Singularities of attractive and repulsive type for p-Laplacian generalized Liénard equation

Abstract

This paper is devoted to an investigation of a kind of p-Laplacian generalized Liénard equations with singularities of attractive and repulsive type, where the nonlinear term g has a singularity at the origin. The novelty of the present article is that we show that singularities of attractive and repulsive type enable the achievement of a new existence criterion of a positive periodic solution through an application of the Manásevich–Mawhin theorem on continuity of the topological degree, recent results in the literature are generalized and significantly improved. Finally, some examples are given to show applications of the theorems.

1 Introduction

In this paper, we consider the following p-Laplacian generalized Liénard equation with singularity:

$$ \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+f\bigl(t,x(t)\bigr)x'(t)+g\bigl(t,x(t) \bigr)=e(t), $$

(1.1)

where $p\geq 2$, $\phi_{p}(x)=\vert x \vert ^{p-2}x$ for $x\neq 0$ and $\phi_{p}(0)=0$, $f:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous function and it is T-periodic about t, $e\in C( \mathbb{R},\mathbb{R})$ is a T-periodic function, $g(t,x)=g_{1}(t,x)+g _{0}(x)$, $g_{1}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous function and it is T-periodic about t, $g_{0}: (0,+ \infty )\to \mathbb{R}$ is a continuous function and has a singularity at the origin, i.e.,

$$ \lim_{x\rightarrow 0^{+}} g_{0}(x)=+\infty \qquad \Bigl(\mbox{or }\lim_{x\rightarrow 0^{+}} g_{0}(x)=-\infty \Bigr). $$

It is said that (1.1) is singularity of attractive type (resp. repulsive type) if $g_{0}(x)\rightarrow +\infty$ (resp. $g_{0}(x) \rightarrow -\infty$) as $x\rightarrow 0^{+}$.

The Liénard equation [10],

$$ x'' + f(x)x' + g(x) = 0, $$

(1.2)

appears as a simplified model in many domains in science and engineering. It was intensively studied during the first half of the 20th century as it can be used to model oscillating circuits or simple pendulums. For example, the Van der Pol oscillator

$$ x''-\mu \bigl(1-x^{2}\bigr)x'+x=0 $$

is a Liénard equation.

From then on, there has appeared some good amount of work on periodic solutions for Liénard equations and the references cited therein. Some classical tools have been used to study the Liénard equation in the literature, including topological degree methods [12, 17], Mawhin’s coincidence degree theorem [1, 3, 4, 11], Massera’s theorem [21], the Manásevich–Mawhin theorem on continuity of the topological degree [23, 26], Schauder’s fixed point theorem [20], generalized polar coordinates [22], and the Poincaré map [27].

At the same time, the study of periodic solution of the Liénard equation with singularity can be traced back to 1996. Zhang in [29] discussed the following singular Liénard equation:

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

(1.3)

where the nonlinear term g has a singularity of repulsive type. The author showed that Eq. (1.3) has at least one periodic solution by applications of coincidence degree theory. Zhang’s work has attracted the attention of many scholars in differential equations and they have contributed to the research of Liénard equation with singularity of repulsive type (see, e.g., [2, 5, 6, 8, 9, 13,14,15,16, 19, 24, 25, 28, 30]). For example, Jebelean and Mawhin [6] in 2004 investigated the following quasi-linear equation of p-Laplacian type:

$$ \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+f(x)x'(t)+g(x)=e(t), $$

(1.4)

where the nonlinear term g satisfied a slightly stronger singularity, i.e.,

$$ \int^{1}_{0}g(u)\,du=-\infty . $$

The authors proven that the above problem has at least one positive periodic solution through a basic application of the Manásevich–Mawhin theorem on continuity of the topological degree. Afterwards, using the Manásevich–Mawhin theorem on continuity of the topological degree again, Lu et al. [15] in 2017 obtained the existence of a positive periodic solution of the following equation with singularity of repulsive type:

$$ x''+f(x)x'- \frac{\alpha (t)}{x^{\mu }}=e(t). $$

(1.5)

All the aforementioned results concern Liénard equations and Liénard equations with singularity of repulsive type. Naturally, a new question arises: how does a generalized Liénard equation work on singularities of attractive and repulsive type? Besides the practical interests, the topic has obvious intrinsic theoretical significance. To answer this question, in this paper, we try to fill this gap and establish the existence of positive periodic solutions of Eq. (1.1) with singularities of attractive and repulsive type. Applying the Manásevich–Mawhin theorem on continuity of the topological degree, we obtain the following conclusions.

Theorem 1.1

Suppose the following conditions hold:

$(H_{1})$ :

There exist constants $0< d_{1}< d_{2}$ such that $g(t,x)-e(t)<0$ for $(t,x)\in [0,T]\times (0,d_{1})$ and $g(t,x)-e(t)>0$ for $(t,x)\in [0,T]\times (d_{2},+\infty )$.

$(H_{2})$ :

There exist positive constants m and n such that

$$ \bigl\vert f(t,x) \bigr\vert \leq m\vert x \vert ^{p-2}+n,\quad \textit{for all } (t,x)\in [0,T]\times \mathbb{R}. $$

$(H_{3})$ :

There exist positive constants a and b such that

$$ g(t,x)\leq a x^{p-1}+b, \quad \textit{for all } (t,x)\in [0,T]\times (0,+\infty ). $$

$(H_{4})$ :

(Repulsive condition) $\lim_{x\to 0^{+}}\int^{1}_{x}g_{0}(s)\,ds=-\infty$.

$(H_{5})$ :

There exists a constant $\alpha >0$ such that $\inf_{x\in \mathbb{R}}\vert f(t,x) \vert \geq \alpha >0$.

Then Eq. (1.1) has a positive T-periodic solution if the one of the following conditions is satisfied:

• (1) $p=2$ and $\frac{aT^{2}}{2}+(m+n)T<1$;

• (2) $p>2$ and $\frac{1}{2^{p-2}}(\frac{aT}{2}+m)T^{\frac{p}{q}}<1$, here $q=\frac{p}{p-1}$.

Remark 1.2

It is worth mentioning that the friction term $f(x)x'(t)$ in Eqs. (1.3), (1.4) and (1.5) satisfy $\int^{T}_{0}f(x(t))x'(t)\,dt=0$, which is crucial to estimate a priori bounds of a positive periodic solution for these equations. However, in this paper, the friction term $f(t,x)x'$ may not satisfy $\int^{T}_{0}f(t,x(t))x'(t)\,dt=0$. For example, let

$$ f(t,x)=\frac{1}{\pi }\bigl(\cos^{2}4t+3\bigr)x^{2}(t)+1. $$

Obviously,

$$ \int^{\frac{\pi }{4}}_{0} \biggl( \frac{1}{\pi }\bigl( \cos^{2}4t+3\bigr)x^{2}(t)+1 \biggr) x'(t)\,dt \neq 0. $$

This implies that our methods to estimate a priori bounds of positive periodic solution for Eq. (1.1) are more complex than Eqs. (1.3), (1.4) and (1.5).

Remark 1.3

From [6, 15, 29], the condition imposed on the external force $e(t)$ is $\int^{T}_{0}e(t)\,dt=0$. But this is unnecessary. For example, let the external force $e(t)=e^{\cos^{2} 4t}$. Obviously, $\int^{T}_{0}e^{\cos^{2} 4t}\,dt\neq 0$. Therefore, our result is more general.

Remark 1.4

If Eq. (1.1) satisfies singularity of attractive type, i.e., $\lim_{x\to 0^{+}}\int^{1}_{x}g_{0}(s)\,ds=+\infty$. Obviously, attractive condition and $(H_{1})$, $(H_{3})$ are in contradiction. Therefore, the above method is no long applicable to the proof of the existence of a positive periodic solution for Eq. (1.1) with singularity of attractive type. Next, we give other conditions to prove the existence of a positive periodic solution for Eq. (1.1) with singularity of attractive type.

