Studies on a 2nth-order p-Laplacian differential equation with singularity

Abstract

In this paper, we consider the 2nth-order p-Laplacian differential equation with singularity

$$ \bigl(\varphi_{p} \bigl(x(t) \bigr)^{(n)} \bigr)^{(n)}+f \bigl(x(t) \bigr)x'(t)+g \bigl(t,x(t-\sigma) \bigr)=e(t). $$

By applications of coincidence degree theory and some analysis techniques, sufficient conditions for the existence of positive periodic solutions are established.

1 Introduction

Generally speaking, differential equations with singularities have been considered from the very beginning of the discipline. The main reason is that singular forces are ubiquitous in applications, gravitational and electromagnetic forces being the most obvious examples. In 1979, Taliaferro [1] discussed the model equation with singularity

$$ y''+\frac{q(t)}{y^{\alpha}}=0,\quad 0< t< 1, $$

(1.1)

subject to

$$ y(0)=0=y(1), $$

and obtained the existence of a solution for the problem. Here $\alpha>0$, $q\in C(0,1)$ with $q>0$ on $(0,1)$ and $\int_{0}^{1} t(1-t)q(t)\,dt<\infty$. We call it the equation with the strong force condition if $\alpha\geq1$ and we call it the equation with the weak force condition if $0<\alpha<1$.

Ding’s work has attracted the attention of many specialists in differential equations. More recently, topological degree theory [2–4], the Schauder fixed point theorem [5, 6], the Krasnoselskii fixed point theorem in a cone [7–9], the Poincaré-Birkhoff twist theorem [10–12], and the Leray-Schauder alternative principle [13–15] have been employed to investigate the existence of positive periodic solutions of singular second-order, third-order and fourth-order differential equations. In 1996, using coincidence degree theory, Zhang [2] 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 $x\rightarrow0^{+}$. In 2007, Torres [5] studied singular forced semilinear differential equation

$$ x''+a(t)x'=f(t,x)+e(t). $$

(1.2)

By the Schauder fixed point theorem, the author has shown that the additional assumption of a weak singularity enabled new criteria for the existence of periodic solutions. Afterwards, Wang [3] investigated the existence and multiplicity of positive periodic solutions of the singular systems (1.2) by the Krasnoselskii fixed point theorem. The conditions he presented to guarantee the existence of positive periodic solutions are beautiful. Recently, Cheng and Ren [14] discussed a kind of fourth-order singular differential equation,

$$ x^{(4)}(t)+ax'''(t)+bx''(t)+cx'(t)+dx(t)=f \bigl(t,x(t) \bigr)+e(t). $$

(1.3)

By application of Green’s function and some fixed point theorems, i.e., the Leray-Schauder alternative principle and Schauder’s fixed point theorem, the authors established two existence results of positive periodic solutions for nonlinear fourth-order singular differential equation.

Motivated by [2, 3, 5, 14], in this paper, we consider the high-order p-Laplacian differential equation with singularity

$$ \bigl(\varphi_{p} \bigl(x(t) \bigr)^{(n)} \bigr)^{(n)}+f \bigl(x(t) \bigr)x'(t)+g \bigl(t,x(t-\sigma) \bigr)=e(t), $$

(1.4)

where $p\geq2$, $\varphi_{p}(x)=|x|^{p-2}x$ for $x\neq0$, and $\varphi_{p}(0)=0$; g is 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 $\int^{T}_{0}e(t)\,dt=0$. T is a positive constant; n is positive integer.

The paper is organized as follows. In Section 2, we introduce some technical tools and present all the auxiliary results; in Section 3, by applying coincidence degree theory and some new inequalities, we obtain sufficient conditions for the existence of positive periodic solutions for (1.4), an example is also given to illustrate our results. Our new results generalize in several aspects some recent results contained in [2, 3, 5].

2 Lemmas

For the sake of convenience, throughout this paper we will adopt the following notation:

$$\begin{aligned}& |u|_{\infty}=\max_{t\in[0,T]}\bigl|u(t)\bigr|,\qquad |u|_{0}=\min _{t\in[0,T]}\bigl|u(t)\bigr|,\\& |u|_{p}= \biggl( \int^{T}_{0}|u|^{p}\,dt \biggr)^{\frac{1}{p}}, \qquad\bar{h}=\frac{1}{T} \int^{T}_{0}h(t)\,dt. \end{aligned}$$

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 [16])

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

• (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 2.2

([17])

If $\omega\in C^{1}(\mathbb{R},\mathbb{R})$ and $\omega(0)=\omega(T)=0$, then

$$\int^{T}_{0}\bigl|\omega(t)\bigr|^{p}\,dt\leq \biggl(\frac{T}{\pi_{p}} \biggr)^{p} \int^{T}_{0}\bigl|\omega'(t)\bigr|^{p} \,dt, $$

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 $x(t)\in C^{n}(\mathbb{R},\mathbb{R})$ and $x^{(j)}(t+T)=x^{(j)}(t)$, $j=0,1,2,\ldots,n-1$, then

$$ \int^{T}_{0}\bigl|x^{(i)}(t)\bigr|^{p} \,dt\leq \biggl(\frac{T}{\pi_{p}} \biggr)^{p(n-i)} \int ^{T}_{0}\bigl|x^{(n)}(t)\bigr|^{p} \,dt, \quad i=1,2,\ldots,n-1, $$

where $\frac{1}{p}+\frac{1}{q}=1$, $p\geq2$.

