A new result on the existence of periodic solutions for Rayleigh equation with a singularity

Abstract

In this paper, we study the existence of periodic solutions for Rayleigh equation with a singularity of repulsive type

$$x''(t)+f\bigl(x'(t)\bigr)+\varphi (t)x(t)-\frac{1}{x^{\alpha }(t)}=p(t), $$

where $\alpha \geqslant 1$ is a constant, and φ and p are T-periodic functions. The proof of the main result relies on a known continuation theorem of coincidence degree theory. The interesting point is that the sign of the function $\varphi (t)$ is allowed to change for $t\in [0,T]$.

1 Introduction

Singular differential equations arise in many disciplines such as physics, fluid dynamics, and ecology (see [1–6] and the references therein). In recent years, the periodic problem of second-order differential equations with singularities has been widely studied. The first study in this area seems to be the paper of Nagumo [7] in 1944. After some works of Forbat and Huaux [8], the interest increased with the pioneering paper of Lazer and Solimini [9]. They considered the existence of periodic solutions suggested by the two fundamental examples ($\alpha >0$, and $h:R\rightarrow R$ is a continuous T-periodic function)

$$ x''(t)+\frac{1}{x^{\alpha }(t)}=h(t) $$

(1.1)

(the singularity of attractive type) and

$$ x''(t)-\frac{1}{x^{\alpha }(t)}=h(t) $$

(1.2)

(the singularity of repulsive type). By using topological degree methods they obtained that a necessary and sufficient condition for the existence of positive periodic solutions for equation (1.1) is $\overline{h}>0$, and if we assume in addition that $\alpha \ge 1$, then a necessary and sufficient condition for the existence of positive periodic solutions for equation (1.2) is $\overline{h}<0$. After that, some methods associated with nonlinear functional analysis theory have been widely applied to the studied problem in many papers such as the variational methods used in [10–13], fixed point theorems used in [14–19], upper and lower solutions methods used in [20, 21], and continuation theorems of coincidence degree used in [22–31]. For example, Torres [14] studied the periodic problem for the equation with singularity of repulsive type

$$ x''+\varphi (t)x-\frac{b(t)}{x^{\mu }}=h(t), $$

(1.3)

where $\varphi,b,h\in L^{1}[0,T]$, and $\mu >0$ is a constant. The function φ is required to satisfy

$$ \varphi (t)\ge 0 \quad \mbox{for all } t\in [0,T]. $$

(1.4)

This is due to the fact that (1.4), together with some other conditions, can guarantee the Green function $G(t,s)$ associated with the boundary value problem for Hill’s equation

$$ x''+\varphi (t)x=h(t),\qquad x(0)=x(T),\qquad x'(0)=x'(T), $$

(1.5)

satisfying $G(t,s)\ge 0$ for all $(t,s)\in [0,T]\times [0,T]$; then, the solution to problem (1.5) is given by

$$ x(t)= \int_{0}^{T}G(t,s)h(s)\,ds. $$

(1.6)

Formula (1.6) is crucial in [14–17] for applying some fixed point theorems on cones. Wang [25] studied the problem of periodic solutions for the singular delay Liénard equation of repulsive type

$$ x''(t)+f\bigl(x(t)\bigr)x'(t)+ \varphi (t)x(t-\tau)-\frac{1}{x^{\mu }(t-\tau)}=h(t), $$

(1.7)

where $f:[0,+\infty)\rightarrow R$ is continuous, $\varphi:R\rightarrow R$ is continuous T-periodic, and $\tau >0$ and $\mu \ge 1$ are constants. To balance the forces of $\varphi (t)x$ at $x=+\infty$ and $\frac{1}{x^{\mu }}$ at $x=0$, φ is also required to satisfy

$$ \varphi (t)\ge 0\quad \mbox{for all } t\in [0,T]. $$

(1.8)

In [26, 28], the authors studied the periodic problem of the equation

$$ x''+f(x)x'+\varphi (t)x- \frac{1}{x^{\mu }}=h(t). $$

(1.9)

In (1.9), the function φ is required to satisfy $\int_{0}^{T}\varphi (s)\,ds>0$, which means that the sign of the function φ is allowed to change. Now, the question is that how to investigate the existence of T-periodic solutions for a Rayleigh equation with a singularity of repulsive type

$$ x''(t)+f\bigl(x'(t)\bigr)+ \varphi (t)x(t)-\frac{1}{x^{\alpha }(t)}=p(t), $$

(1.10)

where $f:R\rightarrow R$ is continuous with $f(0)=0$, $\alpha \geq 1$, and φ, $p:R\rightarrow R$ are continuous and T-periodic.