Theorem 1.5

Assume that conditions $(H_{2})$ and $(H_{5})$ hold. Furthermore, suppose the following conditions hold:

$(H_{6})$ :

There exist constants $0< d_{3}< d_{4}$ such that $g(t,x)-e(t)>0$ for $(t,x)\in [0,T]\times (0,d_{3})$ and $g(t,x)-e(t)<0$ for $(t,x)\in [0,T]\times (d_{4},+\infty )$.

$(H_{7})$ :

(Attractive condition) $\lim_{x\to 0^{+}}\int^{1} _{x}g_{0}(s)\,ds=+\infty$.

$(H_{8})$ :

There exist positive constants β and γ such that

$$ -g(t,x)\leq \beta x^{p-1}+\gamma ,\quad \textit{for all } (t,x)\in [0,T]\times (0,+\infty ). $$

(1.6)

Then Eq. (1.1) has a positive T-periodic solution if one of the following conditions is satisfied:

• (1) $p=2$ and $\frac{\beta T^{2}}{2}+(m+n)T<1$;

• (2) $p>2$ and $\frac{1}{2^{p-2}}(\frac{\beta T}{2}+m)T^{\frac{p}{q}}<1$.

Besides, if the friction term $f(t,x(t))=f(x(t))$, then Eq. (1.1) is rewritten as

$$ \bigl(\phi_{p}(x)' \bigr)'+f\bigl(x(t)\bigr)x'(t)+g\bigl(t,x(t)\bigr)=e(t). $$

(1.7)

Note, if $p=2$ and the external force $e(t)\equiv 0$, the quasi-linear operator $x\mapsto (\phi_{p}(x'))'$ reduces to the linear operator $x\mapsto x''$, then (1.7) is of the differential equation form (1.3). Applying the Manásevich–Mawhin theorem on continuity of the topological degree, we obtain the following conclusions.

Theorem 1.6

Assume that conditions $(H_{1})$, $(H_{3})$ and $(H_{4})$ hold. Then Eq. (1.7) has positive T-periodic solution if $\frac{a}{2^{p-1}}T^{1+\frac{p}{q}}<1$.

Remark 1.7

If the external force $\int^{T}_{0}e(t)\,dt=0$, the result in [24, 29] is included in Theorem 1.6.

Theorem 1.8

Assume that conditions $(H_{6})$, $(H_{7})$ and $(H_{8})$ hold. Then Eq. (1.7) has a positive T-periodic solution if $\frac{a}{2^{p-1}}T^{1+\frac{p}{q}}<1$.

If the nonlinear term $g(t,x(t))=g(x(t))$, then Eq. (1.1) is rewritten as

$$ \bigl(\phi_{p}(x)' \bigr)'+f\bigl(t,x(t)\bigr)x'(t)+g\bigl(x(t)\bigr)=e(t). $$

(1.8)

Applying the Manásevich–Mawhin theorem on continuity of the topological degree, we obtain the following conclusions.

Theorem 1.9

Assume that condition $(H_{5})$ holds. Furthermore, suppose the following conditions hold:

$(H_{1}^{*})$ :

There exist constants $0< d_{1}^{*}< d_{2}^{*}$ such that $g(x)-e(t)<0$ for $x\in (0,d_{1}^{*})$ and $g(x)-e(t)>0$ for $x\in (d_{2}^{*},+\infty )$.

$(H_{4}^{*})$ :

(Repulsive condition) $\lim_{x\to 0^{+}}\int^{1} _{x}g(s)\,ds=-\infty$.

Then Eq. (1.8) has a positive T-periodic solution.

Remark 1.10

If the friction term $f(t,x)\equiv f(x)$ and the external force $\int^{T}_{0}e(t)\,dt=0$, Theorem 4.1 in [25] is included in Theorem 1.9.

Theorem 1.11

Assume that condition $(H_{5})$ holds. Suppose the following conditions hold:

$(H_{6}^{*})$ :

There exist constants $0< d_{3}^{*}< d_{4}^{*}$ such that $g(x)-e(t)>0$ for $x\in (0,d_{3}^{*})$ and $g(x)-e(t)<0$ for $x\in (d_{4}^{*},+\infty )$.

$(H_{7}^{*})$ :

(Attractive condition) $\lim_{x\to 0^{+}}\int ^{1}_{x}g(s)\,ds=+\infty$.

Then Eq. (1.8) has a positive T-periodic solution.

Remark 1.12

We would like to emphasize that the nonlinear term g satisfies the conditions and the work on estimating a priori bounds of positive periodic solutions for Eq. (1.8) is different [25]. In Theorem 3.1 in [25], the nonlinear term g satisfies condition $(H_{3})$, i.e., the semi-linearity condition. In this paper, for the nonlinear term g it has not been required that condition $(H_{3})$ holds, i.e., g may be under a sub-linearity condition, a semi-linearity condition or a super-linearity condition. So, we extend and improve the results in [25].

2 Periodic solution of Eq. (1.1) with singularities of attractive and repulsive type

Firstly, we embed (1.1) into the following equation family with a parameter $\lambda \in (0,1]$:

$$ { } \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+\lambda f\bigl(t,x(t)\bigr)x'(t)+\lambda g \bigl(t,x(t)\bigr)=\lambda e(t). $$

(2.1)

By applications of Theorem 3.1 in [18], we obtain the following result.

Lemma 2.1

Assume that there exist positive constants $E_{1}$, $E_{2}$, $E_{3}$ and $E_{1}< E_{2}$ such that the following conditions hold:

• (1) We have for each possible periodic solution x to Eq. (2.1) that $E_{1}< x(t)< E_{2}$, for all $t\in [0,T]$ and $\Vert x' \Vert < E_{3}$, here $\Vert x' \Vert :=\max_{t\in [0,T]}\vert x'(t) \vert$.

• (2) Each possible solution C to the equation

  $$ g(t,C)-\frac{1}{T} \int^{T}_{0} e(t)\,dt=0 $$

  satisfies $E_{1}< C< E_{2}$.

• (3) We have

  $$ \biggl( g(t,E_{1})-\frac{1}{T} \int^{T}_{0} e(t)\,dt \biggr) \biggl( g(t,E_{2})- \frac{1}{T} \int^{T}_{0} e(t)\,dt \biggr) < 0. $$

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

2.1 Proof of Theorem 1.1

Proof of Theorem 1.1

Firstly, we claim that there exists a point $\xi \in [0,T]$ such that

$$ { } d_{1}\leq x(\xi )\leq d_{2}. $$

(2.2)

In view of $\int^{T}_{0}x'(t)\,dt=0$, we know that there exist two points $t_{1},~t_{2}\in [0,T]$ such that

$$ x'(t_{1})\geq 0\quad \mbox{and}\quad x'(t_{2}) \leq 0. $$

Hence, we have

$$ \phi_{p}\bigl(x'(t_{1})\bigr)\geq 0\quad \mbox{and}\quad \phi_{p}\bigl(x'(t_{2})\bigr)\leq 0. $$

Let $t_{3}$, $t_{4}\in [0,T]$ be, respectively, a global maximum and minimum point of $\phi_{p}(x'(t))$; clearly, we deduce

$$\begin{aligned}& \phi_{p}\bigl(x'(t_{3}) \bigr)\geq 0, \qquad \bigl(\phi_{p}\bigl(x'(t_{3}) \bigr)\bigr)'=0. \end{aligned}$$

(2.3)

$$\begin{aligned}& \phi_{p}\bigl(x'(t_{4}) \bigr)\leq 0, \qquad \bigl(\phi_{p}\bigl(x'(t_{4}) \bigr)\bigr)'=0. \end{aligned}$$

