biggl(\frac{T}{\pi _{p}} \biggr)^{p-1} \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{p-1}{p}}D_{2}T^{\frac{1}{p}}+\cdots \\ &\qquad {}+\frac{1}{2}T^{\frac{1}{q}+\frac{1}{2}} \biggl( \int ^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}}\bigl(\vert g_{\varepsilon} \vert _{2}+\vert e\vert _{2}\bigr) \\ &\qquad {}+ \bigl(\vert \psi \vert _{\infty}+\varepsilon\bigr)D_{2}^{p}T +T^{\frac{1}{2}}D_{2}\bigl(\vert g_{\varepsilon} \vert _{2}+\vert e\vert _{2}\bigr), \end{aligned}$$
(3.13)
where \(\vert g_{\varepsilon} \vert _{2}= (\int^{T}_{0}\vert g_{\varepsilon}(t)\vert ^{2}\,dt )^{\frac{1}{2}}\). Since ε is sufficiently small, from (H5) we know that \((\frac{T}{\pi_{p}} )^{p}\vert \psi \vert _{\infty}<1\). So, it is easy to see that there exists a positive constant \(M'_{1}\) such that
$$\int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt\leq M'_{1}. $$
From (3.7), we have
$$ \begin{aligned}[b] \vert x_{1}\vert _{\infty}&\leq D_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert \,dt\\ &\leq D_{2}+\frac{T^{\frac{1}{q}}}{2} \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac {1}{p}} \\ &\leq D_{2}+\frac{T^{\frac{1}{q}}}{2} \bigl(M_{1}' \bigr)^{\frac{1}{p}}:=M_{1}. \end{aligned} $$
(3.14)
Write
$$I_{+}=\bigl\{ t\in[0,T]:g\bigl(t,x_{1}(t-\sigma)\bigr)\geq0\bigr\} ;\qquad I_{-}= \bigl\{ t\in [0,T]:g\bigl(t,x_{1}(t-\sigma)\bigr)\leq0\bigr\} . $$
Then we get from (3.2) and (3.6)
$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(t,x_{1}(t- \sigma)\bigr)\bigr\vert \,dt&= \int_{I_{+}}g\bigl(t,x_{1}(t-\sigma)\bigr)\,dt- \int _{I_{-}}g\bigl(t,x_{1}(t-\sigma)\bigr)\,dt \\ &=2 \int_{I_{+}}g\bigl(t,x_{1}(t-\sigma)\bigr)\,dt \\ &\leq2 \int_{I_{+}}\bigl(\bigl(\psi(t)+\varepsilon\bigr)x_{1}^{p-1}(t- \sigma)+g_{\varepsilon}(t)\bigr)\,dt \\ &\leq2\bigl(\vert \psi \vert _{\infty}+\varepsilon\bigr) \int^{T}_{0}\bigl\vert x_{1}(t)\bigr\vert ^{p-1}\,dt+2 \int ^{T}_{0}\bigl\vert g_{\varepsilon}(t)\bigr\vert \,dt \\ &\leq2\bigl(\vert \psi \vert _{\infty}+\varepsilon\bigr)TM_{1}^{p-1}+2 \sqrt{T} \vert g_{\varepsilon} \vert _{2}. \end{aligned}$$
(3.15)
By the second equations of (3.3) and (3.15), we obtain
