Periodic solutions for p-Laplacian Rayleigh equations with singularities

Abstract

In this paper, the problem of existence of periodic solutions is studied for p-Laplacian Rayleigh equations with a singularity at $x=0$. By using the topological degree theory, some new results are obtained.

1 Introduction

In recent years, the periodic problem for some types of singular equations has attracted much attention of many researchers because the singular nonlinearity possesses a significant role in many practical situations. For example, the differential equation

$$ x''(t)+cx'(t)-\frac{1}{x(t)}=e(t) $$

described the motion of a piston in a cylinder closed at one extremity. The singular term $-\frac{1}{x}$ in the equation models the restoring force which is caused by a compressed perfect gas (see [1] and the references therein). The interest in studying the equations with a singularity began with some work of Forbat and Huaux [2]. Later, the interest in such problem was renewed by Gordon in [3, 4], and Lazer and Solimini in [5]. For the recent developments on the study of this problem, here, we refer the reader to [6–16], and we notice that the equations studied previously were either of the type of Duffing equations [7–11, 14–16] or of the type of Liénard equations [11, 12, 17, 18]. For example, Jebelean and Mawhin in [1] considered the problem of the existence of positive periodic solutions for the following p-Laplacian Liénard equations with a singularity:

$$ \bigl(\bigl|x'\bigr|^{p-2}x' \bigr)'+f(x)x'+g(x)=h(t) $$

(1.1)

and

$$ \bigl(\bigl|x'\bigr|^{p-2}x' \bigr)'+f(x)x'-g(x)=h(t), $$

(1.2)

where $p>1$ is a constant, $f: [0,+\infty)\rightarrow\mathbb{R}$ is an arbitrary continuous function, $h: \mathbb{R}\rightarrow\mathbb{R}$ is a T-periodic function with $h\in L^{\infty}[0,T]$, $g:(0,+\infty )\rightarrow(0,+\infty)$ is continuous, and singular at $x=0$, this means that $g(x)$ is unbounded as $x\rightarrow0^{+}$. The crucial condition imposed on $g(x)$ is that $g(x)\rightarrow+\infty$ as $x\rightarrow0^{+}$, i.e., equation (1.1) is of attractive type, and equation (1.2) is of repulsive type. Zhang in [17] studied the problem of periodic solutions of the Liénard equation with a repulsive singularity at $x=0$,

$$ x''+f(x)x'+g(t,x)=0. $$

(1.3)

In [18], Wang further studied the existence of positive periodic solutions for a delay Liénard equation with a repulsive singularity at $x=0$,

$$ x''+f(x)x'+g \bigl(t,x(t- \tau) \bigr)=0. $$

(1.4)

The methods in [1] for equation (1.2), in [17] for equation (1.3) and in [18] for equation (1.4) were all based on topological degree theory, and the upper and lower solutions techniques was used in [1] for equation (1.1). But as far as we are aware of, few results appeared on the existence of periodic solutions for p-Laplacian Rayleigh equation with a singularity.

Motivated by this, in this paper, we study the existence of positive T-periodic solutions for p-Laplacian Rayleigh equation with a singularity of the form

$$ \bigl(\bigl|x'\bigr|^{p-2}x' \bigr)'+f \bigl(x' \bigr)-g_{1}(x)+g_{2}(x)=h(t) $$

(1.5)

and

$$ \bigl(\bigl|x'\bigr|^{p-2}x' \bigr)'+f \bigl(x' \bigr)+g_{1}(x)-g_{2}(x)=h(t), $$

(1.6)

where $p>1$ is a constant, $f:\mathbb{R}\rightarrow\mathbb{R}$ is an arbitrary continuous function, $g_{1},g_{2}:(0,+\infty)\rightarrow\mathbb {R}$ are all continuous and $g_{1}(x)$ is unbounded as $x\rightarrow 0^{+}$, $h: \mathbb{R}\rightarrow\mathbb{R}$ is a T-periodic continuous function. Clearly, equation (1.5) and equation (1.6) are all singular at $x=0$. By using Manásevich-Mawhin’s continuation theorem, some new results are obtained.