(2.4)

From condition $(H_{5})$, we can see that the friction term f will not change sign for $(t,x)\in [0,T]\times (0,+\infty )$. Without loss of generality, suppose $f(t,x)>0$ for $(t,x)\in [0,T]\times (0,+\infty )$ and upon substitution of Eq. (2.3) into Eq. (2.1), we obtain

$$ -\lambda g\bigl(t_{3},x(t_{3})\bigr)+\lambda e(t_{3})=\lambda f\bigl(t_{3},x(t_{3}) \bigr)x'(t _{3}). $$

Since $\phi_{p}(x'(t_{3}))=\vert x'(t_{3}) \vert ^{p-2}x'(t_{3})\geq 0$, we know $x'(t_{3})\geq 0$. So we get

$$ g\bigl(t_{3},x(t_{3})\bigr)-e(t_{3})\leq 0. $$

From condition $(H_{1})$, we know that

$$ { } x(t_{3})\leq d_{2}. $$

(2.5)

Similarly, from Eq. (2.4), we see that

$$ g\bigl(t_{4},x(t_{4})\bigr)-e(t_{4})\geq 0, $$

and again by condition $(H_{1})$,

$$ { } x(t_{4})\geq d_{1}. $$

(2.6)

$x(t)$ is a continuous function in $(0,+\infty )$, from Eqs. (2.5) and (2.6), we get Eq. (2.2). Then we have

$$\begin{aligned} \Vert x \Vert &=\max_{t\in [0,T]}\bigl\vert x(t) \bigr\vert =\max_{t\in [\xi ,\xi +T]} \bigl\vert x(t) \bigr\vert \\ &=\max_{t\in [\xi ,\xi +T]}\biggl\vert \frac{1}{2} \bigl( x(t)+x(t-T) \bigr) \biggr\vert \\ &=\max_{t\in [\xi ,\xi +T]}\biggl\vert \frac{1}{2} \biggl( \biggl( x(\xi )+ \int ^{t}_{\xi }x'(s)\,ds \biggr) + \biggl( x(\xi )- \int^{\xi }_{t-T}x'(s)\,ds \biggr) \biggr) \biggr\vert \\ & \leq \max_{t\in [\xi ,\xi +T]} \biggl\{ d_{2}+\frac{1}{2} \biggl( \int ^{t}_{\xi }\bigl\vert x'(s) \bigr\vert \,ds+ \int^{\xi }_{t-T}\bigl\vert x'(s) \bigr\vert \,ds \biggr) \biggr\} \\ &\leq d_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(s) \bigr\vert \,ds. \end{aligned}$$

(2.7)

Multiplying both sides of Eq. (2.1) by $x(t)$ and integrating over the interval $[0,T]$, it is clear that

$$\begin{aligned} &\int^{T}_{0}\bigl(\phi_{p} \bigl(x'(t)\bigr)\bigr)'x(t)\,dt+\lambda \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)x(t)\,dt+ \lambda \int^{T}_{0}g\bigl(t,x(t)\bigr)x(t)\,dt \\ &\quad =\lambda \int^{T}_{0}e(t)x(t)\,dt. \end{aligned}$$

(2.8)

Substituting $\int^{T}_{0}(\phi_{p}(x'(t)))'x(t)\,dt=-\int^{T}_{0}\vert x'(t) \vert ^{p}\,dt$ into Eq. (2.8), we arrive at

$$\begin{aligned}& \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \\& \quad = \lambda \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)x(t)\,dt+ \lambda \int^{T}_{0}g\bigl(t,x(t)\bigr)x(t)\,dt+\lambda \int^{T}_{0}e(t)x(t)\,dt \\& \quad \leq \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \bigl\vert x(t) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x(t) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \bigl\vert x(t) \bigr\vert \,dt \\& \quad \leq \Vert x \Vert \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+\Vert x \Vert \int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \,dt+ \Vert x \Vert \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt. \end{aligned}$$

(2.9)

From condition $(H_{2})$, Eq. (2.9) and the Hölder inequality, we can observe that

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

(2.10)

where $\Vert e \Vert := ( \int^{T}_{0}\vert e(t) \vert ^{2}\,dt ) ^{\frac{1}{2}}$.

Integrating over the interval $[0,T]$ for Eq. (2.1), we conclude that

$$ { } \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt+ \int^{T}_{0}g\bigl(t,x(t)\bigr)\,dt= \int^{T}_{0}e(t)\,dt, $$

(2.11)

From Eq. (2.11), conditions $(H_{2})$ and $(H_{3})$, we have

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \,dt& =\int_{g(t,x(t))\geq 0}g\bigl(t,x(t)\bigr)\,dt- \int_{g(t,x(t))\leq 0}g\bigl(t,x(t)\bigr)\,dt \\ & = 2\int_{g(t,x(t))\geq 0}g^{+}\bigl(t,x(t)\bigr)\,dt+ \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt- \int^{T}_{0}e(t)\,dt \\ & \leq 2a\int^{T}_{0}x^{p-1}(t)\,dt+2bT+ \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ & \leq 2a\Vert x \Vert ^{p-1}T+2bT+m\Vert x \Vert ^{p-2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt+n \int^{T} _{0}\bigl\vert x'(t) \bigr\vert \,dt \\ &\quad {}+T^{\frac{1}{2}} \biggl(\int^{T}_{0}\bigl\vert e(t) \bigr\vert ^{2}\,dt \biggr) ^{\frac{1}{2}}, \end{aligned}$$

(2.12)

where $g^{+}(t,x):=\max \{0,g(t,x)\}$. Substituting Eqs. (2.12) and (2.7) into (2.10), we deduce

$$\begin{aligned} &\int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \\ &\quad \leq 2m\Vert x \Vert ^{p-1} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt +2n \Vert x \Vert \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt+2a\Vert x \Vert ^{p}T+2bT\Vert x \Vert +2T^{\frac{1}{2}}\Vert x \Vert \Vert e \Vert _{2} \\ &\quad \leq 2m \biggl( d_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p-1} \int ^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \\ &\quad \quad {}+2n \biggl( d_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \\ &\quad \quad {}+2aT \biggl( d_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p}+\bigl(2bT+2T ^{\frac{1}{2}}\bigr) \biggl( d_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) \\ & \quad \leq \frac{1}{2^{p-2}}m \biggl( 1+\frac{2d_{2}}{\int^{T}_{0}\vert x'(t) \vert \,dt} \biggr) ^{p-1} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p} \\ &\quad \quad {} +\frac{1}{2^{p-1}}aT \biggl( 1+ \frac{2d_{2}}{\int^{T}_{0}\vert x'(t) \vert \,dt} \biggr) ^{p} \biggl( \int^{T} _{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p} \\ &\quad \quad {}+n \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{2}+\bigl(bT+T^{\frac{1}{2}}\bigr) \int ^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt+2d_{2}\bigl(n+bT+T^{\frac{1}{2}}\bigr). \end{aligned}$$

(2.13)

Next, we introduce classical elementary inequality (see (3.10) in [7]), there exists a $\delta (p)>0$ which is dependent on p only,

$$ (1+x)^{p}\leq 1+(1+p)x, \quad \mbox{for } x\in \bigl[0,\delta (p)\bigr]. $$

(2.14)

Then we consider the following two cases:

Case 1. If $\frac{2d_{2}}{\int^{T}_{0}\vert x_{1}'(t) \vert \,dt}>\delta (p)$, then it is obvious that

$$ \int^{T}_{0}\bigl\vert x_{1}'(t) \bigr\vert \,dt< \frac{2d_{2} }{\delta (p)}. $$

From Eq. (2.7), we deduce

$$\begin{aligned} \Vert x \Vert & \leq d_{2}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \end{aligned}$$