Proof

From $x^{(i-1)}(0)=x^{(i-1)}(T)$, there is a point $t_{i}\in[0,T]$ such that $x^{(i)}(t_{i})=0$. Let $\omega_{i}(t)=x^{(i)}(t+t_{i})$, and then $\omega_{i}(0)=\omega_{i}(T)=0$. From $x^{(i)}(0)=x^{(i)}(T)$, there is a point $t_{i+1}\in[0,T]$ such that $x^{(i+1)}(t_{i+1})=0$. Let $\omega_{i+1}(t)=x^{(i+1)}(t+t_{i+1})$, and then $\omega_{i+1}(0)=\omega_{i+1}(T)=0$. Continuing this way we get from $x^{(n-i)}(0)=x^{(n-i)}(T)$ a point $t_{n-i+1}\in[0,T]$ such that $x^{(n)}(t_{n-i+1})=0$. Let $\omega_{n-i}(t)=x^{(n-i+1)}(t+t_{n-i+1})$, and then $\omega_{n-i}(0)=\omega_{n-i}(T)=0$. From Lemma 2.2, we have

$$\begin{aligned} \int^{T}_{0}\bigl|x^{(i)}(t)\bigr|^{p} \,dt&= \int^{T}_{0}\bigl|\omega_{i}(t)\bigr|^{p} \,dt \\ &\leq \biggl(\frac{T}{\pi_{p}} \biggr)^{p} \int^{T}_{0}\bigl|\omega_{i}'(t)\bigr|^{p} \,dt \\ &= \biggl(\frac{T}{\pi_{p}} \biggr)^{p} \int^{T}_{0}\bigl|x^{(i+1)}(t)\bigr|^{p} \,dt \\ &= \biggl(\frac{T}{\pi_{p}} \biggr)^{p} \int^{T}_{0}\bigl|\omega _{i+1}(t)\bigr|^{p} \,dt \\ &\leq \biggl(\frac{T}{\pi_{p}} \biggr)^{2p} \int^{T}_{0}\bigl|\omega _{i+1}'(t)\bigr|^{p} \,dt \\ &\cdots \\ &\leq \biggl(\frac{T}{\pi_{p}} \biggr)^{p(n-i)} \int^{T}_{0}\bigl|\omega _{n-i-1}'(t)\bigr|^{p} \,dt \\ &= \biggl(\frac{T}{\pi_{p}} \biggr)^{p(n-i)} \int^{T}_{0}\bigl|x^{(n)}(t)\bigr|^{p} \,dt. \end{aligned}$$

(2.1)

 □

In order to apply coincidence degree theorem, we rewrite (1.4) in the form

$$ \textstyle\begin{cases} x_{1}^{(n)}(t)=\varphi_{q}(x_{2}(t)),\\ x_{2}^{(n)}(t)=-f(x_{1}(t))x_{1}'(t)-g(t,x_{1}(t-\sigma))+e(t), \end{cases} $$

(2.2)

where $\frac{1}{p}+\frac{1}{q}=1$. Clearly, if $x(t)=(x_{1}(t),x_{2}(t))^{\top}$ is a T-periodic solution to (2.2), then $x_{1}(t)$ must be a T-periodic solution to (1.4). Thus, the problem of finding a T-periodic solution for (1.4) reduces to finding one for (2.2).

Now, set $X=\{x=(x_{1}(t),x_{2}(t))\in C(\mathbb{R},\mathbb{R}^{2}): x(t+T)\equiv x(t)\}$ with the norm $|x|_{\infty}=\max\{|x_{1}|_{\infty},|x_{2}|_{\infty}\}$; $Y=\{x=(x_{1}(t),x_{2}(t))\in C^{1}(\mathbb{R},\mathbb{R}^{2}): x(t+T)\equiv x(t)\}$ with the norm $\|x\|=\max\{|x|_{\infty},|x'|_{\infty}\}$. Clearly, X and Y are both Banach spaces. Meanwhile, define

$$L:D(L)= \bigl\{ x\in C^{2n} \bigl(\mathbb{R},\mathbb{R}^{2} \bigr): x(t+T) = x(t), t \in \mathbb{R} \bigr\} \subset X\rightarrow Y $$

by

$$ (Lx) (t)= \begin{pmatrix} x_{1}^{(n)}(t)\\ x_{2}^{(n)}(t) \end{pmatrix} $$

and $N: X\rightarrow Y$ by

$$ (Nx) (t)= \begin{pmatrix} \varphi_{q}(x_{2}(t))\\ -f(x_{1})x_{1}'(t)-g(t,x_{1}(t-\sigma))+e(t) \end{pmatrix}. $$