Motivated by this, the aim of this paper is to search for positive T-periodic solutions for (1.10). Using a known continuation theorem of theorem of coincidence degree theory (see [32, 33], and [34]), we obtain a new result on the existence of positive periodic solutions for equation (1.10). In present paper, the sign of φ in (1.10) is allowed to change for $t\in [0,T]$. Although this condition is the same as that in [26, 28], for studying the periodic problem of (1.9), the methods used in [26, 28] for estimating a priori bounds of positive T-periodic solutions to (1.9) cannot be directly applied to (1.10). This is due to the fact that mechanism of the first-order derivative term $f(x'(t))$ influencing a priori bounds of positive T-periodic solutions to (1.10) is different from the corresponding ones of $f(x(t))x'(t)$ in (1.10). For example, if $x(t)$ is a positive T-periodic function such that $x\in C^{1}(R,R)$, then $\int_{0}^{T}f(x(t))x'(t)\,dt=0$, but, generally, $\int_{0}^{T}f(x'(t))\,dt \neq 0$.

2 Preliminary lemmas

Let $C_{T}=\{x\in C(R,R):x(t+T)=x(t), \forall t\in R\}$ with the norm $\vert x\vert _{\infty }=\max_{t\in [0,T]}\vert x(t)\vert$, and let $C^{1}_{T}=\{x' \in C^{1}(R,R):x'(t+T)=x'(t), \forall t\in R\}$ with the norm $\Vert x\Vert =\max \{\vert x\vert _{\infty },\vert x'\vert _{\infty }\}$. Clearly, $C_{T}$ and $C_{T}^{1}$ are both Banach spaces. For any T-periodic solution $\varphi (t)$ with $\varphi \in C_{T}$, by $\varphi_{+}(t)$ and $\varphi_{-}(t)$ we denote $\max \{\varphi (t),0\}$ and $-\min \{ \varphi (t),0\}$, respectively, and $\overline{\varphi }=\frac{1}{T} \int^{T}_{0}\varphi (s)\,ds$. Then $\varphi (t)=\varphi_{+}(t)-\varphi _{-}(t)$ for all $t\in R$, and $\overline{\varphi }=\overline{\varphi _{+}}-\overline{\varphi_{-}}$. Furthermore, for each $u\in C_{T}$, let $\Vert u\Vert _{p}:=(\int_{0}^{T}\vert u(s)\vert ^{p}\,ds)^{1/p}$, $p\in [1,+\infty)$.

The following result can be easily obtained by using Theorem 4 in [32], Chapter 6 of [33], and Theorem 3.1 in [34].

Lemma 2.1

Assume that there exist positive constants $N_{0}$, $N_{1}$, and $N_{2}$ with $0< N_{0}< N_{1}$ such that the following conditions hold.

• 1. For each $\lambda \in (0,1]$, each possible positive T-periodic solution x to the equation

  $$u''+\lambda f\bigl(u'\bigr)+\lambda \varphi (t)u-\frac{\lambda }{u^{\alpha }}= \lambda p(t) $$

  satisfies the inequalities $N_{0}< x(t)< N_{1}$ and $\vert x'(t)\vert < N_{2}$ for all $t\in [0,T]$.

• 2. Each possible solution c to the equation

  $$\frac{1}{c^{\alpha }}-c\overline{\varphi }+\overline{p}=0 $$

  satisfies the inequality $N_{0}< c< N_{1}$.

• 3. The inequality

  $$\biggl(\frac{1}{N_{0}^{\alpha }}-N_{0}\overline{\varphi }+\overline{p} \biggr) \biggl(\frac{1}{N_{1}^{\alpha }}-N_{1}\overline{\varphi }+ \overline{p} \biggr)< 0 $$

  holds.

Then equation (1.10) has at least one positive T-periodic solution u such that $N_{0}< u(t)< N_{1}$ for all $t\in [0,T]$.

Now, we list the following assumptions, which will be used in Section 3 for investigating the existence of positive T-periodic solutions to (1.10).

${[H_{1}]}$ :

There exist constants $L>0$, $\sigma >0$, and $n\geq 1$ such that

$$ \biggl\vert \int^{T}_{0}f\bigl(x'(t)\bigr)\,dt \biggr\vert \leq L \int^{T}_{0}\bigl\vert x'(t)\bigr\vert \,dt,\quad \forall x\in C^{1}_{T} $$

(2.1)

and

$$ yf(y)\geq \sigma \vert y\vert ^{n+1}, \quad \forall y\in R. $$

(2.2)

${[H_{2}]}$ :

The function φ satisfies $\overline{\varphi_{+}}>\overline{ \varphi_{-}}$;

${[H_{3}]}$ :

$\Vert \varphi \Vert _{2}<\sigma T^{-\frac{1}{2}}$ and $(LT^{- \frac{1}{2}}+T^{\frac{1}{2}}\overline{\varphi_{+}})\Vert \varphi \Vert _{2} < \sigma (\overline{\varphi_{+}}-\overline{\varphi_{-}})$.