(2.15)

$$\begin{aligned} & \leq d_{2}+\frac{2d_{2}}{\delta (p)}:=M_{1}''. \end{aligned}$$

(2.16)

Case 2. If $\frac{2d_{2}}{\int^{T}_{0}\vert x_{1}'(t) \vert \,dt}\leq \delta (p)$, from Eq. (2.14), we obtain

$$ { } \biggl( 1+\frac{2d_{2}}{\int^{T}_{0}\vert x'(t) \vert \,dt} \biggr) ^{p-1}\leq 1+ \frac{2d _{2} p}{\int^{T}_{0}\vert x'(t) \vert \,dt} $$

(2.17)

and

$$ { } \biggl( 1+\frac{2d_{2}}{\int^{T}_{0}\vert x'(t) \vert \,dt} \biggr) ^{p}\leq 1+ \frac{2d _{2} (p+1)}{\int^{T}_{0}\vert x'(t) \vert \,dt}. $$

(2.18)

Substituting Eqs. (2.17) and (2.18) into (2.13), we have

$$\begin{aligned} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt & \leq \frac{1}{2^{p-2}}m \biggl( 1+\frac{2d_{2} p}{\int^{T}_{0}\vert x'(t) \vert \,dt} \biggr) \biggl(\int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p} \\ &\quad {}+\frac{1}{2^{p-1}}aT \biggl( 1+\frac{2d_{2}(p+1)}{\int^{T}_{0}\vert x'(t) \vert \,dt} \biggr) \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p} \\ &\quad {}+n \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{2}+\bigl(bT+T^{\frac{1}{2}}\bigr) \int ^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt +N_{1} \\ & = \frac{m}{2^{p-2}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p}+\frac{md _{2} p}{2^{p-3}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p-1} \\ &\quad {}+ \frac{aT}{2^{p-1}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p} \\ &\quad {}+\frac{aTd_{2}(p+1)}{2^{p-2}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p-1}+n \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{2} \\ &\quad {}+\bigl(bT+T^{\frac{1}{2}}\bigr) \int^{T} _{0}\bigl\vert x'(t) \bigr\vert \,dt+N_{1} 1\\ & = \frac{1}{2^{p-2}}(2aT+m) \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p} \\ &\quad {}+ \frac{1}{2^{p-3}}\bigl(2aTd_{2}(p+1)+md_{2} p\bigr) \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{p-1} \\ &\quad {}+n \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt \biggr) ^{2}+\bigl(bT+T^{\frac{1}{2}}\bigr) \int ^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt +N_{1}, \end{aligned}$$

(2.19)

where $N_{1}:=2d_{2}(n+bT+T^{\frac{1}{2}})$. Applying the Hölder inequality, it is easy to verify that

$$\begin{aligned} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt& \leq \frac{1}{2^{p-2}}(2aT+m)T^{\frac{p}{q}} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \\ &\quad {}+\frac{1}{2^{p-3}}\bigl(2aTd_{2}(p+1)+md_{2} p\bigr)T ^{\frac{p-1}{q}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \biggr) ^{\frac{p-1}{p}} \\ &\quad {}+nT^{\frac{2}{q}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \biggr) ^{ \frac{2}{p}}+\bigl(bT+T^{\frac{1}{2}} \bigr)T^{\frac{1}{q}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \biggr) ^{\frac{1}{p}} +N_{1}. \end{aligned}$$

Case (I). If $p>2$ and $\frac{1}{2^{p-2}}(\frac{aT}{2}+m)T^{ \frac{p}{q}}<1$, it is easy to see that there exists a positive $M_{1}'$ (independent of λ) such that

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

(2.20)

Substituting Eq. (2.20) into (2.7), and using the Hölder inequality, we see that

$$ { } \Vert x \Vert \leq d_{2}+\frac{1}{2}T^{\frac{1}{q}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \biggr) ^{\frac{1}{p}}\leq d_{2}+ \frac{1}{2}T^{\frac{1}{q}} \bigl( M_{1}' \bigr) ^{\frac{1}{p}}:=M_{1}'''. $$

Take $M:=\max \{M_{1}'',M_{1}'''\}$, we arrive at

$$ { } \Vert x \Vert \leq M_{1}. $$

(2.21)

In view of $x(0)=x(T)$, there exists a point $t_{0}\in [0,T]$ such that $x'(t_{0})=0$, while $\phi_{p}(0)=0$. Therefore, from Eqs. (2.12), (2.20), (2.21) and condition $(H_{2})$, we have

$$\begin{aligned} \bigl\vert \phi_{p} \bigl(x'(t)\bigr) \bigr\vert & = \biggl\vert \int^{t}_{t_{0}}\bigl(\phi_{p} \bigl(x'(s)\bigr)\bigr)'\,ds\biggr\vert \\ & \leq \lambda \biggl( \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \biggr) \\ & \leq 2m\Vert x \Vert ^{p-2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt+2n \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt+2a \Vert x \Vert ^{p-1}T+2bT+2T^{\frac{1}{2}}\Vert e \Vert _{2} \\ & \leq 2mM_{1}^{p-2}T^{\frac{1}{q}}\bigl(M_{1}' \bigr)^{\frac{1}{p}}+2nT^{ \frac{1}{q}}\bigl(M_{1}' \bigr)^{\frac{1}{p}} +2aM_{1}^{p-1}T+2bT+2T^{ \frac{1}{2}} \Vert e \Vert _{2} \\ &:=M_{2}'. \end{aligned}$$

(2.22)

Besides, we claim that there exists a positive constant $M_{2}>M_{2}'+1$ such that, for all $t\in \mathbb{R}$

$$ { } \bigl\Vert x' \bigr\Vert \leq M_{2}. $$

(2.23)

In fact, if $x'$ is not bounded, there exists a positive constant $M_{2}''$ such that

$$ \bigl\Vert x' \bigr\Vert >M_{2}''. $$

Then we can get

$$ \bigl\Vert \phi_{p}\bigl(x'\bigr) \bigr\Vert =\bigl\Vert x' \bigr\Vert ^{p-1}\geq \bigl(M_{2}'' \bigr)^{p-1}, $$

which is a contradiction. So Eq. (2.23) holds.

Case (II). If $p=2$ and $\frac{aT^{2}}{2}+(m+n)T<1$, it is easy to see that there exists a positive $M_{1}'$ (independent of λ) such that

$$ { } \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{2}\,dt\leq M_{1}'. $$

Similarly, we get $\Vert x \Vert \leq M_{1}$, $\Vert x' \Vert \leq M_{2}$.

On the other hand, it follows from Eq. (2.1) and $g(t,x)=g _{0}(x)+g_{1}(t,x)$ that

$$ { } \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+\lambda f\bigl(t,x(t)\bigr)x'(t))+\lambda \bigl(g_{0}\bigl(x(t)\bigr)+g _{1}\bigl(t,x(t)\bigr)\bigr)= \lambda e(t). $$

(2.24)

Let $t\in [0,\xi ]$ be as in Eq. (2.7), for any $\xi \leq t \leq T$. From Eqs. (2.7) and (2.23), we conclude that

$$ x(\xi )\geq d_{1}. $$

Next, we will show that for any $t\in [\xi ,T]$, there exists a constant $D_{1}\in (0,d_{1})$, such that each positive periodic solution of (1.1) satisfies

$$ x(t)>D_{1}. $$

In fact, multiplying both sides of Eq. (2.24) by $x'(t)$ and integrating on $[\xi ,t]$, we get

$$\begin{aligned} \lambda \int^{x(t)}_{x(\xi )}g_{0}(u)\,du& = \lambda \int^{t}_{\xi }g_{0}\bigl(x(s) \bigr)x'(s)\,ds \\ & = - \int^{t}_{\xi }\bigl(\phi \bigl(x'(s) \bigr)\bigr)'x'(s)\,ds-\lambda \int^{t}_{\xi }f\bigl(s,x(s)\bigr) \bigl(x'(s)\bigr)^{2}\,ds \\ &\quad {}-\lambda \int^{t}_{\xi }g_{1}\bigl(s,x(s) \bigr)x'(s)\,ds+\lambda \int^{t}_{ \xi }e(s)x'(s)\,ds. \end{aligned}$$