(2.3)

Then (2.2) can be converted into 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

$$Px= \begin{pmatrix} x_{1}(0)\\ x_{2}(0) \end{pmatrix} ;\qquad Qy=\frac{1}{T} \int_{0}^{T} \begin{pmatrix} y_{1}(s)\\ y_{2}(s) \end{pmatrix}\,ds, $$

then $\operatorname{Im} P=\operatorname{Ker} L$, $\operatorname{Ker} Q=\operatorname{Im} L$. Setting $L_{P}=L|_{D(L)\cap \operatorname{Ker} P}$ and $L_{P}^{-1}$: $\operatorname{Im} L\rightarrow D(L)$ denoting the inverse of $L_{P}$, then

$$\begin{aligned} & \bigl[L_{P}^{-1}y \bigr](t)= \begin{pmatrix} (Gy_{1})(t)\\ (Gy_{2})(t) \end{pmatrix} , \\ &[Gy_{1}](t)=\sum_{i=1}^{n-1} \frac{1}{i!}x_{1}^{(i)}(0)t^{i}+ \frac {1}{(n-1)!} \int^{t}_{0}(t-s)^{n-1}y_{1}(s) \,ds, \\ &[Gy_{2}](t)=\sum_{i=1}^{n-1} \frac{1}{i!}x_{2}^{(i)}(0)t^{i}+ \frac {1}{(n-1)!} \int_{0}^{t}(t-s)^{n-1}y_{2}(s) \,ds, \end{aligned}$$

(2.4)

where $x_{j}^{(i)}(0)$, $i=1,2,\ldots,n-1$ and $j=1,2$, are defined by the following:

$$E_{1}Z=B, \quad\mbox{where } E_{1}= \left.\begin{pmatrix} 1 & 0 & 0 & \cdots& 0 & 0 \\ c_{1} & 1 & 0 & \cdots& 0 & 0 \\ c_{2} & c_{1} & 1 & \cdots& 0 & 0 \\ \cdots\\ c_{n-3} & c_{n-4} & c_{n-5} & \cdots& 1 & 0 \\ c_{n-2} & c_{n-3} & c_{n-4} & \cdots& c_{1} &0 \end{pmatrix} \right._{(n-1)\times(n-1)}. $$

$Z=(x_{1}^{(n-1)}(0),\ldots,x_{1}''(0),x_{1}'(0))^{\top}$, $B=(b_{1},b_{2},\ldots ,b_{n-1})^{\top}$, $b_{i}=-\frac{1}{i!T}\int^{T}_{0}(T-s)^{i}y_{1}(s)\,ds$, and $c_{k}=\frac{T^{k}}{(k+1)!}$, $k=1,2,\ldots,n-2$.

From (2.3) and (2.4), it is clearly 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 Ω̄.

3 Existence of positive periodic solutions for (1.1)

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

$$ g(t,x)\leq \bigl(\psi(t)+\varepsilon \bigr)x^{p-1}+g_{\varepsilon}(t) $$

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

(H₃) $g(t,x)=g_{0}(x)+g_{1}(t,x)$, 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

$$\bigl|g_{1}(t,x)\bigr|\leq h_{b}(t),\quad \mbox{a.e. }t\in[0,T], \forall 0\leq x \leq b. $$

(H₄) $\int^{1}_{0}g_{0}(x)\,dx=-\infty$.

Theorem 3.1

Assume that conditions (H₁)-(H₄) hold. If $|\psi|_{\infty}\frac{T^{\frac{p}{q}+1}}{2^{p-1}} (\frac{T}{\pi _{p}} )^{p(n-1)}<1$, then (1.4) has at least a 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

$$ \textstyle\begin{cases} x_{1}^{(n)}(t)=\lambda\varphi_{q}(x_{2}(t)),\\ x_{2}^{(n)}(t)=-\lambda f(x_{1}(t))x_{1}'(t)-\lambda g(t,x_{1}(t-\sigma))+\lambda e(t). \end{cases} $$

(3.3)

Substituting $x_{2}(t)=\lambda^{1-p}\varphi_{p}[x_{1}^{(n)}(t)]$ into the second equation of (3.3)

$$ \bigl(\varphi_{p} \bigl(x_{1}^{(n)}(t) \bigr) \bigr)^{(n)}+\lambda^{p}f \bigl(x_{1}(t) \bigr)x_{1}'(t)+\lambda ^{p}g \bigl(t,x_{1}(t-\sigma) \bigr)=\lambda^{p}e(t). $$

(3.4)

Integrating both sides of (3.4) from 0 to T, we have

$$ \int^{T}_{0}g \bigl(t,x_{1}(t-\sigma) \bigr) \,dt=0. $$

(3.5)