Remark 2.1

If assumption $[H_{2}]$ holds, then there are constants $D_{1}$ and $D_{2}$ with $0< D_{1}< D_{2}$ such that

$$\frac{1}{x^{\alpha }}-\overline{\varphi }x+\overline{p}>0 \quad \mbox{for all } x\in (0,D_{1}) $$

and

$$\frac{1}{x^{\alpha }}-\overline{\varphi }x+\overline{p}< 0 \quad \mbox{for all } x\in (D_{2},\infty). $$

Now, we embed equation (1.10) into the following equations family with parameter $\lambda \in (0,1]$:

$$ x''+\lambda f\bigl(x' \bigr)+\lambda \varphi (t)x-\frac{\lambda }{x^{\alpha }}= \lambda p(t),\quad \lambda \in (0,1]. $$

(2.3)

Let

$$ \Omega =\biggl\{ x\in C_{T}:x''+ \lambda f\bigl(x'\bigr)+\lambda \varphi (t)x-\frac{ \lambda }{x^{\alpha }}= \lambda p(t),\lambda \in (0,1];x(t)>0, \forall t\in [0,T]\biggr\} , $$

(2.4)

and let

$$ M_{0}=\max \biggl\{ 1,\frac{LT^{\frac{-1}{n+1}}+T^{\frac{n}{n+1}}\overline{ \varphi_{+}}}{\overline{\varphi_{+}}-\overline{\varphi_{-}}}B + \frac{1+ \overline{p}}{\overline{\varphi_{+}}-\overline{\varphi_{-}}} \biggr\} , $$

(2.5)

where B will be determined by (2.13). Clearly, $M_{0}$ is independent of $(\lambda,x)\in (0,1]\times \Omega$.

Lemma 2.2

Assume that assumptions $[H_{1}]$-$[H_{3}]$ hold. Then for each function $x\in \Omega$, there exists a point $t_{0}\in [0,T]$ such that

$$x(t_{0})\leq M_{0}, $$

where $M_{0}$ is defined by (2.5)

Proof

If the conclusion does not hold, then there is a function $x_{0}\in \Omega$ satisfying

$$ x_{0}(t)>M_{0} \quad \mbox{for all } t\in [0,T]. $$

(2.6)

From (2.4) we get

$$ x''_{0}+\lambda f \bigl(x'_{0}\bigr)+\lambda \varphi (t)x_{0}- \frac{\lambda }{x ^{\alpha }_{0}}=\lambda p(t). $$

(2.7)

Integrating (2.7) over the interval $[0,T]$, we get

$$\begin{aligned}& \int^{T}_{0}f\bigl(x'_{0}(t) \bigr)\,dt+ \int^{T}_{0}\varphi_{+}(t)x_{0}(t)\,dt \\& \quad = \int ^{T}_{0}\varphi_{-}(t)x_{0}(t)\,dt+ \int^{T}_{0} \frac{1}{x^{\alpha }_{0}(t)}\,dt+ \int^{T}_{0}p(t)\,dt, \end{aligned}$$

that is,

$$\begin{aligned}& \int^{T}_{0}\varphi_{+}(t)x_{0}(t)\,dt \\& \quad =- \int^{T}_{0}f\bigl(x'_{0}(t) \bigr)\,dt+ \int ^{T}_{0}\varphi_{-}(t)x_{0}(t)\,dt+ \int^{T}_{0} \frac{1}{x^{\alpha }_{0}(t)}\,dt+ \int^{T}_{0}p(t)\,dt. \end{aligned}$$

Since $\varphi_{+}(t)\geq 0$ and $\varphi_{-}(t)\geq 0$ for all $t\in [0,T]$, it follows from the integral mean value theorem and condition (2.1) in $[H_{1}]$ that there are two points $\xi,\zeta \in [0,T]$ such that

$$x_{0}(\xi)T\overline{\varphi_{+}}\leq L \int^{T}_{0}\bigl\vert x'_{0}(t) \bigr\vert \,dt+x _{0}(\zeta)T\overline{\varphi_{-}}+M_{0}^{-\alpha }T+T \overline{p}, $$

which, together with the fact of $M_{0}\ge 1$ in (2.5), yields

$$x_{0}(\xi)T\overline{\varphi_{+}}\leq L \int^{T}_{0}\bigl\vert x'_{0}(t) \bigr\vert \,dt+\vert x_{0}\vert _{\infty }T\overline{ \varphi_{-}}+T+T\overline{p}, $$

that is,

$$ x_{0}(\xi)\leq \frac{LT^{\frac{-1}{n+1}}}{\overline{\varphi_{+}}} \biggl( \int^{T}_{0}\bigl\vert x'_{0}(t) \bigr\vert ^{n+1}\,dt \biggr)^{\frac{1}{n+1}} +\frac{\overline{ \varphi_{-}}}{\overline{\varphi_{+}}}\vert x_{0}\vert _{\infty }+\frac{1+ \overline{p}}{\overline{\varphi_{+}}}. $$

(2.8)

Since

$$ \vert x_{0}\vert _{\infty }\leq x_{0}(\xi)+T^{\frac{n}{n+1}} \biggl( \int^{T}_{0}\bigl\vert x'_{0}(s) \bigr\vert ^{n+1}\,ds \biggr)^{\frac{1}{n+1}}, $$