$$\begin{aligned} &\int^{T}_{0}\bigl\vert x_{2}'(t) \bigr\vert \,dt \\ &\quad \leq\lambda \int^{T}_{0}\bigl\vert f\bigl(x_{1}(t) \bigr)\bigr\vert \bigl\vert x_{1}'(t)\bigr\vert \, dt+\lambda \int^{T}_{0}\bigl\vert g\bigl(t,x_{1}(t- \sigma)\bigr)\bigr\vert \,dt+\lambda \int^{T}_{0}\bigl\vert e(t)\bigr\vert \,dt \\ &\quad \leq\lambda \vert f\vert _{M_{1}}T^{\frac{1}{q}} \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac{1}{p}} +\lambda\bigl(2\bigl( \vert \psi \vert _{\infty}+\varepsilon\bigr)TM_{1}^{p-1}+2 \sqrt{T} \vert g_{\varepsilon} \vert _{2}\bigr)+\lambda\sqrt{T} \vert e\vert _{2} \\ &\quad \leq\lambda \vert f\vert _{M_{1}}T^{\frac{1}{q}} \bigl(M_{1}' \bigr)^{\frac{1}{p}} +\lambda\bigl(2\bigl( \vert \psi \vert _{\infty}+\varepsilon\bigr)TM_{1}^{p-1}+2 \sqrt{T} \vert g_{\varepsilon} \vert _{2}\bigr)+\lambda\sqrt{T} \vert e\vert _{2} \\ &\quad :=\lambda M_{2}', \end{aligned}$$
(3.16)
where \(\vert f\vert _{M_{1}}=\max_{0< x_{1}\leq M_{1}}\vert f(x_{1}(t))\vert \). By the first equation of (3.3), we have
$$\int^{T}_{0}\bigl\vert x_{2}(s)\bigr\vert ^{q-2}x_{2}(s)\,ds=0, $$
which implies that there is a constant \(t_{2}\in[0,T]\) such that \(x_{2}(t_{2})=0\), so
$$ \vert x_{2}\vert _{\infty}\leq \frac{1}{2} \int^{t_{2}}_{0}\bigl\vert x_{2}'(s)\bigr| \,ds\leq\frac{1}{2} \int ^{T}_{0}\bigl\vert x_{2}'(s)\bigr| \,ds\leq\frac{\lambda}{2}M_{2}':=\lambda M_{2}. $$
(3.17)
On the other hand, it follows from (3.4) that
$$ \bigl(\varphi_{p}\bigl(x_{1}'(t+ \sigma)\bigr)\bigr)'+\lambda^{p}\bigl(f \bigl(x_{1}(t+\sigma)\bigr)x_{1}'(t+\sigma )+g \bigl(t+\sigma,x_{1}(t)\bigr)\bigr)=\lambda^{p} e(t+\sigma). $$
(3.18)
Namely,
$$\begin{aligned} & \bigl(\varphi_{p}\bigl(x_{1}'(t+ \sigma)\bigr)\bigr)'+\lambda^{p}f\bigl(x_{1}(t+ \sigma)\bigr)x_{1}'(t+\sigma ) \\ &\qquad{} +\lambda^{p}g_{0} \bigl(x_{1}(t)\bigr)+g_{1}\bigl(t+\sigma,x_{1}(t) \bigr)=\lambda^{p} e(t+\sigma). \end{aligned}$$
(3.19)
Multiplying both sides of (3.19) by \(x_{1}'(t)\), we get
$$ \begin{aligned}[b] &\bigl(\varphi_{p}\bigl(x_{1}'(t+ \sigma)\bigr)\bigr)'x_{1}'(t)+ \lambda^{p}f\bigl(x_{1}(t+\sigma )\bigr)x_{1}'(t+ \sigma)x_{1}'(t)\\ &\qquad{}+\lambda^{p}g_{0} \bigl(x_{1}(t)\bigr)x_{1}'(t)+\lambda ^{p}g_{1}\bigl(t+\sigma,x_{1}(t) \bigr)x_{1}'(t)=\lambda^{p} e(t+ \sigma)x_{1}'(t). \end{aligned} $$
(3.20)
Let \(\tau\in[0,T]\), for any \(\tau\leq t\leq T\), we integrate (3.20) on \([\tau, t]\) and get
$$\begin{aligned} \lambda^{p} \int^{x_{1}(t)}_{x_{1}(\tau)}g_{0}(u)\,du={}& \lambda^{p} \int^{t}_{\tau }g_{0}\bigl(x_{1}(s) \bigr)x_{1}'(s)\,ds \\ ={}&{-} \int^{t}_{\tau}\bigl(\varphi_{p} \bigl(x_{1}'(s+\sigma)\bigr)\bigr)'x_{1}'(s) \,ds-\lambda^{p} \int ^{t}_{\tau}f\bigl(x_{1}(s+\sigma) \bigr)x_{1}'(s+\sigma)x_{1}'(s)\,ds \\ &{}-\lambda^{p} \int^{t}_{\tau}g_{1}\bigl(s+ \sigma,x_{1}(s)\bigr)x_{1}'(s)\,ds+ \lambda^{p} \int ^{t}_{\tau}e(s+\sigma)x_{1}'(s) \,ds. \end{aligned}$$