In view of (H₁), there exist positive constants $D_{1}$, $D_{2}$, and $\xi\in[0,T]$ such that

$$D_{1} \leq \bigl|x_{1}(\xi) \bigr|\leq D_{2}. $$

Then we have

$$\bigl|x_{1}(t)\bigr|= \biggl\vert x_{1}(\xi)+ \int^{t}_{\xi}x_{1}'(s)\,ds \biggr\vert \leq D_{2}+ \int^{t}_{\xi}\bigl|x_{1}'(s)\bigr|\,ds, \quad t\in[\xi,\xi+T], $$

and

$$\bigl|x_{1}(t)\bigr|=\bigl|x_{1}(t-T)\bigr|= \biggl\vert x_{1}(\xi)- \int_{t-T}^{\xi}x_{1}'(s)\,ds \biggr\vert \leq D_{2} + \int_{t-T}^{\xi}\bigl|x_{1}'(s)\bigr| \,ds, \quad t\in[\xi,\xi+T]. $$

Combing the above two inequalities, we obtain

$$ \begin{aligned}[b] |x_{1}|_{\infty}&=\max _{t\in[0,T]}\bigl|x_{1}(t)\bigr|=\max_{t\in[\xi,\xi+T]}\bigl|x_{1}(t)\bigr| \\ &\leq\max_{t\in[\xi,\xi+T]} \biggl\{ D_{2}+\frac{1}{2} \biggl( \int^{t}_{\xi}\bigl|x_{1}'(s)\bigr| \,ds+ \int^{\xi}_{t-T}\bigl|x_{1}'(s)\bigr|\,ds \biggr) \biggr\} \\ &\leq D_{2}+\frac{1}{2} \int^{T}_{0}\bigl|x_{1}'(s)\bigr|\,ds. \end{aligned} $$

(3.6)

Multiplying both sides of (3.4) by $x_{1}(t)$ and integrating over interval $[0,T]$, we get

$$\begin{aligned} & \int^{T}_{0} \bigl(\varphi_{p} \bigl(x_{1}^{(n)}(t) \bigr) \bigr)^{(n)}x_{1}(t) \,dt+\lambda^{p} \int ^{T}_{0}f \bigl(x_{1}(t) \bigr)x_{1}'(t)x_{1}(t)\,dt+ \lambda^{p} \int^{T}_{0}g \bigl(t,x_{1}(t-\sigma) \bigr)x_{1}(t)\,dt \\ &\quad=\lambda^{p} \int^{T}_{0}e(t)x_{1}(t)\,dt. \end{aligned}$$

(3.7)

Substituting $\int^{T}_{0}(\varphi_{p}(x_{1}^{(n)}(t)))^{(n)}x_{1}(t)\,dt=(-1)^{n}\int ^{T}_{0}|x_{1}^{(n)}(t)|^{p}\,dt$, $\int^{T}_{0}f(x_{1}(t))x_{1}'(t)x_{1}(t)\,dt=0$ into (3.7), we have

$$(-1)^{n} \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt=-\lambda^{p} \int^{T}_{0}g \bigl(t,x_{1}(t-\sigma ) \bigr)x_{1}(t)\,dt+\lambda^{p} \int^{T}_{0}e(t)x_{1}(t)\,dt. $$

Namely,

$$\begin{aligned} \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt&\leq \int^{T}_{0}\bigl|g \bigl(t,x_{1}(t-\sigma ) \bigr)\bigr|\bigl|x_{1}(t)\bigr|\,dt+ \int^{T}_{0}\bigl|e(t)\bigr|\bigl|x_{1}(t)\bigr|\,dt \\ &\leq|x_{1}|_{\infty}\int^{T}_{0}\bigl|g \bigl(t,x_{1}(t-\sigma) \bigr)\bigr|\,dt+|x_{1}|_{\infty}|e|_{\infty}T. \end{aligned}$$

(3.8)

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 (3.5)

$$\begin{aligned} \int^{T}_{0}\bigl|g \bigl(t,x_{1}(t-\sigma) \bigr)\bigr|\,dt&= \int_{I_{+}}g \bigl(t,x_{1}(t-\sigma) \bigr)\,dt- \int _{I_{-}}g \bigl(t,x_{1}(t-\sigma) \bigr)\,dt \\ &=2 \int_{I_{+}}g \bigl(t,x_{1}(t-\sigma) \bigr)\,dt \\ &\leq2 \int_{I_{+}} \bigl( \bigl(\psi(t)+\varepsilon \bigr)x_{1}^{p-1}(t- \sigma)+g_{\varepsilon}(t) \bigr)\,dt \\ &\leq2\bigl(|\psi|_{\infty}+\varepsilon\bigr) \int^{T}_{0}\bigl|x_{1}(t)\bigr|^{p-1} \,dt+2 \int ^{T}_{0}\bigl|g_{\varepsilon}(t)\bigr|\,dt. \end{aligned}$$

(3.9)