The interesting thing is that the singular term in equation (1.6) (or in equation (1.5)) is not required to have $g_{1}(x)\rightarrow +\infty$ (or $-g_{1}(x)\rightarrow-\infty$) as $x\rightarrow0^{+}$. For example, let

$$ g_{1}(u)=\frac{1}{u^{\mu}}\biggl|\sin\frac{1}{u}\biggr|, $$

where $\mu\ge1$ is a constant. It is easy to verify that $g_{1}(x)$ does not approach +∞ as $x\rightarrow0^{+}$. Furthermore, if $x\in C^{1}(\mathbb{R},\mathbb{R})$ with T-periodic, then the first order derivative term $f(x)x'$ in equations (1.1)-(1.4) satisfies $\int_{0}^{T}f(x(t))x'(t)\,dt=0$, which is crucial for obtaining an a priori bounds of all the possible T-periodic solutions for equation (1.1)-equation (1.4). However, the first order derivative term in equation (1.5) and equation (1.6) is $f(x')$; generally, $\int_{0}^{T}f(x'(t))\,dt=0$ does not hold. This means that the method for estimating an a priori bounds of all the possible T-periodic solutions to equation (1.5) and equation (1.6) is different from the corresponding ones in [1, 17, 18].

2 Preliminary lemmas

The following two lemmas (Lemma 2.1 and Lemma 2.2) are all consequences of Theorem 3.1 in [19].

Lemma 2.1

Assume that there exist constants $0<\eta _{0}<\eta_{1}$, $M_{2}>0$, such that the following conditions hold.

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

  $$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'+ \lambda f \bigl(u' \bigr)-\lambda g_{1}(u)+\lambda g_{2}(t,u)=\lambda h(t) $$

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

• (2) Each possible solution c to the equation

  $$ g_{1}(c)-g_{2}(c)+\bar{h}=0 $$

  satisfies the inequality $\eta_{0}< c<\eta_{1}$.

• (3) It holds

  $$ \bigl(g_{1}(\eta_{0})-g_{2}( \eta_{0})+\bar{h} \bigr) \bigl(g_{1}(\eta _{1})-g_{2}(\eta_{1})+\bar{h} \bigr)< 0. $$

Then equation (1.5) has at least one T-periodic solution u such that $\eta_{0}< u(t)<\eta_{1}$ for all $t\in[0,T]$.

Lemma 2.2

Assume that there exist constants $0<\eta _{0}<\eta_{1}$, $M_{2}>0$, such that the following conditions hold:

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

  $$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'+ \lambda f \bigl(u' \bigr)+\lambda g_{1}(u)-\lambda g_{2}(u)=\lambda h(t) $$

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

• (2) Each possible solution c to the equation

  $$ g_{1}(c)-g_{2}(c)-\bar{h}=0 $$

  satisfies the inequality $\eta_{0}< c<\eta_{1}$.

• (3) It holds

  $$ \bigl(g_{1}(\eta_{0})-g_{2}( \eta_{0})-\bar{h} \bigr) \bigl(g_{1}(\eta _{1})-g_{2}(\eta_{1})-\bar{h} \bigr)< 0. $$

Then equation (1.6) has at least one T-periodic solution u such that $\eta_{0}< u(t)<\eta_{1}$ for all $t\in[0,T]$.

In order to study the existence of positive periodic solutions to equation (1.5) and equation (1.6), we list the following assumptions:

(H₁) there are positive constants $m_{0}$ and $m_{1}$ with $m_{0}< m_{1}$ such that

$$ g_{1}(x)-g_{2}(x)-|h|_{\infty}>0\quad \mbox{for all } x\in(0,m_{0}] $$

(2.1)

and

$$ g_{1}(x)-g_{2}(x)+|h|_{\infty}< 0\quad \mbox{for all }x\in[m_{1},\infty), $$

(2.2)

where $|h|_{\infty}=\max_{t\in[0,T]}|h(t)|$;

(H₂) $g_{1}(x)\ge0$ for all $x\in(0,+\infty)$, and $\int _{0}^{1}g_{1}(s)\,ds=+\infty$;

(H₃) $f(0)=0$;

(H₄) there are constants n, $\sigma_{0}$ and $\sigma_{1}$ with $n>1$, $0<\sigma_{0}\le\sigma_{1}$ such that

$$ xf(x)\ge\sigma_{0}|x|^{n} \quad\mbox{or}\quad xf(x)\le- \sigma_{0}|x|^{n}\quad \mbox{for all } x\in\mathbb{R} $$

and

$$ \bigl|f(x)\bigr|\le\sigma_{1}|x|^{n-1} \quad\mbox{for all } x\in\mathbb{R}. $$