Furthermore, we get

$$\begin{aligned} \lambda \biggl\vert \int^{x(t)}_{x(\xi )}g_{0}(u)\,du\biggr\vert & \leq \biggl\vert \int ^{t}_{\xi }\bigl(\phi \bigl(x'(s) \bigr)\bigr)'x'(s)\,ds\biggr\vert +\lambda \biggl\vert \int^{t}_{ \xi }f\bigl(s,x(s)\bigr) \bigl(x'(s)\bigr)^{2}\,ds\biggr\vert \\ &\quad {}+\lambda \biggl\vert \int^{t}_{\xi }g_{1}\bigl(s,x(s) \bigr)x'(s)\,ds\biggr\vert +\lambda \biggl\vert \int^{t}_{\xi }e(s)x'(s)\,ds\biggr\vert . \end{aligned}$$

(2.25)

By Eqs. (2.1), (2.22) and (2.23), we arrive at

$$\begin{aligned} \biggl\vert \int^{t}_{\xi }\bigl(\phi_{p} \bigl(x'(t)\bigr)\bigr)'x'(s)\,ds\biggr\vert & \leq \bigl\Vert x' \bigr\Vert \int ^{T}_{0}\bigl\vert \bigl(\phi \bigl(x'(s)\bigr)\bigr)' \bigr\vert \,ds \\ & \leq \lambda \bigl\Vert x' \bigr\Vert \biggl( \int^{T}_{0}\bigl\vert f\bigl(s,x'(s) \bigr) \bigr\vert \,ds+ \int^{T}_{0}\bigl\vert g\bigl(s,x(s)\bigr) \bigr\vert \,ds+ \int^{T}_{0}\bigl\vert e(s) \bigr\vert \,ds \biggr) \\ & \leq 2\lambda M_{2}M_{2}' . \end{aligned}$$

Moreover, from Eq. (2.25), we deduce

$$\begin{aligned} &\biggl\vert \int^{t}_{\xi }f\bigl(s,x(s)\bigr) \bigl(x'(s)\bigr)^{2}\biggr\vert \leq \bigl\Vert x' \bigr\Vert ^{2}T\bigl(m\Vert x \Vert ^{p-2}+n\bigr)\leq M_{2}^{2}T\bigl(mM_{1}^{p-2}+n \bigr), \\ &\biggl\vert \int^{t}_{\xi }g_{1}\bigl(s,x(s) \bigr)x'(s)\,ds\biggr\vert \leq \Vert x \Vert \int^{T} _{0}\bigl\vert g_{1} \bigl(s,x(s)\bigr) \bigr\vert \,ds\leq M_{2}\sqrt{T} \Vert g_{M_{1}} \Vert _{2}, \\ & \biggl\vert \int^{t}_{\xi }e(s)x'(s)\,dt\biggr\vert \leq M_{2}\sqrt{T} \Vert e \Vert _{2}, \end{aligned}$$

where $\Vert g_{M_{1}} \Vert :=\max_{0< x\leq M_{1}}\vert g_{1}(t,x) \vert \in L ^{2}(0,T)$. Form these inequalities we can derive from equation (2.25) that

$$ { } \biggl\vert \int^{x(t)}_{x(\xi )}g_{0}(u)\,du\biggr\vert \leq M_{2}\bigl(2M_{2}'+M _{2}T \bigl(mM_{1}^{p-2}+n\bigr)+\sqrt{T} \Vert g_{M_{1}} \Vert _{2}+\sqrt{T} \Vert e \Vert _{2}\bigr):=M _{3}. $$

(2.26)

In view of the repulsive condition $(H_{4})$ and $x(\xi )\geq d_{1}$, there exists $D_{1}\in (0,d_{1})$ such that

$$ \biggl\vert \int^{D_{1}}_{d_{1}}g_{0}(u)\,du\biggr\vert >M_{3}. $$

Thus, there exists a point $\eta \in [\xi ,T]$ such that $x(\eta ) \leq D_{1}$, then

$$ \biggl\vert \int^{x(\eta )}_{x(\xi )}g_{0}(u)\,du\biggr\vert \geq \biggl\vert \int^{D _{1}}_{d_{1}}g_{0}(u)\,du\biggr\vert >M_{3}, $$

which contradicts Eq. (2.26). Therefore, we can obtain that

$$ x(t)\geq D_{1},\quad \mbox{for all }t\in [\xi ,T]. $$

Similarly, we can consider $t\in [0,\xi ]$.

Let $E_{1}<\min \{D_{1},M_{3}\}$, $E_{2}>\max \{d_{2}, M_{1}\}$, $E_{3}>M_{2}$ are constants, from Eqs. (2.7), (2.21) and (2.23), we see that periodic solution x to Eq. (2.1) satisfies

$$ E_{1}< x(t)< E_{2}, \qquad \bigl\Vert x' \bigr\Vert < E_{3}. $$

Then condition (1) of Lemma 2.1 is satisfied. For a possible solution C in the equation

$$ g(t,C)-\frac{1}{T} \int^{T}_{0}e(t)\,dt=0, $$

satisfies $E_{1}< C<E_{2}$. Therefore, condition (2) of Lemma 2.1 holds. Finally, we consider that condition (3) of Lemma 2.1 is also satisfied. In fact, from $(H_{1})$, we have

$$ g(t,E_{1})-\frac{1}{T} \int^{T}_{0}e(t)\,dt< 0, $$

and

$$ g(t,E_{2})-\frac{1}{T} \int^{T}_{0}e(t)\,dt>0, $$

So condition (3) is also satisfied. Applying Lemma 2.1, we see that Eq. (1.1) has at least one positive periodic solution. □

2.2 Proof of Theorem 1.5

Proof of Theorem 1.5

Let $\underline{t}$, t̅, respectively, the global minimum and maximum points $x(t)$ on $[0,T]$; then $x'(\underline{t})=0$ and $x'(\overline{t})=0$, and we claim that

$$ { } \bigl(\phi_{p}\bigl(x'(\overline{t}) \bigr)\bigr)'\leq 0. $$

(2.27)

In fact, if Eq. (2.27) does not hold, then $(\phi_{p}(x'( \overline{t})))'>0$ and there exists $\varepsilon >0$ such that $(\phi_{p}(x'(t)))'>0$ for $t\in (\overline{t}-\varepsilon , \overline{t}+\varepsilon )$. Therefore $\phi_{p}(x'(t))$ is strictly increasing for $t\in (\overline{t}-\varepsilon ,\overline{t}+\varepsilon )$. Then we know that $x'(t)$ is strictly increasing for $t\in ( \overline{t}-\varepsilon ,\overline{t}+\varepsilon )$. This contradicts the definition of t̅. Thus, equation (2.27) is true. From Eqs. (2.1) and (2.27), we get

$$ { } g\bigl(\overline{t}+x(\overline{t})\bigr)-e(\overline{t})\leq 0. $$

(2.28)

Similarly, we deduce

$$ g\bigl(\underline{t}+x(\underline{t})\bigr)-e(\underline{t})\geq 0. $$

(2.29)

By condition $(H_{6})$, Eqs. (2.28) and (2.29), we see that

$$ x(\underline{t})\geq d_{3},\quad \mbox{and}\quad x(\overline{t})\leq d_{4}. $$

In view of x being a continuous function, we see that there exists a point $\xi^{*}\in (0,T)$ such that

$$ d_{3}\leq x\bigl(\xi^{*}\bigr)\leq d_{4}. $$

(2.30)