Substituting (3.9) into (3.8), we have

$$\begin{aligned} \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt\leq{}& 2|x_{1}|_{\infty}\bigl(|\psi|_{\infty}+\varepsilon\bigr) \int^{T}_{0}\bigl|x_{1}(t)\bigr|^{p-1}\,dt \\ &{}+ |x_{1}|_{\infty}\biggl(2 \int^{T}_{0}\bigl|g_{\varepsilon}(t)\bigr| \,dt+|e|_{\infty}T \biggr) \\ \leq{}&2\bigl(|\psi|_{\infty}+\varepsilon\bigr)T|x_{1}|^{p}_{\infty}+|x_{1}|_{\infty}\biggl(2T^{\frac{1}{2}} \biggl( \int^{T}_{0}\bigl|g_{\varepsilon}(t)\bigr|^{2}\,dt \biggr)^{\frac {1}{2}}+|e|_{\infty}T \biggr) \\ ={}&2\bigl(|\psi|_{\infty}+\varepsilon\bigr)T|x_{1}|^{p}_{\infty}+|x_{1}|_{\infty}\bigl(2T^{\frac{1}{2}}|g_{\varepsilon}|_{2}+|e|_{\infty}T \bigr). \end{aligned}$$

(3.10)

From (3.6) and Lemma 2.3, we have

$$\begin{aligned} |x_{1}|_{\infty}&\leq D_{2}+ \frac{1}{2} \int^{T}_{0}\bigl|x_{1}'(t)\bigr|\,dt \leq D_{2}+\frac{T^{\frac{1}{q}}}{2} \biggl( \int^{T}_{0}\bigl|x_{1}'(t)\bigr|^{p} \,dt \biggr)^{\frac {1}{p}} \\ &\leq D_{2}+\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt \biggr)^{\frac{1}{p}}. \end{aligned}$$

(3.11)

Substituting (3.11) into (3.10), we have

$$\begin{aligned} &\int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt\\ &\quad\leq2\bigl(|\psi|_{\infty}+\varepsilon\bigr)T \biggl(D_{2}+ \frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt \biggr)^{\frac{1}{p}} \biggr)^{p} \\ &\qquad{}+ \biggl(D_{2}+\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt \biggr)^{\frac{1}{p}} \biggr) \bigl(2T^{\frac{1}{2}}|g_{\varepsilon}|_{2}+|e|_{\infty}T \bigr) \\ &\quad=2\bigl(|\psi|_{\infty}+\varepsilon\bigr)T \biggl(\frac{T^{\frac{p}{q}}}{2^{p}} \biggl( \frac{T}{\pi_{p}} \biggr)^{p(n-1)} \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt +pD_{2}\frac{T^{\frac{p-1}{q}}}{2^{p-1}} \biggl(\frac{T}{\pi_{p}} \biggr)^{(p-1)(n-1)} \\ &\qquad{}\cdot \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|\,dt \biggr)^{\frac{p-1}{p}}+\cdots +pD_{2}^{p-1}\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \biggl( \int^{T}_{0}\bigl|x_{1}'(t)\bigr|^{p} \,dt \biggr)^{\frac{1}{p}}+D_{2}^{p} \biggr) \\ &\qquad{}+ \biggl(D_{2}+\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt \biggr)^{\frac{1}{p}} \biggr) \bigl(2T^{\frac{1}{2}}|g_{\varepsilon}|_{2}+|e|_{\infty}T \bigr) \\ &\quad=\bigl(|\psi|_{\infty}+\varepsilon\bigr)\frac{T^{\frac{p}{q}+1}}{2^{p-1}} \biggl( \frac{T}{\pi_{p}} \biggr)^{p(n-1)} \int^{T}_{0}\bigl|x_{1}^{(n)}\bigr|^{p} \,dt \\ &\qquad+\bigl(|\psi|_{\infty}+\varepsilon\bigr)pD_{2}\frac{T^{\frac {p-1}{q}+1}}{2^{p-2}} \biggl( \frac{T}{\pi_{p}} \biggr)^{(p-1)(n-1)} \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt \biggr)^{\frac{p-1}{p}}+\cdots \\ &\qquad{}+ \bigl(2\bigl(|\psi|_{\infty}+ \varepsilon\bigr)TpD_{2}^{p-1}+2T^{\frac {1}{2}}|g_{\varepsilon}|_{2}+|e|_{\infty}T \bigr)\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \\ &\qquad{}\cdot \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt \biggr)^{\frac{1}{p}}+2\bigl(|\psi |_{\infty}+\varepsilon\bigr)TD_{2}^{p}+D_{2} \bigl(2T^{\frac{1}{2}}|g_{\varepsilon}|_{2}+|e|_{\infty}T \bigr). \end{aligned}$$

Since ε sufficiently small, we know that $|\psi|_{\infty}\frac{T^{\frac{p}{q}+1}}{2^{p-1}} (\frac{T}{\pi _{p}} )^{p(n-1)}<1$. So, it is easy to see that there exists a positive constant $M_{1}'$ such that

$$\int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt\leq M_{1}'. $$

From (3.11), we have

$$ \begin{aligned}[b] |x_{1}|_{\infty}&\leq D_{2}+ \frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}(t)\bigr|^{p} \,dt \biggr)^{\frac{1}{p}}\\ &\leq D_{2}+\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{n-1} \bigl(M_{1}' \bigr)^{\frac{1}{p}}:=M_{1}. \end{aligned} $$