Now, we embed equation (1.5) and equation (1.6) into the following two equations family with a parameter $\lambda\in(0,1]$, respectively:

$$ \bigl(\bigl|x'\bigr|^{p-2}x' \bigr)'+\lambda f \bigl(x' \bigr)-\lambda g_{1}(x)+\lambda g_{2}(x)=\lambda h(t), \quad\lambda\in(0,1], $$

(2.3)

and

$$ \bigl(\bigl|x'\bigr|^{p-2}x' \bigr)'+\lambda f \bigl(x' \bigr)+\lambda g_{1}(x)-\lambda g_{2}(x)=\lambda h(t), \quad\lambda\in(0,1]. $$

(2.4)

Lemma 2.3

Assume that assumptions (H₁) and (H₃) hold, let $m_{0}$ and $m_{1}$ be positive constants determined in assumption (H₁). Then the following conclusions hold:

• (1) for each possible positive T-periodic solution $u(t)$ of equation (2.3) there exists $\tau\in[0,T]$ such that

  $$ m_{0}< u(\tau)< m_{1}; $$

  (2.5)

• (2) each possible solution c to the equation

  $$ g_{1}(c)-g_{2}(c)+\bar{h}=0 $$

  satisfies the inequality $m_{0}< c< m_{1}$;

• (3) $g_{1}(u)-g_{2}(u)+\bar{h}>0$ for all $u\in(0,m_{0}]$, and $g_{1}(u)-g_{2}(u)+\bar{h}<0$ for all $u\in[m_{1},+\infty)$.

Proof

(1) Suppose that $u(t)$ be an arbitrary positive T-periodic solution to equation (2.3), then

$$ \bigl(\bigl|u'(t)\bigr|^{p-2}u'(t) \bigr)'+\lambda f \bigl(u'(t) \bigr)-\lambda g_{1} \bigl(u(t) \bigr)+\lambda g_{2} \bigl(u(t) \bigr)= \lambda h(t),\quad \lambda\in(0,1]. $$

(2.6)

Let $t_{0}$ and $t_{1}$ be the maximum point and the minimum point of $u(t)$ on $[0,T]$, respectively, then $u'(t_{0})=0$ and $u'(t_{1})=0$. We can prove that

$$ \bigl(\bigl|u'(t)\bigr|^{p-2}u'(t) \bigr)' |_{t=t_{0}}\le0. $$

(2.7)

In fact, if (2.7) does not hold, then $(|u'|^{p-2}u')' |_{t=t_{0}}>0$. By the continuity of $(|u'(t)|^{p-2}u'(t))'$ for $t\in [t_{0},t_{0}+T]$, we see that there is a constant $\delta\in(0,T)$ such that $(|u'(t)|^{p-2}u'(t))'>0$ for $t\in(t_{0},t_{0}+\delta)$, and then $|u'(t)|^{p-2}u'(t)>|u'(t)|^{p-2}u'(t) |_{t=t_{0}}=0$ for $t\in (t_{0},t_{0}+\delta)$, i.e., $u'(t)>0$ for $t\in(t_{0},t_{0}+\delta )$, which results in $u(t)>u(t_{0})$ for $t\in(t_{0},t_{0}+\delta)$. So $u(t_{0})<\max_{t\in[t_{0},t_{0}+T]}u(t)=\max_{t\in[0,T]}u(t)$, which contradicts the fact that $t_{0}$ is the maximum point of $u(t)$ on $[0,T]$. This contradiction implies that (2.7) holds. Similarly, we have

$$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'|_{t=t_{1}}\ge0. $$

(2.8)

It follows from (2.6) and (2.7) that

$$ f \bigl(u'(t_{0}) \bigr)-g_{1} \bigl(u(t_{0}) \bigr)+g_{2} \bigl(u(t_{0}) \bigr) \ge-|h|_{\infty}. $$

(2.9)

By using assumption (H₃), we have $f(u'(t_{0}))=f(0)=0$, which together with (2.9) yields

$$ g_{1} \bigl(u(t_{0}) \bigr)-g_{2} \bigl(u(t_{0}) \bigr)\le|h|_{\infty}, $$

and, by using condition (2.1) in assumption (H₁), we have

$$ u(t_{0})> m_{0}. $$

(2.10)

Similarly, condition (2.2) in assumption (H₁), together (2.6) and (2.8), implies that

$$ u(t_{1})< m_{1}. $$

(2.11)

Without loss of generality, suppose $u(t_{0})>m_{1}$, then by (2.10) and (2.11), we obtain from the intermediate value property of the continuous function $u(t)$ that (2.5) holds.