(3.21)
By (3.14), (3.15), (3.16), (3.17), and (3.18), we have
$$\begin{aligned} &\biggl\vert \int^{t}_{\tau}\bigl(\varphi_{p} \bigl(x_{1}'(t+\sigma)\bigr)\bigr)'x_{1}'(s) \,ds\biggr\vert \\ &\quad \leq \int^{t}_{\tau}\bigl\vert \bigl(\varphi_{p} \bigl(x_{1}'(t+\sigma)\bigr)\bigr)'\bigr\vert \bigl\vert x_{1}'(s)\bigr\vert \,ds \\ &\quad \leq\bigl\vert x_{1}'\bigr\vert _{\infty}\int^{T}_{0}\bigl\vert \bigl(\varphi_{p} \bigl(x_{1}'(t+\sigma)\bigr)\bigr)'\bigr\vert \,dt \\ &\quad \leq\lambda^{p}\bigl\vert x_{1}'\bigr\vert _{\infty}\biggl( \int^{T}_{0}\bigl\vert f\bigl(x_{1}(t) \bigr)\bigr\vert \bigl\vert x_{1}'(t)\bigr\vert \,dt+ \int^{T}_{0}\bigl\vert g\bigl(t,x_{1}(t- \sigma)\bigr)\bigr\vert \, dt+ \int^{T}_{0}\bigl\vert e(t)\bigr\vert \,dt \biggr) \\ &\quad \leq\lambda^{p} M_{2}^{p-1}\bigl(\vert f\vert _{M_{1}} M_{1}^{\prime\frac{1}{p}}T^{\frac{1}{q}}+2 \bigl(\vert \psi \vert _{\infty}+\varepsilon \bigr)TM_{1}^{p-1}+2T^{\frac{1}{2}} \bigl\vert g_{\varepsilon}^{+}\bigr\vert _{2}+T^{\frac{1}{2}}\vert e\vert _{2}\bigr). \end{aligned}$$
We have
$$\begin{aligned}& \begin{aligned} \biggl\vert \int^{t}_{\tau}f\bigl(x_{1}(s+\sigma) \bigr)x_{1}'(s+\sigma)x_{1}'(s)\,ds \biggr\vert &\leq \vert f\vert _{M_{1}} \biggl( \int^{T}_{0}\bigl\vert x_{1}'(s) \bigr\vert \,ds \biggr)^{2}\\ &\leq \vert f\vert _{M_{1}}T^{\frac{2}{q}} \biggl( \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert ^{p}\,dt \biggr)^{\frac {2}{p}} \\ &\leq \vert f\vert _{M_{1}}T^{\frac{2}{q}}\bigl(M_{1}' \bigr)^{\frac{2}{p}}, \end{aligned} \\& \biggl\vert \int^{t}_{\tau}g\bigl(s+\sigma,x_{1}(s) \bigr)x_{1}'(s)\,ds\biggr\vert \leq\bigl\vert x_{1}'\bigr\vert \int ^{T}_{0}\bigl\vert g\bigl(t,x(t-\sigma)\bigr)\bigr\vert \,dt\leq M_{2}^{p-1}\sqrt{T} \vert g_{M_{1}} \vert _{2}, \end{aligned}$$
where \(g_{M_{1}}=\max_{0\leq x\leq M_{1}}\vert g_{1}(t,x)\vert \in L^{2}(0,T)\) is as in (H3). We have
$$\biggl\vert \int^{t}_{\tau}e(t+\sigma)x_{1}'(t) \,dt\biggr\vert \leq M_{2}^{p-1}T^{\frac{1}{2}}\vert e \vert _{2}. $$
From these inequalities we can derive from (3.21) that
$$ \biggl\vert \int^{x_{1}(t)}_{x_{1}(\tau)}g_{0}(u)\,du\biggr\vert \leq M_{3}', $$
(3.22)
for some constant \(M_{3}'\) which is independent on λ, x, and t. In view of the strong force condition (H4), we know that there exists a constant \(M_{3}>0\) such that
$$ x_{1}(t)\geq M_{3},\quad \forall t\in[\tau,T]. $$
(3.23)