(3.12)

Since $x_{1}(0)=x_{1}(T)$, there exists a point $\eta_{1}\in[0,T]$ such that $x_{1}'(\eta_{1})=0$. From Lemma 2.3, we can easily get

$$\begin{aligned} \bigl|x_{1}'\bigr|_{\infty}&\leq \frac{1}{2} \int^{T}_{0}\bigl|x_{1}''(t)\bigr| \,dt \\ &\leq\frac{T^{\frac{1}{q}}}{2} \biggl( \int^{T}_{0}\bigl|x_{1}''(t)\bigr|^{p} \,dt \biggr)^{\frac {1}{p}} \\ &\leq\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{(n-2)} \biggl( \int^{T}_{0}\bigl|x_{1}^{(n)}\bigr|^{p} \biggr)^{\frac{1}{p}} \\ &\leq\frac{T^{\frac{1}{q}}}{2} \biggl(\frac{T}{\pi_{p}} \biggr)^{(n-2)} \bigl(M_{1}' \bigr)^{\frac{1}{p}}:=M_{2}. \end{aligned}$$

(3.13)

On the other hand, form $x_{2}^{(n-2)}(0)=x_{2}^{(n-2)}(T)$, there exists a point $\eta_{2}\in[0,T]$ such that $x_{2}^{(n-1)}(\eta_{2})=0$, from the second equation of (3.3) and (3.9), we have

$$\begin{aligned} \bigl|x_{2}^{(n-1)}\bigr|_{\infty}&\leq\frac{1}{2}\max \biggl\vert \int ^{T}_{0}x_{2}^{(n)}(t)\,dt \biggr\vert \\ &\leq\frac{\lambda}{2} \int^{T}_{0}\bigl|-f \bigl(x_{1}(t) \bigr)x_{1}'(t)-g(t,x_{1} \bigl(t,x_{1}(t- \sigma ) \bigr)+e(t)\bigr|\,dt \\ &\leq\frac{\lambda}{2} \bigl(|f|_{M_{1}}TM_{2}+2\bigl(| \psi|_{\infty}+\varepsilon \bigr)TM_{1}^{p-1}+2 \sqrt{T}|g_{\varepsilon}|_{2}+T|e|_{\infty}\bigr):=\lambda M_{n-1}, \end{aligned}$$

where $|f|_{M_{1}}=\max\limits_{0< x_{1}(t)\leq M_{1}}|f(x_{1}(t))|$. Since $x_{2}(0)=x_{2}(T)$, there exists a point $\eta_{3}\in[0,T]$ such that $x_{2}'(\eta_{3})=0$. From the Wirtinger inequality (see [18], Lemma 2.4), we can easily get

$$\begin{aligned} \bigl|x_{2}'\bigr|_{\infty}&\leq \frac{1}{2} \int^{T}_{0}\bigl|x_{2}''(t)\bigr| \,dt\leq \frac{T^{\frac{1}{2}}}{2} \biggl( \int^{T}_{0}\bigl|x_{2}''(t)\bigr|^{2} \,dt \biggr)^{\frac{1}{2}} \\ &\leq\frac{T}{2} \biggl(\frac{T}{2\pi} \biggr)^{(n-3)}\bigl|x_{2}^{(n-1)}\bigr|_{\infty} \\ &\leq\frac{T}{2} \biggl(\frac{T}{2\pi} \biggr)^{(n-3)} ( \lambda M_{n-1}):=\lambda M_{3}. \end{aligned}$$

(3.14)

By the first equation of (3.3), we have

$$\int^{T}_{0} \bigl|x_{2}(t) \bigr|^{q-2}x_{2}(t) \,dt=0, $$

which implies that there is a constant $\eta_{4}\in[0,T]$ such that $x_{2}(\eta_{4})=0$, so

$$ |x_{2}|_{\infty}\leq\frac{1}{2} \int^{T}_{0}\bigl|x_{2}'(t)\bigr|\,dt \leq\frac {T}{2}\bigl|x_{2}'\bigr|_{\infty}\leq \frac{\lambda T}{2}M_{3}:=\lambda M_{4}. $$

(3.15)

Next, it follows from (3.4) that

$$ \bigl(\varphi_{p} \bigl(x_{1}^{(n)}(t+ \sigma) \bigr) \bigr)^{(n)}+\lambda^{p} \bigl(f \bigl(x_{1}(t+\sigma ) \bigr)x_{1}'(t+\sigma)+g \bigl(t+\sigma,x_{1}(t) \bigr) \bigr)=\lambda^{p} e(t+\sigma). $$

(3.16)