(2) Conclusion (2), as well as conclusion (3), follows directly from assumption (H₁). □

Similar to the proof of Lemma 2.3, we obtain the following result.

Lemma 2.4

Assume that assumptions (H₁) and (H₃) hold, let $m_{0}$ and $m_{1}$ be positive constants determined in assumption (H₁). Then the following conclusions hold:

• (1) each possible positive T-periodic solution $u(t)$ to equation (2.4) satisfies

  $$ m_{0}< u(t)< m_{1} \quad\textit{for all } t\in[0,T]; $$

• (2) each possible solution c to the equation

  $$ g_{1}(c)-g_{2}(c)-\bar{h}=0 $$

  satisfies the inequality $m_{0}< c< m_{1}$;

• (3) $g_{1}(u)-g_{2}(u)-\bar{h}>0$ for all $u\in(0,m_{0}]$, and $g_{1}(u)-g_{2}(u)-\bar{h}<0$ for all $u\in[m_{1},+\infty)$.

3 Main results

Theorem 3.1

Assume that assumptions (H₁), (H₂), and (H₄) hold, then equation (1.5) has at least one positive T-periodic solution.

Proof

First of all, we will show that there exist $M_{1}$, $M_{2}$ with $M_{1}>m_{1}$ and $M_{2}>0$ such that each positive T-periodic solution $u(t)$ of equation (2.3) satisfies the inequalities

$$ u(t)< M_{1}, \qquad\bigl|u'(t)\bigr|< M_{2}. $$

(3.1)

In fact, if u is a positive T-periodic solution of equation (2.3), then

$$ \bigl(\bigl|u'(t)\bigr|^{p-2}u'(t) \bigr)'+\lambda f \bigl(u'(t) \bigr)-\lambda g_{1} \bigl(u(t) \bigr)+\lambda g_{2} \bigl(u(t) \bigr)= \lambda h(t),\quad \lambda\in(0,1]. $$

(3.2)

It is easy to see that assumption (H₃) can be deduced from (H₄), so by using Lemma 2.3, we see that there is a point $\tau\in [0,T]$ such that

$$ m_{0}< u(\tau)< m_{1}. $$

(3.3)

Multiplying (3.2) with $u'(t)$ and integrating over the interval $[0,T]$, we have

$$ \int_{0}^{T} u'(t)f \bigl(u'(t) \bigr)\,dt= \int_{0}^{T}u'(t)h(t)\,dt. $$

(3.4)

It follows from assumption (H₄) that

$$ \sigma_{0} \int_{0}^{T}\bigl|u'(t)\bigr|^{n}\,dt \le \int_{0}^{T}\bigl|u'(t)f \bigl(u'(t) \bigr)\bigr|\,dt=\biggl| \int _{0}^{T}u'(t)f \bigl(u'(t) \bigr)\,dt\biggr|, $$

which together with (3.4) yields

$$ \sigma_{0} \int_{0}^{T}\bigl|u'(t)\bigr|^{n}\,dt \le \biggl( \int_{0}^{T}\bigl|h(t)\bigr|^{\frac {n}{n-1}}\,dt \biggr)^{\frac{n-1}{n}} \biggl( \int_{0}^{T}\bigl|u'(t)\bigr|^{n}\,dt \biggr)^{\frac{1}{n}}, $$

i.e.,

$$ \int_{0}^{T}\bigl|u'(t)\bigr|^{n}\,dt \le\sigma_{0}^{\frac{n}{1-n}} \int_{0}^{T}\bigl|h(t)\bigr|^{\frac {n}{n-1}}\,dt, $$

(3.5)

and then by (3.4), we get

$$\begin{aligned} u(t)&\le u(\tau)+T^{\frac{n-1}{n}} \biggl( \int_{0}^{T}\bigl|u'(t)\bigr|^{n}\,dt \biggr)^{\frac{1}{n}} \\ &< m_{1}+T^{\frac{n-1}{n}}\sigma_{0}^{\frac{1}{1-n}} \biggl( \int _{0}^{T}\bigl|h(t)\bigr|^{\frac{n}{n-1}}\,dt \biggr)^{\frac{n-1}{n}} :=M_{1}. \end{aligned}$$

