- Research
- Open access
- Published:

# Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

*Boundary Value Problems*
**volume 2024**, Article number: 85 (2024)

## Abstract

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type,

where \(f:(0,+\infty )\rightarrow R\) is continuous, which may have a singularity at the origin, the sign of \(\varphi (t)\), \(e(t)\) is allowed to change, and *μ*, *γ* are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive *T*-periodic solution when \(\mu \in [0,+\infty )\).

## 1 Introduction

Since singular equations have a wide range of application in physics, engineering, mechanics, and other subjects (see [1–7]), the periodic problem for a certain second order differential equation has attracted much attention from many researchers. In the past years, lots of papers (see [8–14]) were concerned with the problem of periodic solutions to the second order singular equation without the first derivative term,

where \(f:[0,\infty )\rightarrow \mathbb{R}\) is continuous, \(\varphi ,b, h\in L^{1}[0,T]\), and \(\mu >0\) is a constant. Among these papers, we notice that the coefficient function \(\varphi (t)\) is required to be

This is because (1.2), together with other conditions, can ensure that the function \(G(t,s)\ge 0\) for \((t,s)\in [0,T]\times [0,T]\), where the \(G(t,s)\) is the Green function associated with the boundary value problem for Hill’s equation

The condition \(G(t,s)\ge 0\) for \((t,s)\in [0,T]\times [0,T]\) is crucial for obtaining the positive periodic solutions to (1.1) by means of some fixed point theorems on cones. Beginning with the paper of Habets–Sanchez [15], many works (see [16–21]) discussed the existence of a periodic solution for Liénard equations with singularities,

where \(\varphi (t)\) and \(e(t)\) are *T*-periodic with \(\varphi , e\in L^{1}[0,T]\), while *γ* is a constant with \(\gamma > 0\). However, in those papers, the conditions of \(\varphi (t)\ge 0\) for a.e. \(t\in [0,T]\), the strong singularity \(\gamma \in [1,+\infty )\), and \(f(x)\) being continuous on \([0,+\infty )\) are needed. To the best of our knowledge, there are fewer papers dealing with the equation where the function \(f(x)\) possesses a singularity at \(x=0\). We find that Hakl, Torres, and Zamora in [22] considered the periodic problem for the singular equation of repulsive type,

where \(\mu \in (0,1]\) is a constant, *φ* is a *T*-periodic function with \(\varphi \in L^{1}([0,T], R)\), and the sign of \(\varphi (t)\) can change, while \(f\in C((0,+\infty ),R)\) may be singular at \(x=0\) and \(g\in C((0,+\infty ),R)\) has a repulsive singularity at \(x=0\), i.e., \(\lim _{x\rightarrow 0^{+}}g(x)=-\infty \). By using Schauder’s fixed point theorem, some results on the existence of positive *T*-periodic solutions were obtained. However, the strong singularity condition \(\int ^{1}_{0}g(s)ds = -\infty \) is also required. In a recent paper [23], the authors consider the periodic problem to (1.4) for the special case \(g(x)=\frac{1}{x^{\gamma}}\), where \(\gamma \in (0,+\infty )\). But, in [23], the function \(\varphi (t)\) is required to satisfy \(\varphi (t)\ge 0\) a.e. \(t\in [0,T]\) for the case \(\mu >1\) (see Theorem 3.1, [23]). Motivated by this, in the present paper, we continue to study the periodic problem for the singular equation,

where *f*, *φ* are as same as those in (1.4); \(\mu >0\) and \(\gamma >0\) are constants, *e* is a *T*-periodic function with \(e\in L^{1}([0,T],R)\), and \(\int _{0}^{T}e(s)ds=0\). By means of a continuation theorem of coincidence degree principle developed by Manásevich and Mawhin, as well as the techniques of a priori estimates, some new results on the existence of positive periodic solutions are obtained. The interesting point in this paper is that the function \(f(x)\) has a singularity at \(x=0\), the sign of \(\varphi (t)\) is allowed to change, and \(\mu ,\gamma \in (0,+\infty )\). Compared with [22], we allow the singular term \(\frac{1}{x^{\gamma}}\) to have a weak singularity, i.e., \(\gamma \in (0,1)\). Also, for the case of \(\mu >1\), the sign of \(\varphi (t)\) is allowed to change, which is essentially different from the condition \(\varphi (t)\ge 0\) for a.e. \(t\in [0,T]\) in [23].