The case \(t\in[0,\tau]\) can be treated similarly.
From (3.14), (3.17), and (3.23), we let
$$\Omega=\bigl\{ x=(x_{1},x_{2})^{\top}: E_{1}\leq \vert x_{1}\vert _{\infty}\leq E_{2}, \vert x_{2}\vert _{\infty}\leq E_{3}, \forall t\in[0,T]\bigr\} , $$
where \(0< E_{1}<\min(M_{3}, D_{1})\), \(E_{2}>\max(M_{1}, D_{2}) \), \(E_{3}>M_{2}\). \(\Omega_{2}=\{x:x\in\partial\Omega\cap \operatorname {Ker}L\}\) then \(\forall x\in \partial\Omega\cap \operatorname {Ker}L\)
$$ QNx=\frac{1}{T} \int^{T}_{0} \begin{pmatrix}\varphi_{q}(x_{2}(t)) \\ -f(x_{1}(t))x_{1}'(t)-g(t,x_{1}(t-\sigma))+e(t) \end{pmatrix} \,dt. $$
If \(QNx=0\), then \(x_{2}(t)=0\), \(x_{1}=E_{2}\) or \(-E_{2}\). But if \(x_{1}(t)=E_{2}\), we know
$$0= \int^{T}_{0}\bigl\{ g(t,E_{2})-e(t)\bigr\} \,dt. $$
From assumption (H2), we have \(x_{1}(t)\leq D_{2}\leq E_{2}\), which yields a contradiction. Similarly if \(x_{1}=-E_{2}\). We also have \(QNx\neq0\), i.e., \(\forall x\in\partial\Omega\cap \operatorname {Ker}L\), \(x\notin \operatorname {Im}L\), so conditions (1) and (2) of Lemma 2.1 are both satisfied. Define the isomorphism \(J:\operatorname {Im}Q\rightarrow \operatorname {Ker}L\) as follows:
$$J(x_{1},x_{2})^{\top}=(x_{2},-x_{1})^{\top}. $$
Let \(H(\mu,x)=-\mu x+(1-\mu)JQNx\), \((\mu,x)\in[0,1]\times\Omega\), then \(\forall (\mu,x)\in(0,1)\times(\partial\Omega\cap \operatorname {Ker}L)\),
$$ H(\mu,x)= \begin{pmatrix}-\mu x_{1}-\frac{1-\mu}{T}\int^{T}_{0}[g(t,x_{1})-e(t)]\,dt\\ -\mu x_{2}-(1-\mu)\vert x_{2}\vert ^{p-2}x_{2} \end{pmatrix} . $$
We have \(\int^{T}_{0}e(t)\,dt=0\). So, we can get
$$\begin{aligned}& H(\mu,x)= \begin{pmatrix}-\mu x_{1}-\frac{1-\mu}{T}\int^{T}_{0}g(t,x_{1})\,dt\\ -\mu x_{2}-(1-\mu)\vert x_{2}\vert ^{p-2}x_{2} \end{pmatrix} , \\& \quad \forall (\mu,x)\in(0,1)\times(\partial\Omega\cap \operatorname {Ker}L). \end{aligned}$$
From (H2), it is obvious that \(x^{\top}H(\mu,x)<0\), \(\forall (\mu,x)\in(0,1)\times(\partial\Omega\cap \operatorname {Ker}L)\). Hence
$$\begin{aligned} \deg\{JQN,\Omega\cap \operatorname {Ker}L,0\}&=\deg\bigl\{ H(0,x),\Omega\cap \operatorname {Ker}L,0\bigr\} \\ &=\deg\bigl\{ H(1,x),\Omega\cap \operatorname {Ker}L,0\bigr\} \\ &=\deg\{I,\Omega\cap \operatorname {Ker}L,0\}\neq0. \end{aligned}$$
So condition (3) of Lemma 2.1 is satisfied. By applying Lemma 2.1, we conclude that the equation \(Lx=Nx\) has a solution \(x=(x_{1},x_{2})^{\top}\) on \(\bar{\Omega}\cap D(L)\), i.e., (2.1) has an T-periodic solution \(x_{1}(t)\). □