Namely,

$$\begin{aligned} &\bigl(\varphi_{p} \bigl(x_{1}^{(n)}(t+ \sigma) \bigr) \bigr)^{(n)}+\lambda^{p}f \bigl(x_{1}(t+ \sigma ) \bigr)x_{1}'(t+\sigma)+\lambda^{p}(g_{0} \bigl(x_{1}(t) \bigr)+g_{1} \bigl(t+\sigma,x_{1}(t) \bigr) \\ &\quad=\lambda^{p} e(t+\sigma). \end{aligned}$$

(3.17)

Multiplying both sides of (3.17) by $x_{1}'(t)$, we get

$$\begin{aligned} &\bigl(\varphi_{p} \bigl(x_{1}^{(n)}(t+ \sigma) \bigr) \bigr)^{(n)}x_{1}'(t)+ \lambda^{p}f \bigl(x_{1}(t+\sigma ) \bigr)x_{1}'(t+ \sigma)x_{1}'(t) \\ &\qquad{}+\lambda^{p}g_{0} \bigl(x_{1}(t) \bigr)x_{1}'(t)+\lambda ^{p}g_{1} \bigl(t+\sigma,x_{1}(t) \bigr)x_{1}'(t) \\ &\quad=\lambda^{p} e(t+ \sigma)x_{1}'(t). \end{aligned}$$

(3.18)

Let $\tau\in[0,T]$, for any $\tau\leq t\leq T$, we integrate (3.18) on $[\tau, t]$ and get

$$\begin{aligned} &\lambda^{p} \int^{x_{1}(t)}_{x_{1}(\tau)}g_{0}(u)\,du \\ &\quad=\lambda^{p} \int^{t}_{\tau }g_{0} \bigl(x_{1}(s) \bigr)x_{1}'(s)\,ds \\ &\quad=- \int^{t}_{\tau} \bigl(\varphi_{p} \bigl(x_{1}^{(n)}(s+\sigma) \bigr) \bigr)^{(n)}x_{1}'(s) \,ds-\lambda ^{p} \int^{t}_{\tau}f \bigl(x_{1}(s+\sigma) \bigr)x_{1}'(s+\sigma)x_{1}'(s) \,ds \\ &\qquad{}-\lambda^{p} \int^{t}_{\tau}g_{1} \bigl(s+ \sigma,x_{1}(s) \bigr)x_{1}'(s)\,ds+ \lambda^{p} \int ^{t}_{\tau}e(s+\sigma)x_{1}'(s) \,ds. \end{aligned}$$

(3.19)

By (3.12), (3.13), and (3.16), we have

$$\begin{aligned} &\biggl\vert \int^{t}_{\tau} \bigl(\varphi_{p} \bigl(x_{1}^{(n)}(s+\sigma ) \bigr) \bigr)^{(n)}x_{1}'(s) \,ds \biggr\vert \\ &\quad\leq \int^{t}_{\tau}\bigl| \bigl(\varphi _{p} \bigl(x_{1}^{(n)}(s+\sigma) \bigr) \bigr)^{(n)}\bigr|\bigl|x_{1}'(s)\bigr| \,ds \\ &\quad\leq\bigl|x_{1}'\bigr|_{\infty}\int^{T}_{0}\bigl| \bigl(\varphi_{p} \bigl(x_{1}^{(n)}(t+\sigma) \bigr) \bigr)^{(n)}\bigr|\,dt \\ &\quad\leq\lambda^{p}\bigl|x_{1}'\bigr|_{\infty}\biggl( \int^{T}_{0}\bigl|f \bigl(x_{1}(t) \bigr)\bigr|\bigl|x_{1}'(t)\bigr|\,dt+ \int^{T}_{0}\bigl|g \bigl(t,x_{1}(t-\sigma ) \bigr)\bigr|\,dt+ \int^{T}_{0}\bigl|e(t)\bigr|\,dt \biggr) \\ &\quad\leq\lambda^{p} M_{2} \bigl(|f|_{M_{1}}M_{2} +2\bigl(|\psi|_{\infty}+\varepsilon\bigr)TM_{1}^{p-1}+2T^{\frac{1}{2}} \bigl|g_{\varepsilon}^{+}\bigr|_{2} +T|e|_{\infty}\bigr). \end{aligned}$$

Also we have

$$\begin{aligned} & \biggl\vert \int^{t}_{\tau}f \bigl(x_{1}(s+\sigma) \bigr)x_{1}'(s+\sigma)x_{1}'(s) \,ds \biggr\vert \leq |f|_{M_{1}}M_{2}^{2}T, \\ & \biggl\vert \int^{t}_{\tau}g \bigl(s+\sigma,x_{1}(s) \bigr)x_{1}'(s)\,ds \biggr\vert \leq\bigl|x_{1}'\bigr|_{\infty}\int^{T}_{0}\bigl|g(t,x(t-\sigma)\bigr|\,dt\leq M_{2}\sqrt{T}|g_{M_{1}}|_{2}, \end{aligned}$$

where $g_{M_{1}}=\max\limits_{0\leq x\leq M_{1}}|g_{1}(t,x)|\in L^{2}(0,T)$ is as in (H₃).

$$\biggl\vert \int^{t}_{\tau}e(t+\sigma)x_{1}'(t) \,dt \biggr\vert \leq M_{2}T|e|_{\infty}. $$

From these inequalities we can derive form (3.19) that

$$ { } \biggl\vert \int^{x_{1}(t)}_{x_{1}(\tau)}g_{0}(u)\,du \biggr\vert \leq M_{5}', $$

(3.20)

for some constant $M_{5}'$ which is independent on λ, x, and t. In view of the strong force condition (H₄), we know that there exists a constant $M_{5}>0$ such that

$$ x_{1}(t)\geq M_{5},\quad \forall t\in[ \tau,T]. $$

(3.21)

The case $t\in[0,\tau]$ can be treated similarly.