From Eq. (2.7), we have

$$ x(t)\leq d_{4}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt. $$

We follow the same strategy and notation as in the proof of Theorem 1.1. From Eqs. (2.11), (2.12), condition $(H_{2})$ and $(H_{8})$, we obtain

$$\begin{aligned} &\int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \,dt \\ & \quad = \int_{g(t,x(t))\geq 0}g\bigl(t,x(t)\bigr)\,dt- \int_{g(t,x(t))\leq 0}g\bigl(t,x(t)\bigr)\,dt \\ & \quad = -2 \int_{g(t,x(t))\leq 0}g^{-}\bigl(t,x(t)\bigr)\,dt- \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt+ \int^{T}_{0}e(t)\,dt \\ & \quad \leq 2\beta \int^{T}_{0}x^{p-1}(t)\,dt+2\gamma T+ \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ & \quad \leq 2\beta T \Vert x \Vert ^{p-1}+2\gamma T+ \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt, \end{aligned}$$

(2.31)

where $g^{-}(t,x):=\min \{g(t,x),0\}$. The remaining part of the proof is the same as that of Theorem 1.1. □

3 Periodic solution of Eq. (1.7) with singularities of attractive and repulsive type

3.1 Proof of Theorem 1.6

Proof of Theorem 1.6

Consider the homotopic equation

$$ \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+\lambda f\bigl(x(t)\bigr)x'(t)+\lambda g \bigl(t,x(t)\bigr)=e(t). $$

(3.1)

We follow the same strategy and notation as in the proof of Theorem 1.1. From Eq. (2.7), we have

$$ \int^{T}_{0}\bigl(\phi_{p} \bigl(x'(t)\bigr)\bigr)'x(t)\,dt+\lambda \int^{T}_{0}g\bigl(t,x(t)\bigr)x(t)\,dt= \lambda \int^{T}_{0}e(t)x(t)\,dt, $$

(3.2)

since $\int^{T}_{0}f(x(t))x'(t)x(t)\,dt=0$. Substituting $\int^{T}_{0}( \phi_{p}(x'(t)))'x(t)\,dt=-\int^{T}_{0}\vert x'(t) \vert ^{p}\,dt$ into equation (3.2), we get

$$ \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \leq \Vert x \Vert \int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \,dt+\Vert x \Vert \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt. $$

(3.3)

Integrating over the interval $[0,T]$ for Eq. (3.1), we obtain

$$ \int^{T}_{0}g\bigl(t,x(t)\bigr)\,dt= \int^{T}_{0}e(t)\,dt. $$

(3.4)

From Eq. (3.4) and condition $(H_{3})$, we see that

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \,dt & = \int_{g(t,x(t))\geq 0}g\bigl(t,x(t)\bigr)\,dt- \int_{g(t,x(t))\leq 0}g\bigl(t,x(t)\bigr)\,dt \\ & = 2 \int_{g(t,x(t))\geq 0}g^{+}\bigl(t,x(t)\bigr)\,dt- \int^{T}_{0}e(t)\,dt \\ & \leq 2a \int^{T}_{0}x^{p-1}(t)\,dt+2bT+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ & \leq 2a\Vert x \Vert ^{p-1}T+2bT+T^{\frac{1}{2}} \biggl( \int^{T}_{0}\bigl\vert e(t) \bigr\vert ^{2}\,dt \biggr) ^{\frac{1}{2}}. \end{aligned}$$

(3.5)

Substituting Eqs. (3.5) into (3.3), and from Eq. (2.19), we conclude that

$$\begin{aligned} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt& \leq \frac{2aT^{1+\frac{p}{q}}}{2^{p-2}} \int ^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt +\frac{ad_{2}(p+1)T^{1+\frac{p-1}{q}}}{2^{p-2}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \biggr) ^{\frac{p-1}{p}} \\ &\quad {}+\bigl(bT+T^{\frac{1}{2}}\bigr)T^{\frac{1}{q}} \biggl( \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{p}\,dt \biggr) ^{\frac{1}{p}} +N_{1}. \end{aligned}$$

Since $\frac{2aT^{1+\frac{p}{q}}}{2^{p-2}}<1$, it is easy to see that there exists a positive $M_{1}'$ (independent of λ) such that

$$ \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt\leq M_{1}'. $$

The remaining part of the proof is the same as that of Theorem 1.1. □

3.2 Proof of Theorem 1.8

Proof of Theorem 1.8

We follow the same strategy and notation as in the proof of Theorem 1.5 and 1.6. We only consider $\int^{T}_{0}\vert g(t,x(t)) \vert \,dt$.

From Eqs. (2.31), (3.4) and condition $(H_{8})$, we have

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(t,x(t)\bigr) \bigr\vert \,dt & = \int_{g(t,x(t))\geq 0}g\bigl(t,x(t)\bigr)\,dt- \int_{g(t,x(t))\leq 0}g\bigl(t,x(t)\bigr)\,dt \\ & = -2 \int_{g(t,x(t))\leq 0}g^{-}\bigl(t,x(t)\bigr)\,dt+ \int^{T}_{0}e(t)\,dt \\ & \leq 2\beta \int^{T}_{0}x^{p-1}(t)\,dt+2\gamma T+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ & \leq 2\beta T \Vert x \Vert ^{p-1}+2\gamma T+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt. \end{aligned}$$

(3.6)

The remaining part of the proof is the same as that of Theorem 1.1. □

4 Periodic solution of Eq. (1.8) with singularities of attractive and repulsive type

4.1 Proof of Theorem 1.9

Proof of Theorem 1.9

Consider the homotopic equation

$$ \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+\lambda f\bigl(t,x(t)\bigr)x'(t)+\lambda g \bigl(x(t)\bigr)=e(t). $$

(4.1)

We follow the same strategy and notation as in the proof of Theorem 1.1. From Eq. (2.7), we deduce

$$ x(t)\leq d_{2}^{*}+\frac{1}{2} \int^{T}_{0}\bigl\vert x'(t) \bigr\vert \,dt. $$

(4.2)

Multiplying both sides of Eq. (4.1) by $x'(t)$ and integration over the interval $[0,T]$, we get

$$\begin{aligned} &\int^{T}_{0}\bigl(\phi_{p} \bigl(x'(t)\bigr)\bigr)'x'(t)\,dt+\lambda \int^{T}_{0}f\bigl(t,x(t)\bigr) \bigl(x'(t)\bigr)^{2}\,dt+ \lambda \int^{T}_{0} g\bigl(x(t)\bigr)x'(t)\,dt \\ &\quad = \lambda \int^{T}_{0}e(t)x'(t)\,dt. \end{aligned}$$

Since $\int^{T}_{0}(\phi_{p}(x'(t)))'x'(t)\,dt=0$ and $\int^{T}_{0}g(x(t))x'(t)\,dt=0$, we obtain

$$ { } \int^{T}_{0}f\bigl(t,x(t)\bigr)\bigl\vert x'(t) \bigr\vert ^{2}\,dt= \int^{T}_{0}e(t)x'(t)\,dt. $$

(4.3)

From Eq. (4.3), we have

$$ { } \biggl\vert \int^{T}_{0}f\bigl(t,x(t)\bigr)\bigl\vert x'(t) \bigr\vert ^{2}\,dt\biggr\vert =\biggl\vert \int^{T}_{0}e(t)x'(t)\,dt\biggr\vert . $$

(4.4)

From condition $(H_{5})$, we see that

$$ \biggl\vert \int^{T}_{0}f\bigl(t,x(t)\bigr)\bigl\vert x'(t) \bigr\vert ^{2}\,dt\biggr\vert = \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert ^{2}\,dt \geq \alpha \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{2}\,dt. $$

(4.5)