Finally, we present an example to illustrate our result.

Example 3.1

Consider the p-Laplacian Liénard type differential equation with singularity and deviating argument:
$$ \bigl(\varphi_{p}\bigl(x'(t)\bigr) \bigr)'+f\bigl(x(t)\bigr)x'(t)+\frac{1}{5}( \cos2t+2)x^{3}(t-\sigma )-\frac{1}{x^{\kappa}(t-\sigma)}=\sin 2t, $$
(3.24)
where \(\kappa\geq1\) and \(p=4\), f is a continuous function, σ is a constant, and \(0\leq\sigma< T\).
It is clear that \(T=\pi\), \(g(t,x)=\frac{1}{5}(\cos2t+2)x^{3}(t-\sigma)-\frac{1}{x^{\kappa}(t-\sigma)}\), \(\psi(t)=\frac{1}{5}(\cos2t+2)\). It is obvious that (H1)-(H4) hold. Now we consider the assumption (H5). Since \(\vert \psi \vert _{\infty}\leq\frac{3}{5}\), we have
$$\begin{aligned} \biggl(\frac{T}{\pi_{p}} \biggr)^{p}\vert \psi \vert _{\infty}= \biggl(\frac{T}{\frac{2\pi (p-1)^{1/p}}{p\sin(\pi/p)}} \biggr)^{p}\vert \psi \vert _{\infty}\leq \biggl(\frac{\pi}{\frac{2\pi(4-1)^{1/4}}{4\sin{\pi/4}}} \biggr)^{4}\times \frac{3}{5}=\frac{4}{5}< 1. \end{aligned}$$

So by Theorem 3.1, we know (3.24) has at least one positive π-periodic solution.

Declarations

Acknowledgements

YX and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC Project (No. 11501170), Fundamental Research Funds for the Universities of Henan Province (NSFRF140142), Henan Polytechnic University Outstanding Youth Fund (J2015-02) and Henan Polytechnic University Doctor Fund (B2013-055).

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)
College of Computer Science and Technology, Henan Polytechnic University
(2)
School of Mathematics and Information Science, Henan Polytechnic University

References

  1. Atslega, S, Sadyrbaev, F: On periodic solutions of Liénard type equations. Math. Model. Anal. 18, 708-716 (2013) View ArticleMathSciNetMATHGoogle Scholar
  2. Cheung, WS, Ren, JL: On the existence of periodic solutions for p-Laplacian generalized Liénard equation. Nonlinear Anal. TMA 60, 65-75 (2005) MathSciNetMATHGoogle Scholar
  3. Feng, MQ: Periodic solutions for prescribed mean curvature Liénard equation with a deviating argument. Nonlinear Anal., Real World Appl. 13, 1216-1223 (2012) View ArticleMathSciNetMATHGoogle Scholar
  4. Gao, FB, Lu, SP: Existence of periodic solutions for a Liénard type p-Laplacian differential equation with a deviating argument. Nonlinear Anal. 69, 4754-4763 (2008) View ArticleMathSciNetMATHGoogle Scholar
  5. Gutiérrez, A, Torres, P: Periodic solutions of Liénard equation with one or two weak singularities. Differ. Equ. Appl. 3, 375-384 (2011) MathSciNetMATHGoogle Scholar
  6. Liu, WB: Existence and uniqueness of periodic solutions for a kind of Liénard type p-Laplacian equation. Nonlinear Anal. TMA 69, 724-729 (2008) View ArticleMATHGoogle Scholar
  7. Liu, WB, Liu, JY, Zhang, HX, Hu, ZG, Wu, YQ: Existence of periodic solutions for Liénard-type p-Laplacian systems with variable coefficients. Ann. Pol. Math. 109, 109-119 (2013) View ArticleMathSciNetMATHGoogle Scholar
  8. Ma, TT, Wang, ZH: Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete Contin. Dyn. Syst. 33, 1563-1581 (2013) MathSciNetMATHGoogle Scholar
  9. Meng, H, Long, F: Periodic solutions for a Liénard type p-Laplacian differential equation. J. Comput. Appl. Math. 224, 696-701 (2009) View ArticleMathSciNetMATHGoogle Scholar
  10. Tiryaki, A, Zafer, A: Global existence and boundedness for a class of second-order nonlinear differential equations. Appl. Math. Lett. 26, 939-944 (2013) View ArticleMathSciNetMATHGoogle Scholar
  11. Zhang, MR: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203, 254-269 (1996) View ArticleMathSciNetMATHGoogle Scholar
  12. Wang, ZH: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal., Real World Appl. 16, 227-234 (2014) View ArticleMathSciNetMATHGoogle Scholar
  13. Gaines, R, Mawhin, J: Coincidence Degree and Nonlinear Differential Equations. Springer, Berlin (1977) MATHGoogle Scholar
  14. Zhang, MR: Nonuniform nonresonance at the first eigenvalue of the p-Laplacian. Nonlinear Anal. TMA 29, 41-51 (1996) View ArticleGoogle Scholar

Copyright

© Xin and Cheng 2016