From (3.12), (3.13), (3.14), (3.15), and (3.21), we get

$$\begin{aligned} \Omega={}& \bigl\{ x=(x_{1},x_{2})^{\top}: E_{1}\leq|x_{1}|_{\infty}\leq E_{2}, \bigl|x_{1}'\bigr|_{\infty}\leq E_{3}, |x_{2}|_{\infty}\leq E_{4} \mbox{ and}\\ &{} \bigl|x_{2}'\bigr|_{\infty}\leq E_{5} , \forall t \in[0,T] \bigr\} , \end{aligned}$$

where $0< E_{1}<\min(M_{5}, D_{1})$, $E_{2}>\max(M_{1}, D_{2})$, $E_{3}>M_{2}$, $E_{4}>M_{4}$, and $E_{5}>M_{3}$. $\Omega_{2}=\{x:x\in\partial\Omega\cap \operatorname{Ker} L\}$, 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 $\forall (\mu,x)\in(0,1)\times(\partial\Omega\cap \operatorname{Ker} L)$,

$$ H(\mu,x)= \begin{pmatrix}-\mu x_{1}-\frac{1-\mu}{T}\int^{T}_{0}[g(t,x_{1})-e(t)]\,dt\\ -\mu x_{2}-(1-\mu)|x_{2}|^{q-2}x_{2} \end{pmatrix} . $$

We have $\int^{T}_{0}e(t)\,dt=0$. So, we can get

$$\begin{aligned} H(\mu,x)= \begin{pmatrix}-\mu x_{1}-\frac{1-\mu}{T}\int^{T}_{0}g(t,x_{1})\,dt\\ -\mu x_{2}-(1-\mu)|x_{2}|^{q-2}x_{2} \end{pmatrix} , \quad\forall (\mu,x)\in(0,1)\times(\partial\Omega\cap \operatorname{Ker} L). \end{aligned}$$

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

$$\begin{aligned}[b] \deg\{JQN,\Omega\cap \operatorname{Ker} L,0\}&=\deg \bigl\{ H(0,x),\Omega\cap \operatorname{Ker} L,0 \bigr\} \\ &=\deg \bigl\{ H(1,x),\Omega\cap \operatorname{Ker} L,0 \bigr\} \\ &=\deg\{I,\Omega\cap \operatorname{Ker} L,0\}\neq0. \end{aligned} $$

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., (1.4) has a positive T-periodic solution $x_{1}(t)$. □

Example 3.1

Consider the high-order p-Laplacian differential equation with singularity

$$ \bigl(\varphi_{p} \bigl(x(t)''' \bigr)\bigr)''''+f \bigl(x(t) \bigr)x'(t)+\frac{1}{6}(\sin2t+3)x^{3}(t-\sigma )- \frac{1}{x^{\kappa}(t-\sigma)}=\cos 2t, $$

(3.22)

where $\kappa\geq1$ and $p=4$, f is continuous function, σ is a constant, and $0\leq\sigma< T$.

It is clear that $T=\pi$, $n=3$, $g(t,x)=\frac{1}{6}(\sin2t+3)x^{3}(t-\sigma)-\frac{1}{x^{\kappa}(t-\sigma)}$, $\psi(t)=\frac{1}{6}(\sin2t+3)$, $|\psi|_{\infty}=\frac{2}{3}$. It is obvious that (H₁)-(H₄) hold. Now we consider the assumption condition

$$\begin{aligned} &|\psi|_{\infty}\frac{T^{\frac{p}{q}+1}}{2^{p-1}} \biggl(\frac{T}{\pi _{p}} \biggr)^{p(n-1)} \\ &\quad=|\psi|_{\infty}\frac{T^{\frac{p}{q}+1}}{2^{p-1}} \biggl(\frac{T}{\frac {2\pi(p-1)^{1/p}}{p\sin(\pi/p)}} \biggr)^{p(n-1)} \\ &\quad=\frac{2}{3}\cdot\frac{\pi^{\frac{4}{3}}}{2^{3}} \biggl(\frac{\pi}{\frac {2\pi(4-1)^{1/4}}{4\sin{\pi/4}}} \biggr)^{8} \\ &\quad=\frac{4\pi^{\frac{4}{3}}}{27}< 1. \end{aligned}$$

So by Theorem 3.1, we know (3.22) has at least one positive π-periodic solution.