Substituting Eqs. (4.5) into (4.4), and using the Hölder inequality, we arrive at

$$\begin{aligned} \alpha \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{2}\,dt& \leq \int^{T}_{0}\bigl\vert e(t) \bigr\vert \bigl\vert x(t) \bigr\vert \,dt \\ & \leq \biggl( \int^{T}_{0}\bigl\vert e(t) \bigr\vert ^{2}\,dt \biggr) ^{\frac{1}{2}} \biggl( \int ^{T}_{0}\bigl\vert x(t) \bigr\vert ^{2}\,dt \biggr) ^{\frac{1}{2}} \\ & = \Vert e \Vert _{2} \biggl( \int^{T}_{0}\bigl\vert x(t) \bigr\vert ^{2}\,dt \biggr) ^{\frac{1}{2}}. \end{aligned}$$

It is easy to see that there exists a positive constant $M_{1}'$ (independent of λ) such that

$$ \int^{T}_{0}\bigl\vert x'(t) \bigr\vert ^{2}\,dt\leq M_{1}'. $$

From Eq. (2.21), it is obvious that

$$ x(t)\leq M_{1}. $$

Integrating both sides of Eq. (4.1) over $[0,T]$, it is clear that

$$ { } \int^{T}_{0}\bigl[f\bigl(t,x(t)\bigr)x'(t)+g \bigl(x(t)\bigr)-e(t)\bigr]\,dt=0. $$

(4.6)

From Eq. (4.6), we have

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt&= \int_{g(x(t))\geq 0}g\bigl(x(t)\bigr)\,dt- \int_{g(x(t))\leq 0}g\bigl(x(t)\bigr)\,dt \\ &=2 \int_{g(x(t))\geq 0}g^{+}\bigl(x(t)\bigr)\,dt+ \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt- \int^{T}_{0}e(t)\,dt. \end{aligned}$$

(4.7)

Case (I). If $\overline{e}:=\frac{1}{T}\int^{T}_{0}e(t)\,dt\leq 0$, from (4.7), we get

$$ \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt \leq 2 \int^{T}_{0}\bigl(g^{+}\bigl(x(t) \bigr)-e(t)\bigr)\,dt+ \int ^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt. $$

Since $g^{+}(x(t))-e(t)\geq 0$, from condition $(H_{1}^{*})$, we know $x(t)\geq d_{2}^{*}$. Then we deduce

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt& \leq 2 \int^{T}_{0}g^{+}\bigl(x(t)\bigr)\,dt+ \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert + \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ & \leq 2T\bigl\Vert g^{+}_{M_{1}} \bigr\Vert + \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert + \int^{T} _{0}\bigl\vert e(t) \bigr\vert \,dt, \end{aligned}$$

(4.8)

where $\Vert g^{+}_{M_{1}} \Vert :=\max_{d_{2}^{*}\leq x\leq M_{1}}g ^{+}(x)$. From Eqs. (2.22) and (4.8), we have

$$\begin{aligned} \bigl\Vert \phi_{p} \bigl(x'\bigr)\bigr\Vert & \leq \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int ^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ & \leq 2 \biggl( T\bigl\Vert g^{+}_{M_{1}} \bigr\Vert + \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int ^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \biggr) \\ & \leq 2 \bigl( T\bigl\Vert g^{+}_{M_{1}} \bigr\Vert + \Vert f_{M_{1}} \Vert _{2}\bigl(M_{1}' \bigr)^{ \frac{1}{2}}+T^{\frac{1}{2}}\Vert e \Vert _{2} \bigr) \\ & :=M_{2}', \end{aligned}$$

(4.9)

where $f_{M_{1}}:=\max_{0< x(t)\leq M_{1}}\vert f(t,x(t)) \vert$, $\Vert f_{M_{1}} \Vert _{2}:= ( \int^{T}_{0}\vert f(t,x(t)) \vert ^{2}\,dt ) ^{\frac{1}{2}}$.

Case (II). If $\overline{e}>0$, from (4.7), we have

$$ \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt\leq 2 \int^{T}_{0}g^{+}\bigl(x(t)\bigr)\,dt+ \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt. $$

Since $g^{+}(x(t))\geq 0$, from condition $(H_{1}^{*})$, we know that there exists a positive constant $d_{2}^{**}$ such that $x(t)\geq d _{2}^{**}$. Therefore, we have

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt& \leq 2 \int^{T}_{0}g^{+}\bigl(x(t)\bigr)\,dt+ \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt \\ & \leq 2T\bigl\Vert g^{+}_{M} \bigr\Vert + \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt, \end{aligned}$$

where $\Vert g^{+}_{M} \Vert :=\max_{d_{2}^{**}\leq x\leq M_{1}}g^{+}(x)$. Similarly, we can get $\vert \phi_{p}(x'(t)) \vert \leq M_{2}'$.

The remaining part of the proof is the same as that of Theorem 1.1. □

4.2 Proof of Theorem 1.11

Proof of Theorem 1.11

We follow the same strategy and notation as in the proof of Theorem 1.5 and 1.6. We can get

$$ x(t)\leq M_{1}. $$

From Eq. (4.6), we deduce

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt&= \int_{g(x(t))\geq 0}g\bigl(x(t)\bigr)\,dt- \int_{g(x(t))\leq 0}g\bigl(x(t)\bigr)\,dt \\ &=-2 \int_{g(x(t))\leq 0}g^{-}\bigl(x(t)\bigr)\,dt- \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt+ \int^{T}_{0}e(t)\,dt. \end{aligned}$$

(4.10)

Case (I). If $\overline{e}:=\frac{1}{T}\int^{T}_{0}e(t)\,dt\geq 0$, from Eq. (4.10), we have

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt & \leq -2\int^{T}_{0}\bigl(g^{-}\bigl(x(t) \bigr)-e(t)\bigr)\,dt- \int ^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt. \end{aligned}$$

Since $g^{-}(x(t))-e(t)\leq 0$, from condition $(H_{6}^{*})$, we know $x(t)\geq d_{4}^{*}$. Then we get

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt& \leq -2 \int^{T}_{0}g^{-}\bigl(x(t)\bigr)\,dt+ \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert + \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ & \leq 2T\bigl\Vert g^{-}_{M_{1}} \bigr\Vert + \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert + \int^{T} _{0}\bigl\vert e(t) \bigr\vert \,dt, \end{aligned}$$

(4.11)

where $\Vert g^{-}_{M_{1}} \Vert :=\max_{d_{4}^{*}\leq x\leq M_{1}}(-g ^{-}(x))$. From Eqs. (2.22) and (4.11), we see that

$$\begin{aligned} \bigl\Vert \phi_{p} \bigl(x'\bigr)\bigr\Vert &\leq \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int ^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt+ \int^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \\ &\leq 2 \biggl( T\bigl\Vert g^{-}_{M_{1}} \bigr\Vert + \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt+ \int ^{T}_{0}\bigl\vert e(t) \bigr\vert \,dt \biggr) \\ & \leq 2 \bigl( T\bigl\Vert g^{-}_{M_{1}} \bigr\Vert + \Vert f_{M_{1}} \Vert _{2}\bigl(M_{1}' \bigr)^{ \frac{1}{2}}+T^{\frac{1}{2}}\Vert e \Vert _{2} \bigr) :=M_{2}'. \end{aligned}$$

(4.12)

Case (II). If $\overline{e}<0$, from Eq. (4.10), we arrive at

$$ \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt\leq -2 \int^{T}_{0}g^{-}\bigl(x(t)\bigr)\,dt- \int^{T}_{0}f\bigl(t,x(t)\bigr)x'(t)\,dt. $$