(3.6)

Let $G=\max_{x\in[0,M_{1}]}|g_{2}(x)|$, then it follows from (3.2) and the fact of $g_{1}\in C((0,\infty),[0,+\infty))$ that

$$\begin{aligned} \int_{0}^{T}\bigl| \bigl(\bigl|u'(t)\bigr|^{p-2}u'(t) \bigr)'\bigr|\,dt\le{}&\lambda \int_{0}^{T}\bigl|f \bigl(u'(t) \bigr)\bigr| \,dt+ \lambda \int_{0}^{T}g_{1} \bigl(u(t) \bigr)\,dt+ \lambda TG \\ &{}+\lambda \int_{0}^{T}\bigl|h(t)\bigr|\,dt. \end{aligned}$$

(3.7)

Furthermore, by integrating (3.2) over the interval $[0,T]$, we have

$$\begin{aligned} \int_{0}^{T}g_{1} \bigl(u(t) \bigr)\,dt&= \int_{0}^{T}g_{2} \bigl(u(t) \bigr)\,dt+ \int_{0}^{T}f \bigl(u'(t) \bigr)\,dt- \int _{0}^{T}h(t)\,dt \\ &\le \int_{0}^{T}\bigl|f \bigl(u'(t) \bigr)\bigr| \,dt+TG+ \int_{0}^{T}\bigl|h(t)\bigr|\,dt. \end{aligned}$$

Substituting it into (3.7), and by using assumption (H₃), we have

$$\begin{aligned} \int_{0}^{T}\bigl| \bigl(\bigl|u'(t)\bigr|^{p-2}u'(t) \bigr)'\bigr|\,dt \le&2\lambda \int _{0}^{T}\bigl|f \bigl(u'(t) \bigr)\bigr| \,dt+2 \lambda TG+2\lambda \int_{0}^{T}\bigl|h(t)\bigr|\,dt \\ \le&2\lambda\sigma_{1} \int_{0}^{T}\bigl|u'(t)\bigr|^{n-1} \,dt+2 \lambda TG+2\lambda \int _{0}^{T}\bigl|h(t)\bigr|\,dt \\ \le&2\lambda\sigma_{1}T^{\frac{1}{n}} \biggl( \int_{0}^{T}\bigl|u'(t)\bigr|^{n}\,dt \biggr)^{\frac{n-1}{n}}+2\lambda TG+2\lambda \int_{0}^{T}\bigl|h(t)\bigr|\,dt, \end{aligned}$$

which together with (3.5) yields

$$\begin{aligned} \int_{0}^{T}\bigl| \bigl(\bigl|u'(t)\bigr|^{p-2}u'(t) \bigr)'\bigr|\,dt\le{}&2\lambda \biggl[\sigma_{1}\sigma _{0}^{-1}T^{\frac{1}{n}} \biggl( \int_{0}^{T}\bigl|h(t)\bigr|^{\frac{n}{n-1}}\,dt \biggr)^{\frac{n-1}{n}}+TG \\ &{}+ \int_{0}^{T}\bigl|h(t)\bigr|\,dt \biggr]. \end{aligned}$$

(3.8)

Since $\max_{t\in[0,T]}|u'(t)|^{p-1}\le\int _{0}^{T}|(|u'(t)|^{p-2}u'(t))'|\,dt$, it follows from (3.8) that

$$ \max_{t\in[0,T]}\bigl|u'(t)\bigr|^{p-1}< \lambda M_{2}, $$

(3.9)

i.e.,

$$ \max_{t\in[0,T]}\bigl|u'(t)\bigr|< M_{2}, $$

(3.10)

where $M_{2}=1+2^{\frac{1}{p-1}} [\sigma_{1}\sigma_{0}^{-1}T^{\frac {1}{n}} (\int_{0}^{T}|h(t)|^{\frac{n}{n-1}}\,dt )^{\frac {n-1}{n}}+TG+\int_{0}^{T}|h(t)|\,dt ]^{\frac{1}{p-1}}$.