(2.9)

it follows from (2.8), (2.9), and $[H_{2}]$ that

$$ \vert x_{0}\vert _{\infty }\leq \frac{LT^{\frac{-1}{n+1}}+T^{\frac{n}{n+1}}\overline{ \varphi_{+}}}{\overline{\varphi_{+}}-\overline{\varphi_{-}}} \biggl( \int ^{T}_{0}\bigl\vert x'_{0}(s) \bigr\vert ^{n+1}\,ds \biggr)^{\frac{1}{n+1}} +\frac{1+ \overline{p}}{\overline{\varphi_{+}}-\overline{\varphi_{-}}}. $$

(2.10)

On the other hand, multiplying both sides of (2.7) by $x_{0}'(t)$ and integrating it over the interval $[0,T]$, we get

$$\lambda \int^{T}_{0}f\bigl(x'_{0}(t) \bigr)x_{0}'(t)\,dt=-\lambda \int^{T}_{0} \varphi (t)x_{0}(t)x_{0}'(t)\,dt+ \lambda \int^{T}_{0}p(t)x_{0}'(t)\,dt. $$

From condition (2.2) in $[H_{1}]$ we have

$$\begin{aligned}& \sigma \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \\& \quad \leq - \int^{T}_{0}\varphi (t)x_{0}(t)x_{0}'(t)\,dt+ \int^{T}_{0}p(t)x _{0}'(t)\,dt \\& \quad \leq \vert x_{0}\vert _{\infty } \int^{T}_{0}\bigl\vert \varphi (t)\bigr\vert \bigl\vert x_{0}'(t)\bigr\vert \,dt+ \int ^{T}_{0}\bigl\vert p(t)\bigr\vert \bigl\vert x_{0}'(t)\bigr\vert \,dt \\& \quad \leq \vert x_{0}\vert _{\infty } \biggl( \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \biggr)^{ \frac{1}{n+1}} \biggl( \int^{T}_{0}\vert \varphi \vert ^{\frac{n+1}{n}}\,dt \biggr)^{ \frac{n}{n+1}} \\& \qquad {} + \biggl( \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \biggr)^{ \frac{1}{n+1}} \biggl( \int^{T}_{0}\bigl\vert p(t)\bigr\vert ^{\frac{n+1}{n}}\,dt \biggr)^{ \frac{n}{n+1}}, \end{aligned}$$

that is,

$$\begin{aligned} \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \leq& \sigma^{-1}\vert x_{0} \vert _{\infty }\Vert \varphi \Vert _{\frac{n+1}{n}} \biggl( \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \biggr)^{ \frac{1}{n+1}} \\ &{}+\sigma^{-1}\Vert p\Vert _{\frac{n+1}{n}} \biggl( \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \biggr)^{\frac{1}{n+1}}. \end{aligned}$$

(2.11)

We infer from (2.10) and (2.11) that

$$\begin{aligned}& \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \\& \quad \leq \frac{LT^{\frac{-1}{n+1}}+T^{\frac{n}{n+1}}\overline{\varphi _{+}}}{\sigma (\overline{\varphi_{+}}-\overline{\varphi_{-}})} \Vert \varphi \Vert _{\frac{n+1}{n}} \biggl( \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \biggr)^{ \frac{2}{n+1}} \\& \qquad {}+\sigma^{-1} \biggl(\frac{1+\overline{p}}{\overline{\varphi_{+}}-\overline{ \varphi_{-}}} \Vert \varphi \Vert _{\frac{n+1}{n}}+\Vert p\Vert _{\frac{n+1}{n}} \biggr) \biggl( \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \biggr)^{\frac{1}{n+1}}. \end{aligned}$$

(2.12)

According to (2.12), we list two cases.

Case 1::

If $n>1$, then we see that there exists $B_{0}>0$ such that $(\int^{T}_{0}\vert x_{0}'(t)\vert ^{n+1}\,dt)^{\frac{1}{n+1}}\leq B_{0}$;

Case 2::

If $n=1$, then by assumption $[H_{3}]$ there exists $B_{1}>0$ such that $(\int^{T}_{0}\vert x_{0}'(t)\vert ^{2}\,dt)^{ \frac{1}{2}}\leq B_{1}$.

Letting $B=\max \{B_{0},B_{1}\}$, it follows from Case 1 or Case 2 that

$$ \biggl( \int^{T}_{0}\bigl\vert x_{0}'(t) \bigr\vert ^{n+1}\,dt \biggr)^{\frac{1}{n+1}}\leq B. $$

(2.13)

Substituting (2.13) into (2.10), we have

$$\vert x_{0}\vert _{\infty }\le \frac{LT^{\frac{-1}{n+1}}+T^{\frac{n}{n+1}}\overline{ \varphi_{+}}}{\overline{\varphi_{+}}-\overline{\varphi_{-}}}B + \frac{1+ \overline{p}}{\overline{\varphi_{+}}-\overline{\varphi_{+}}}. $$