Since $g^{-}(x(t))\leq 0$, from condition $(H_{6}^{*})$, we know that there exists a positive constant $d_{4}^{**}$ such that $x(t)\geq d _{4}^{**}$. Therefore, we conclude that

$$\begin{aligned} \int^{T}_{0}\bigl\vert g\bigl(x(t)\bigr) \bigr\vert \,dt& \leq -2 \int^{T}_{0}g^{-}\bigl(x(t)\bigr)\,dt+ \int^{T} _{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt \\ & \leq 2T\bigl\Vert g^{+}_{M} \bigr\Vert + \int^{T}_{0}\bigl\vert f\bigl(t,x(t)\bigr) \bigr\vert \bigl\vert x'(t) \bigr\vert \,dt, \end{aligned}$$

where $\Vert g^{-}_{M} \Vert :=\max_{d_{4}^{**}\leq x\leq M_{1}}(-g^{-}(x))$. Similarly, we can get $\vert \phi_{p}(x'(t)) \vert \leq M_{2}'$. □

5 Examples

Example 5.1

Consider the following p-Laplacian generalized Liénard equation with singularity of attractive type:

$$ { } \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+ \biggl( \frac{1}{\pi }\bigl(\cos^{2} 4t+3 \bigr)x^{2}+1 \biggr) x'(t)-\frac{1}{3 \pi^{2}}(\sin 8t+2)x^{3}+\frac{1}{x^{\kappa }}=e^{\cos^{2} 4t}, $$

(5.1)

where $p=4$, κ is a constant and $\kappa \geq 1$.

Comparing Eq. (5.1) to (1.1), it is easy to see that $f(t,x)=\frac{1}{\pi }(\cos^{2} 4t+3)x^{2}+1$, $g(t,x)=-\frac{1}{3\pi ^{2}}(\sin 8t+2)x^{3}+\frac{1}{x^{\kappa }}$, $e(t)=e^{\cos^{2} 4t}$, $T=\frac{\pi }{4}$. Obviously, there exist constants $d_{3}=0.1$ and $d_{4}=1$ such that $(H_{6})$ holds. $1\leq \vert f(t,x) \vert \leq \frac{4}{ \pi }x^{2}+1$, $\alpha =1$, $m=\frac{4}{\pi }$, $n=1$, then conditions $(H_{2})$ and $(H_{5})$ hold. $-g(t,x)\leq \frac{1}{\pi^{2}}x^{3}+1$, $\beta =\frac{1}{\pi^{2}}$, $\gamma =1$. $\lim_{x\to 0^{+}}\int ^{1}_{x}g_{0}(s)\,ds=\lim_{x\to 0^{+}}\int^{1}_{x}\frac{-1}{s ^{\kappa }}\,ds=+\infty$, thus, conditions $(H_{7})$ and $(H_{8})$ hold. Next, it is verified that

$$\begin{aligned} \frac{1}{2^{p-2}} \biggl( \frac{\beta T}{2}+m \biggr) T^{\frac{p}{q}}& = \frac{1}{2^{2}}\times \biggl( \frac{1}{2\pi }+\frac{4}{\pi } \biggr) \times \biggl( \frac{\pi }{4} \biggr) ^{3}= \frac{9\pi^{2}}{2^{9}}< 1. \end{aligned}$$

Therefore, by Theorem 1.5, we know that Eq. (5.1) has at least one positive $\frac{\pi }{4}$-periodic solution.

Example 5.2

Consider the following Liénard equation with singularity of repulsive type:

$$ { } x''(t)+6x^{10}x'(t)+ \frac{1}{6\pi }\bigl(\sin^{2} 2t+5\bigr)x-\frac{1}{x^{ \kappa }}=e^{\cos^{2} 2t}, $$

(5.2)

where $p=2$, κ is a constant and $\kappa \geq 1$.

Comparing Eq. (5.2) to (1.7), it is easy to see that $f(x)=6x^{10}(t)$, $g(t,x)=\frac{1}{6\pi }(\sin^{2} 2t+5)x-\frac{1}{x ^{\kappa }}$, $e(t)=e^{\cos^{2} 2t}$, $T=\frac{\pi }{2}$. Obviously, there exist constants $d_{1}=0.1$ and $d_{2}=1$ such that $(H_{1})$ holds. $g(t,x)\leq \frac{1}{\pi }x+1$, $a=\frac{1}{\pi }$, $b=1$. $\lim_{x\to 0^{+}}\int^{1}_{x}g_{0}(s)\,ds=\lim_{x\to 0^{+}} \int^{1}_{x}\frac{1}{s^{\kappa }}\,ds=-\infty$, thus, conditions $(H_{3})$ and $(H_{4})$ hold. Next, it is verified that

$$\begin{aligned} \frac{1}{2^{p-1}}T^{1+\frac{p}{q}}& = \frac{1}{2}\times \frac{1}{\pi }\times \biggl( \frac{\pi }{2} \biggr) ^{2} \\ & = \frac{\pi }{8}< 1. \end{aligned}$$

Therefore, by Theorem 1.6, we know that Eq. (5.2) has at least one positive $\frac{\pi }{2}$-periodic solution.

Example 5.3

Consider the following p-Laplacian generalized Liénard equation with singularity of repulsive type:

$$ { } \bigl(\phi_{p}\bigl(x'(t)\bigr) \bigr)'+ \bigl( (\cos t+2)x^{6}+3 \bigr) x'(t)+\sum_{i=1} ^{n}x^{2i}(t)- \frac{1}{x^{\mu }}=e^{\sin t}, $$

(5.3)

where $p=10$, μ is a constant and $\mu \geq 1$, n is a integer.

Comparing Eqs. (5.3) to (1.8), it is easy to see that $f(t,x)=(\cos t+2)x^{6}+3$, $g(x)=\sum_{i=1}^{n}x^{2i}(t)-\frac{1}{x ^{\mu }}$, $e(t)=e^{\cos t}$, $T=2\pi$. Obviously, there exist constants $d_{1}^{*}=0.1$ and $d_{2}^{*}=1$ such that $(H_{1}^{*})$ holds. $\vert f(t,x) \vert \geq 3$, $\alpha =3$, then condition $(H_{2})$ holds. Next, we prove that condition $(H_{4}^{*})$ holds. In fact,

$$ \lim_{x\to 0^{+}} \int^{1}_{x}g(s)\,ds=\lim_{t\to 0^{+}} \int^{1}_{x} \Biggl( \sum _{i=1}^{n}s^{2i}-\frac{1}{s^{\mu }} \Biggr)\,ds=-\infty . $$

Therefore, by Theorem 1.9, we know that Eq. (5.3) has at least one positive 2π-periodic solution.

6 Conclusions

In this article we introduce the existence of periodic solution for p-Laplacian generalized Liénard equation with singularity of attractive and repulsive type. Due to the friction term $f(t,x)x'(t)$ may not satisfy $\int^{T}_{0}f(t,x(t))x'(t)\,dt=0$. This implies that the work on estimating a priori bounds of periodic solutions for generalized Liénard Eq. (1.1) is more difficult than the corresponding work on Liénard equation in [6, 15, 24, 25, 29]. Secondly, attractive conditions $(H_{7})$ and $(H_{8})$ are contradicted with the repulsive conditions $(H_{3})$ and $(H_{4})$, the methods of [6, 15, 24, 25, 29] are no longer applicable to the proof of periodic solution for Eq. (1.1) with singularity of attractive singularity. In this paper, using the Manásevich–Mawhin theorem on continuity of the topological degree and conditions ($H_{1}$)–($H_{5}$), we prove the existence of a periodic solution for equation (1.1) with singularity of repulsive type; by conditions $(H_{2})$, ($H_{5}$) ($H_{6}$)–($H_{8}$), we obtain the existence of a periodic solution for Eq. (1.1) with singularity of attractive type. Moreover, we investigate the existence of a periodic solution for Eqs. (1.7) and (1.8) with singularities of attractive and repulsive type.