Below, we will show that there exists a constant $M_{0}\in (0,m_{0})$, such that

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

(3.11)

Let τ be determined as in Lemma 2.3. Multiplying (3.2) by $u'(t)$ and integrating over the interval $[\tau,t]$ (or $[t,\tau]$), we get

$$\begin{aligned} & \int_{\tau}^{t} \bigl(\bigl|u'(s)\bigr|^{p-2}u'(s) \bigr)'u'(s)\,ds \\ &\qquad{}+\lambda \int_{t_{0}}^{t} f \bigl(u'(s) \bigr)u'(s)\,ds-\lambda \int_{\tau}^{t} g_{1} \bigl(u(s) \bigr)u'(s)\,ds+\lambda \int_{\tau}^{t}g_{2} \bigl(u(s) \bigr)u'(s)\,ds \\ &\quad=\lambda \int_{\tau}^{t} h(s)u'(s)\,ds,\quad \lambda \in(0,1). \end{aligned}$$

(3.12)

Set $y(t)=|u'(t)|^{p-2}u'(t)$, then $y(t)$ is absolutely continuous and $u'(t)=|y(t)|^{q-2}y(t)$, where $q\in(1,+\infty)$ with $\frac {1}{p}+\frac{1}{q}=1$. So

$$\begin{aligned} \int_{\tau}^{t} \bigl(\bigl|u'(s)\bigr|^{p-2}u'(s) \bigr)'u'(s)\,ds =& \int_{\tau }^{t}\bigl|y(s)\bigr|^{q-2}y(s)y'(s) \,ds \\ =&\frac{|y(t)|^{q}}{q}-\frac{|y(\tau)|^{q}}{q}=\frac{|u'(t)|^{p}}{q}-\frac {|u'(\tau)|^{p}}{q}. \end{aligned}$$

Substituting it into (3.12), we get

$$ \frac{|u'(t)|^{p}}{q}-\frac{|u'(\tau)|^{p}}{q}+\lambda \int_{\tau }^{t}f \bigl(u' \bigr)u'\,dt=\lambda \int_{\tau}^{t}g_{1}(u)u' \,dt- \lambda \int_{\tau }^{t}g_{2}(u)u' \,dt+ \lambda \int_{\tau}^{t} h(t)u'\,dt, $$

which yields the estimate

$$\begin{aligned} \lambda \int_{u(t)}^{u(\tau)}g_{1}(s)\,ds \leq& \frac{|u'(t)|^{p}}{q}+\frac {|u'(\tau)|^{p}}{q} \\ &{}+\lambda \int_{0}^{T}\bigl|f \bigl(u' \bigr)\bigr|\bigl|u'\bigr|\,dt+\lambda \int _{0}^{T}\bigl|g_{2}(u)\bigr|\bigl|u'\bigr| \,dt+ \lambda \int_{0}^{T} \bigl|h(t)u'\bigr|\,dt. \end{aligned}$$

From (3.9), we get

$$ \lambda \int_{u(t)}^{u(\tau)}g_{1}(s)\,ds\leq \frac{2\lambda M_{2}^{\frac {p}{p-1}}}{q}+\lambda \Bigl(\max_{0\leq u\leq M_{2}}\bigl|f(u)\bigr| \Bigr)TM_{2}^{\frac {1}{p-1}}+\lambda GTM_{2}^{\frac{1}{p-1}}+ \lambda\|h\| _{L_{1}}M_{2}^{\frac{1}{p-1}}, $$

which gives

$$ \int_{u(t)}^{u(\tau)}g_{1}(s)\,ds\leq M_{3} \quad \mbox{for all } t\in[\tau ,\tau+T] $$

(3.13)

with

$$ M_{3}= \frac{2M_{2}^{\frac{p}{p-1}}}{q}+ \Bigl(\max_{0\leq u\leq M_{2}}\bigl|f(u)\bigr| \Bigr)TM_{2}^{\frac{1}{p-1}}+TGM_{2}^{\frac{1}{p-1}}+\|h\| _{L_{1}}M_{2}^{\frac{1}{p-1}}. $$

From (H₂) there exists $M_{0}\in(0,m_{0})$ such that

$$ \int_{M_{0}}^{m_{0}}g_{1}(s) \,ds>M_{3}. $$

(3.14)