By the definition of $M_{0}$ in (2.5) we have

$$\vert x_{0}\vert _{\infty }\le M_{0}, $$

that is,

$$ x_{0}(t)\le M_{0}\quad \mbox{for all } t\in [0,T], $$

which contradicts (2.6). This contradiction proves Lemma 2.2. □

Lemma 2.3

Assume that $[H_{2}]$ holds. Then there exists a positive constant $\gamma >0$ such that, for each $x\in \Omega$, there is a point $t_{1}\in [0,T]$ satisfying

$$x(t_{1})\geq \gamma. $$

Proof

Let $x(t_{1})=\max_{t\in [0,T]}x(t)$. Then $x''(t_{1}) \leq 0$ and $x'(t_{1})=0$, which, together with (2.3), yields

$$\lambda f(0)+\lambda \varphi (t_{1})x(t_{1})- \frac{\lambda }{x^{ \alpha }(t_{1})}\ge \lambda p(t_{1}). $$

Since $f(0)=0$, we have

$$ \begin{aligned} x(t_{1})\max _{t\in [0,T]}\varphi (t)-\frac{1}{x^{\alpha }(t_{1})} & \geq p(t_{1}) \geq -\vert p\vert _{\infty }. \end{aligned} $$

(2.14)

Multiplying both sides of (2.14) by $x^{\alpha }(t_{1})$, we get

$$ x^{\alpha +1}(t_{1})\max_{t\in [0,T]} \varphi (t)+x^{\alpha }(t_{1})\vert p\vert _{ \infty }-1\geq 0. $$

(2.15)

Set $S(u)=u^{\alpha +1}\max \varphi (t)+u^{\alpha }\vert p\vert _{\infty }-1$ for $u\in [0,+\infty)$. By assumption $[H_{2}]$ we have

$$\begin{aligned}& S(0)=-1< 0, \\& \lim_{u\rightarrow +\infty }S(u)=+\infty. \end{aligned}$$

So $S(u)$ has zero points on $(0,+\infty)$. Let γ be the minimum zero point of $S(u)$ on $(0,+\infty)$. Then $S(\gamma)=0$. It follows from (2.15) that

$$x(t_{1})\geq \gamma. $$

The proof is complete. □

3 Main result

Theorem 3.1

Assume that $[H_{1}]$-$[H_{3}]$ hold. Then equation (1.10) has at least one positive T-periodic solution.

Proof

Firstly, we will show that there exist $N_{1}>0$ and $N_{2}>0$ such that each positive T-periodic solution $x(t)$ of equation (2.3) satisfying

$$ x(t)< N_{1} \quad \mbox{and}\quad \bigl|x'(t)\bigr| < N_{2}\quad \mbox{for all } t\in [0,T]. $$

(3.1)

Suppose that x is an arbitrary positive T-periodic solution of equation (2.3). Then

$$ x''+\lambda f\bigl(x' \bigr)+\lambda \varphi (t)x-\frac{\lambda }{x^{\alpha }}= \lambda p(t),\quad \lambda \in (0,1]. $$

(3.2)

This implies that $x\in \Omega$. So by Lemma 2.2 there exists a point $t_{0}\in [0,T]$ such that

$$x(t_{0})\leq M_{0}, $$

and then

$$ \vert x\vert _{\infty }\leq M_{0}+T^{\frac{n}{n+1}} \biggl( \int^{T}_{0}\bigl\vert x'(s)\bigr\vert ^{n+1}\,ds \biggr)^{\frac{1}{n+1}}. $$

(3.3)

Integrating (3.2) over the interval $[0,T]$, we get

$$ \int^{T}_{0}f\bigl(x'(t)\bigr)\,dt+ \int^{T}_{0}\varphi (t)x(t)\,dt- \int^{T}_{0}\frac{1}{x ^{\alpha }(t)}\,dt= \int^{T}_{0}p(t)\,dt. $$

(3.4)

On the other hand, similarly to the proof of (2.11), we have

$$\begin{aligned} \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{n+1}\,dt \leq& \sigma^{-1}\vert x\vert _{\infty }\Vert \varphi \Vert _{ \frac{n+1}{n}} \biggl( \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{n+1}\,dt \biggr)^{\frac{1}{n+1}} \\ &{}+\sigma^{-1}\Vert p\Vert _{\frac{n+1}{n}} \biggl( \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{n+1}\,dt \biggr)^{\frac{1}{n+1}}. \end{aligned}$$

(3.5)

Substituting (3.3) into (3.5), we have

$$\begin{aligned}& \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{n+1}\,dt \\& \quad \leq \sigma^{-1}\Vert \varphi \Vert _{\frac{n+1}{n}}T^{\frac{n}{n+1}} \biggl( \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{n+1}\,dt \biggr)^{\frac{2}{n+1}} \\& \qquad {} +\bigl(\sigma^{-1}\Vert \varphi \Vert _{\frac{n+1}{n}}M_{0}+ \sigma^{-1}\Vert p\Vert _{ \frac{n+1}{n}} \bigr) \biggl( \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{n+1}\,dt \biggr)^{ \frac{1}{n+1}}. \end{aligned}$$

(3.6)

According to (3.6), we list two cases.