Therefore, if there is a $t^{*}\in[\tau,\tau+T]$ such that $u(t^{*})\le M_{0}$, then from (3.14) we get

$$ \int_{u(t^{*})}^{u(\tau)}g_{1}(s)\,ds\ge \int_{M_{0}}^{m_{0}}g_{1}(s)\,ds> M_{3}, $$

which contradicts (3.13). This contradiction shows that $u(t)>M_{0}$ for all $t\in[0,T]$. So (3.11) holds. Let $\eta_{0}\in(0, M_{0})$ and $\eta_{1}\in(M_{1}, +\infty)$ be two constants, then from (3.6), (3.10), and (3.11), we see that each possible positive T-periodic solution u to equation (2.3) satisfies

$$ \eta_{0}< u(t)< \eta_{1},\qquad\bigl|u'(t)\bigr|< M_{2}. $$

This implies that condition (1) of Lemma 2.1 is satisfied. We can deduce from conclusion (2) of Lemma 2.3 that each possible solution c to the equation

$$ g_{1}(c)-g_{2}(c)+\bar{h}=0 $$

satisfies the inequality $\eta_{0}< c<\eta_{1}$, and from conclusion (3) of Lemma 2.3, we obtain

$$ g_{1}(c)-g_{2}(c)+\bar{h}>0\quad \mbox{for } c\in(0, \eta_{0}] $$

and

$$ g_{1}(c)-g_{2}(c)+\bar{h}< 0\quad \mbox{for } c\in[ \eta_{1},+\infty), $$

which results in

$$ \bigl(g_{1}(\eta_{0})-g_{2}( \eta_{0})+\bar{h} \bigr) \bigl(g_{1}(\eta _{1})-g_{2}(\eta_{1})+\bar{h} \bigr)< 0. $$

So condition (3) of Lemma 2.1 holds. By using Lemma 2.1, we see that equation (1.5) has at least one positive T-periodic solution. The proof is complete. □

Theorem 3.2

Assume that assumptions (H₁) and (H₃) hold, then equation (1.6) has at least one positive T-periodic solution.

Proof

Suppose that $u(t)$ be an arbitrary T-periodic solution to equation (2.4), then

$$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'+\lambda f \bigl(u' \bigr)+\lambda g_{1}(u)-\lambda g_{2}(u)=\lambda h(t),\quad \lambda\in(0,1]. $$

(3.15)

By using Lemma 2.4, we see that

$$ m_{0}< u(t)< m_{1} \quad\mbox{for all } t \in[0,T], $$

(3.16)

where $m_{0}$ and $m_{1}$ are constants determined in assumption (H₁). Multiplying equation (3.15) with $(|u'|^{p-2}u')'$ and integrating over the interval $[0,T]$, we have

$$\begin{aligned} & \int_{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|^{2}\,dt+\lambda \int _{0}^{T} \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'f \bigl(u'(t) \bigr)\,dt \\ &\qquad{}+\lambda \int_{0}^{T} \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'g_{1}(u)\,dt-\lambda \int _{0}^{T} \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'g_{2}(u)\,dt \\ &\quad=\lambda \int_{0}^{T} \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'h(t)\,dt. \end{aligned}$$

(3.17)

Let $G_{1}=\max_{0\le x\le m_{1}}|g_{2}(x)|$, $G_{2}=\max_{m_{0}\le x\le m_{1}}|g_{1}(x)|$. Take $y(t)=|u'(t)|^{p-2}u'(t)$, then $u'(t)=|y(t)|^{q-2}y(t)$, where $q\in(1,+\infty)$ is a constant with $\frac{1}{p}+\frac{1}{q}=1$, then

$$ \int_{0}^{T} \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'f \bigl(u'(t) \bigr)\,dt= \int_{0}^{T}f \bigl(\bigl|y(t)\bigr|^{q-2}y(t) \bigr)y'(t)\,dt=0. $$

So, it follows from (3.16) and (3.17) that

$$\begin{aligned} & \int_{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|^{2}\,dt \\ &\quad\le G_{1} \int_{0}^{T}\bigl| \bigl(\bigr|u'\bigr|^{p-2}u' \bigr)'\bigl|\,dt+G_{2} \int _{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|\,dt+|h|_{\infty} \int_{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|\,dt \\ &\quad\le T^{1/2} \bigl(G_{1}+G_{2}+|h|_{\infty} \bigr) \biggl( \int _{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|^{2}\,dt \biggr)^{1/2}, \end{aligned}$$

which results in

$$ \int_{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|^{2}\,dt\le T \bigl(G_{1}+G_{2}+|h|_{\infty} \bigr)^{2}. $$

(3.18)