Case 1::

If $n>1$, then there exists $\rho_{0}>0$ such that $(\int ^{T}_{0}\vert x'(t)\vert ^{n+1}\,dt)^{\frac{1}{n+1}}\leq \rho_{0}$;

Case 2::

If $n=1$, then by assumption $[H_{3}]$ there exists $\rho_{1}>0$ such that $(\int^{T}_{0}\vert x'(t)\vert ^{2}\,dt)^{ \frac{1}{2}}\leq \rho_{1}$.

Letting $\rho =\max \{\rho_{0},\rho_{1}\}$, it follows from Case 1 or Case 2 that

$$ \biggl( \int^{T}_{0}\bigl\vert x'(t)\bigr\vert ^{n+1}\,dt \biggr)^{\frac{1}{n+1}}\leq \rho, $$

(3.7)

and according to (3.3), we have

$$ x(t)\leq M_{0}+T^{\frac{n}{n+1}}\rho:=N_{1} \quad \mbox{for all } t\in [0,T]. $$

(3.8)

Clearly, there is a point $t_{2}\in [0,T]$ such that $x'(t_{2})=0$. Multiplying both sides of (3.2) by $x'(t)$ and integrating it over the interval $[t_{2},t]$, we get

$$\begin{aligned}& \int^{t}_{t_{2}}x''(t)x'(t)\,dt \\& \quad =\lambda \int^{t}_{t_{2}}\biggl[-f\bigl(x'(t) \bigr)x'(t)-\varphi (t)x(t)x'(t)+\frac{x'(t)}{x ^{\alpha }(t)}+p(t)x'(t) \biggr]\,dt \\& \qquad \mbox{for all } t\in [t_{2},t_{2}+T], \end{aligned}$$

and then

$$\begin{aligned} \frac{\vert x'(t)\vert ^{2}}{2} \leq& \lambda \bigl\vert x'\bigr\vert _{\infty } \biggl[\vert x\vert _{\infty } \int^{t_{2}+T}_{t_{2}}\bigl\vert \varphi (t)\bigr\vert \,dt+ \int^{t_{2}+T}_{t_{2}}\frac{1}{x ^{\alpha }(t)}\,dt+ \int^{t_{2}+T}_{t_{2}}\bigl\vert p(t)\bigr\vert \,dt \biggr] \\ =&\lambda \bigl\vert x'\bigr\vert _{\infty } \biggl[ \vert x\vert _{\infty } \int^{T}_{0}\bigl\vert \varphi (t)\bigr\vert \,dt+ \int^{T}_{0}\frac{1}{x^{\alpha }(t)}\,dt+ \int^{T}_{0}\bigl\vert p(t)\bigr\vert \,dt \biggr] \\ =&\lambda \bigl\vert x'\bigr\vert _{\infty } \biggl[N_{1}T\overline{\vert \varphi \vert }+ \int^{T} _{0}\frac{1}{x^{\alpha }(t)}\,dt+T\overline{\vert p\vert } \biggr]\quad \mbox{for all } t \in [t_{2},t_{2}+T]. \end{aligned}$$

(3.9)

Since

$$\bigl\vert x'\bigr\vert _{\infty }=\max _{t\in [0,T]}\bigl\vert x'(t)\bigr\vert =\max _{t\in [t_{2},t_{2}+T]}\bigl\vert x'(t)\bigr\vert , $$

it follows from (3.9) that

$$\frac{\vert x'\vert ^{2}_{\infty }}{2}\leq \lambda \bigl\vert x'\bigr\vert _{\infty } \biggl[N_{1}T\overline{\vert \varphi \vert }+ \int^{T}_{0}\frac{1}{x^{\alpha }(t)}\,dt+T\overline{\vert p \vert } \biggr], $$

that is,

$$\frac{\vert x'\vert _{\infty }}{2}\leq \lambda \biggl[N_{1}T\overline{\vert \varphi \vert }+ \int^{T}_{0}\frac{1}{x^{\alpha }(t)}\,dt+T\overline{\vert p \vert } \biggr], $$

which implies that

$$ \frac{\vert x'(t)\vert }{2}\le \frac{\vert x'\vert _{\infty }}{2}\le \lambda \biggl[N_{1}T\overline{\vert \varphi \vert }+ \int^{T}_{0}\frac{1}{x^{\alpha }(t)}\,dt+T\overline{\vert p \vert } \biggr]\quad \mbox{for all } t\in [0,T]. $$

(3.10)