Since

$$\begin{aligned} \max_{t\in[0,T]}\bigl|u'(t)\bigr|^{p-1}&=\max _{t\in[0,T]}\bigl|u'(t)\bigr|^{p-2}\bigl|u'(t)\bigr| \le \int_{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|\,dt\\ &\le T^{1/2} \biggl( \int _{0}^{T}\bigl| \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'\bigr|^{2}\,dt \biggr)^{1/2}, \end{aligned}$$

it follows from (3.18) that

$$ \max_{t\in[0,T]}\bigl|u'(t)\bigr|^{p-1}\le T \bigl(G_{1}+G_{2}+|h|_{\infty} \bigr)^{1/2}, $$

i.e.,

$$ \max_{t\in[0,T]}\bigl|u'\bigr| \le T^{\frac{1}{p-1}} \bigl(G_{1}+G_{2}+|h|_{\infty } \bigr)^{\frac{1}{2(p-1)}}. $$

(3.19)

Let $M_{2}= T^{\frac{1}{p-1}} (G_{1}+G_{2}+|h|_{\infty} )^{\frac {1}{2(p-1)}}+1$, and $u(t)$ be an arbitrary T-periodic solution to equation (2.4). Then from (3.16) and (3.19), we see that

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

This implies that condition (1) of Lemma 2.2 is satisfied, and it is easy to see from assumption (H₁) that conditions (2)-(3) in Lemma 2.2 also hold. By using Lemma 2.2, we see that equation (1.6) possesses a T-periodic solution $u(t)$ such that $m_{0}\le u(t)\le m_{1}$ for all $t\in[0,T]$. The proof is complete. □

Now, if the singular restoring force term $g_{1}(x)$ in equation (1.5) (or in equation (1.6)) satisfies

$$ \lim_{x\rightarrow0^{+}}g_{1}(x)=+\infty $$

(3.20)

and $g_{2}(x)$ satisfies

$$ \lim_{x\rightarrow+\infty}g_{2}(x)=+\infty, $$

(3.21)

then assumption (H₁) holds. Thus, by applying Theorem 3.1 and Theorem 3.2, respectively, we can obtain the following results.

Corollary 3.1

Assume that (3.20), (3.21), and assumptions (H₂) and (H₄) hold, then equation (1.5) has at least one positive T-periodic solution.

Corollary 3.2

Assume that (3.20), (3.21), and assumption (H₃) hold, then equation (1.6) has at least one positive T-periodic solution.

Example 3.1

Considering the following equation:

$$ x''(t)+ \bigl(x'(t) \bigr)^{3}-\frac{1}{x^{2}(t)}\biggl|\sin\frac{1}{x(t)}\biggr|+x^{2}(t)= \cos t+2. $$

(3.22)

Corresponding to equation (1.5), $f(u)=u^{3}$, $g_{1}(u)$ can be regarded as $g_{1}(u)=\frac{1}{u^{2}}|\sin\frac{1}{u}|$, $g_{2}(u)=u^{2}-2$, and $h(t)=\cos t$. Since

$$\int_{0}^{1} \biggl(\frac{1}{u^{2}}\biggl|\sin \frac{1}{u}\biggr| \biggr)\,du=+\infty, $$

it follows that assumption (H₂) holds. By simple calculating, we can chose $\sigma_{0}=\sigma_{1}=1$, $n=4$, $m_{0}=\frac{1}{2}$, and $m_{1}=2$ such that verifying assumptions (H₁) and (H₄). Thus, by using Theorem 3.1, we see that equation (3.22) has at least one positive 2π-periodic solution.

Example 3.2

Consider the following equation:

$$ x''(t)+ \bigl(x'(t) \bigr)^{4}+\frac{1}{x^{\frac{1}{2}}(t)}-\frac{1}{x^{\frac {1}{4}}(t)}-x^{3}(t)=\cos t. $$

(3.23)

Corresponding to equation (1.6), $f(u)=u^{4}$, $g_{1}(u)=\frac {1}{u^{\frac{1}{2}}}-\frac{1}{u^{\frac{1}{4}}}$, $g_{2}(u)=u^{3}$, and $h(t)=\cos t$. It is easy to see that conditions (3.20) and (3.21), and assumption (H₃) are all satisfied. By using Corollary 3.2, we see that equation (3.23) has at least one positive 2π-periodic solution.

Remark 3.1

The first order derivative term in equation (3.23) is $(x'(t))^{4}$, and then $\int_{0}^{T}(x'(t))^{4}\,dt\neq0$ for all T-periodic continuous function x, generally. This implies that the methods used for studying equation (1.1) in [1] is not valid for equation (3.23).