On the other hand, from (3.4) and condition (2.1) in $[H_{1}]$ we have

$$\begin{aligned} \int^{T}_{0}\frac{1}{x^{\alpha }(t)}\,dt =& \int^{T}_{0}f\bigl(x'(t)\bigr)\,dt+ \int ^{T}_{0}\varphi (t)x(t)\,dt- \int^{T}_{0}p(t)\,dt \\ \leq& L \int^{T}_{0}\bigl\vert x'(t)\bigr\vert \,dt+N_{1}T\overline{\vert \varphi \vert }+T \overline{\vert p\vert } \\ \leq& L\rho T^{\frac{n}{n+1}}+N_{1}T\overline{\vert \varphi \vert }+T \overline{\vert p\vert }, \end{aligned}$$

where ρ is determined in (3.7). Substituting this formula into (3.10), we obtain

$$ \bigl\vert x'(t)\bigr\vert \leq \lambda \bigl[2L \rho T^{\frac{n}{n+1}}+4N_{1}T\overline{\vert \varphi \vert }+4T \overline{\vert p\vert } \bigr]:=\lambda N_{2}\quad \mbox{for all } t \in [0,T]. $$

(3.11)

So we have

$$ \bigl\vert x'(t)\bigr\vert \leq N_{2} \quad \mbox{for all } t\in [0,T]. $$

(3.12)

We further show that there exists a constant $\gamma_{0}\in (0,\gamma)$ such that each positive T= periodic solution of (2.3) satisfies

$$ x(t)>\gamma_{0}\quad \mbox{for all } t\in [0,T]. $$

(3.13)

In fact, suppose that $x(t)$ is an arbitrary positive T-periodic solution of (2.3). Then

$$ x''+\lambda f\bigl(x' \bigr)+\lambda \varphi (t)x-\frac{\lambda }{x^{\alpha }}= \lambda p(t), \quad \lambda \in (0,1]. $$

(3.14)

By Lemma 2.3 we see that there is a point $t_{1}\in [0,T]$ such that

$$x(t_{1})\geq \gamma. $$

For $t\in [t_{1},t_{1}+T]$, multiplying both sides of (3.14) with $x'(t)$ and integrating it over the interval $[t_{1},t]$ (or $[t,t_{1}]$), we get

$$\frac{\vert x'(t)\vert ^{2}}{2}-\frac{\vert x'(t_{1})\vert ^{2}}{2}+\lambda \int^{t}_{t _{1}}f\bigl(x' \bigr)x'\,dt=\lambda \int^{t}_{t_{1}}\frac{1}{x^{\alpha }}x'\,dt - \lambda \int^{t}_{t_{1}}\varphi (t)xx'\,dt+\lambda \int^{t}_{t_{1}}p(t)x'\,dt, $$

which results in

$$\begin{aligned}& \lambda \int^{x(t)}_{x(t_{1})}\frac{1}{s^{\alpha }}\,ds \\& \quad = \frac{\vert x'(t)\vert ^{2}}{2}-\frac{\vert x'(t_{1})\vert ^{2}}{2}+\lambda \int^{t} _{t_{1}}f\bigl(x'(s) \bigr)x'(s)\,ds +\lambda \int^{t}_{t_{1}}\varphi (s)x(s)x'(s)\,ds- \lambda \int^{t}_{t_{1}}p(s)x'(s)\,ds, \end{aligned}$$

that is,

$$\begin{aligned} \lambda \int^{x(t_{1})}_{x(t)}\frac{1}{s^{\alpha }}\,ds = &- \frac{\vert x'(t)\vert ^{2}}{2}+\frac{\vert x'(t_{1})\vert ^{2}}{2}-\lambda \int^{t}_{t _{1}}f\bigl(x'(s) \bigr)x'(s)\,ds \\ &{} -\lambda \int^{t}_{t_{1}}\varphi (s)x(s)x'(s)\,ds+ \lambda \int^{t} _{t_{1}}p(s)x'(s)\,ds. \end{aligned}$$

According to (2.2) in $[H_{1}]$, we get $\int^{t}_{t_{1}}f(x'(s))x'(s)\,ds \ge 0$. Thus, it follows from the last formula that

$$\begin{aligned} \lambda \int^{x(t_{1})}_{x(t)}\frac{1}{s^{\alpha }}\,ds \le& - \frac{\vert x'(t)\vert ^{2}}{2}+\frac{\vert x'(t_{1})\vert ^{2}}{2}-\lambda \int^{t}_{t _{1}}\varphi (s)x(s)x'(s)\,ds+ \lambda \int^{t}_{t_{1}}p(s)x'(s)\,ds \\ \le& \bigl\vert x'\bigr\vert _{\infty }^{2}+ \lambda \int_{0}^{T}\bigl\vert \varphi (s)x(s)x'(s)\bigr\vert \,ds+ \lambda \int^{T}_{0}\bigl\vert p(s)x'(s)\bigr\vert \,ds, \end{aligned}$$

which, together with (3.8) and (3.11), yields

$$\lambda \int^{x(t_{1})}_{x(t)}\frac{1}{s^{\alpha }}\,ds\le \lambda^{2} N ^{2}_{2}+\lambda^{2} N_{1}N_{2}T\overline{\vert \varphi \vert }+ \lambda^{2} N _{2}T\overline{\vert p\vert }, $$

that is,

$$ \int^{x(t_{1})}_{x(t)}\frac{1}{s^{\alpha }}\,ds\leq N^{2}_{2}+ N_{1}N _{2}T\overline{ \vert \varphi \vert }+ N_{2}T\overline{\vert p\vert }:=N_{3}. $$

(3.15)

Since $\alpha \ge 1$, it follows that there exists $\gamma_{0}\in (0, \gamma)$ such that

$$\int^{\gamma }_{\eta }\frac{1}{x^{\alpha }(t)}\,dt>N_{3} \quad \mbox{for all } \eta \in (0,\gamma_{0}), $$

which, together with (3.15), implies that

$$x(t)>\gamma_{0}\quad \mbox{for all } t\in [0,T]. $$

So (3.13) holds.

Let $n_{0}=\min \{D_{1},\gamma_{0}\}$ and $n_{1}\in (N_{1}+D_{2},+ \infty)$ be two constants. Then from (3.8), (3.12), and (3.13) we see that each possible positive T-periodic solution x to (2.3) satisfies

$$n_{0}< x(t)< n_{1},\qquad \bigl\vert x'(t)\bigr\vert < N_{2}. $$

This implies that condition 1 and condition 2 of Lemma 2.1 hold. In addition, from Remark 2.1 we can infer that

$$\frac{1}{c^{\alpha }}-c\overline{\varphi }+\overline{p}>0 \quad \mbox{for } c \in (0,n_{0}] $$

and

$$\frac{1}{c^{\alpha }}-c\overline{\varphi }+\overline{p}< 0 \quad \mbox{for } c \in [n_{1},+\infty), $$

which results in

$$\biggl(\frac{1}{n^{\alpha }_{0}}-n_{0}\overline{\varphi }+\overline{p} \biggr) \biggl(\frac{1}{n^{\alpha }_{1}}-n_{1}\overline{\varphi }+ \overline{p} \biggr)< 0. $$

Therefore, condition 3 of Lemma 2.1 holds. Thus, by Lemma 2.1 we see that equation (1.10) has at least one positive T-periodic solution. The proof is complete. □

Example 3.1

Consider the equation

$$ x''(t)+10x'(t)- \frac{(x'(t))^{3}}{1+(x'(t))^{2}}+a(1+2\sin t)x(t)-\frac{1}{x ^{2}(t)}=\cos t, $$

(3.16)

where $a\in (0,\infty)$. Corresponding to (1.10), we see that $f(x)=10x-\frac{x^{3}}{1+x^{2}}$, $\varphi (t)=a(1+2\sin t)$, $p(t)=\cos t$, and $T=2\pi$.

Firstly, from (3.16) we see that $f(0)=0$ and

$$\overline{\varphi_{+}}=\frac{1}{T} \int_{0}^{T}\varphi_{+}(t)\,dt= \frac{\frac{2 \pi }{3}+\sqrt{3}}{\pi }a,\qquad \overline{\varphi_{-}}=\frac{1}{T}\varphi _{-}(t)\,dt=\frac{-\frac{\pi }{3}+\sqrt{3}}{\pi }a. $$

Obviously, $[H_{2}]$ is satisfied. Secondly, integrating $f(x')$ over the internal $[0,T]$, we get

$$\begin{aligned} \biggl\vert \int^{T}_{0}f\bigl(x'\bigr)\,dt \biggr\vert =& \biggl\vert \int^{T}_{0}\biggl[10x'(t)- \frac{(x'(t))^{3}}{1+(x'(t))^{2}}\biggr]\,dt \biggr\vert \\ =& \biggl\vert - \int^{T}_{0}\frac{(x'(t))^{3}}{1+(x'(t))^{2}}\,dt \biggr\vert \\ =& \biggl\vert \int^{T}_{0}\frac{\vert x'(t)\vert ^{3}}{1+(x'(t))^{2}}\,dt \biggr\vert \\ \leq& \int^{T}_{0}\bigl\vert x'(t)\bigr\vert \,dt, \end{aligned}$$

which implies that we can chose $L=1$ such that assumption $[H_{1}]$ holds. Besides, from

$$yf(y)=10y^{2}-\frac{y^{4}}{1+y^{2}} \geq 9y^{2} $$

we see that the constant σ can be chosen as $\sigma =9$ such that assumption $[H_{1}]$ is satisfied. Last, let $L=1$, $\sigma =9$, $n=1$. Then we get

$$\begin{aligned}& 1-\frac{LT^{\frac{-1}{2}}+T^{\frac{1}{2}}\overline{\varphi_{+}}}{ \sigma (\overline{\varphi_{+}}-\overline{\varphi_{-}})} \Vert \varphi \Vert _{2}=1-\frac{ \sqrt{3}}{9}-\frac{18+4\sqrt{3}\pi }{27}a>0, \\& 1-\sigma^{-1}\Vert \varphi \Vert _{2}T^{\frac{1}{2}}=1- \frac{2\pi }{3 \sqrt{3}}a>0. \end{aligned}$$

If

$$a< \frac{27-3\sqrt{3}}{18+4\sqrt{3}\pi }, $$

then $[H_{3}]$ holds. Thus, by Theorem 3.1 we have that equation (3.16) has at least one positive 2π-periodic solution.