## 2 Essential definitions and lemmas

Throughout this paper, let \(C_{T}=\{x\in C(R,R) :x(t+T)=x(t),\forall t\in R\}\) with the norm \(|x|_{\infty} = \max _{t\in [0,T]}|x(t)|\). Clearly, \(C_{T}\) is a Banach space. For any *T*-periodic function \(x(t)\), we denote \(\bar{x}=\frac{1}{T}\int _{0}^{T}x(s)ds\), \(x_{+}(t)= \max \{ x(t),0\}\), and \(x_{-}= -\min \{x(t),0\}\). Thus, \(x(t)=x_{+}(t)- x_{-}(t)\) for all \(t\in R\), and \(\overline{x}= \overline{x_{+}}-\overline{x_{-}}\). Furthermore, for each \(u\in C_{T}\), let \(\|u\|_{p}=(\int ^{T}_{0}|u(s)|^{p}ds)^{\frac{1}{p}}\), \(p\in [1,+\infty )\).

### Lemma 2.1

([24])

*Assume that there exit positive constants* \(M_{0}\) *and* \(M_{1}\), *with* \(0 < M_{0} < M_{1}\), *such that the following conditions hold*:

(1) *for each* \(\lambda \in (0,1]\), *each possible positive* *T*-*periodic solution* *u* *to the equation*

*satisfies the inequality* \(M_{0}< u(t)< M_{1}\) *for all* \(t\in [0,T]\);

(2) *each possible solution* \(c\in (0,+\infty )\) *to the equation*

*satisfies the inequality* \(M_{0}< c< M_{1}\);

(3) *the inequality*

*holds*.

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

### Lemma 2.2

([22])

*Let* \(u(t):[0,\omega ]\rightarrow R\) *be an arbitrary absolutely continuous function with* \(u(0)=u(\omega )\). *Then the inequality*

*holds*.

### Remark 2.3

If \(\overline{\varphi}>0\), then there are constants \(C_{1}\) and \(C_{2}\) with \(0< C_{1}< C_{2}\) such that

and

Now, we embed equation (1.5) into the following equation family with a parameter \(\lambda \in (0,1]\):

Let

and

### Lemma 2.4

*Assume* \(\overline{\varphi}>0\) *and* \(\overline{e}=0\), *then there are two constants* \(\tau _{1}, \tau _{2} \in [0,T]\) *for each* \(u\in D\), *such that*

*and*

### Proof

Let \(u \in D\), then

Dividing both sides of (2.8) by \(u^{\mu}(t)\) and integrating over the interval \([0,T]\), we obtain

Since the inequality \(\int ^{T}_{0}\frac{u''(t)}{u^{\mu}(t)}dt \geq 0\) holds, it is easy to see that

i.e.,

From this, we can verify (2.6). In fact, if (2.6) does not hold, then

which together with (2.9) gives

i.e.,

On the other hand, (2.10) implies that \(u(\tau _{1}) > 1\). It follows from (2.11) that \((\frac{2}{\overline{\varphi}})^{\frac{1}{\mu}} > 1\), i.e., \(\frac{2}{\overline{\varphi}} >1\). By using (2.11) again, we get

which contradicts with (2.10), verifying (2.6).

Integrating both sides of (2.8) over the interval \([0,T]\), we obtain

Since \(\int ^{T}_{0}e(t)dt=T\bar{e}=0\), it follows that \(\int ^{T}_{0}\varphi (t)u^{\mu}(t)dt = \int ^{T}_{0} \frac{1}{u^{\gamma}(t)}dt\). If

then

By using the mean value theorem for integrals, we get that there is a point \(\xi \in [0,T]\) such that

i.e.,

Thus (2.7) immediately follows from (2.13) and (2.14). □

### Lemma 2.5

*Assume* \(\overline{\varphi}>0\) *and* \(\overline{e}=0\) *for a*.*e*. \(t\in [0,T]\) *and suppose that the following assumptions*:

*and*

*hold*, *where* \(H(x)=F(x)-T\overline{\varphi _{-}}x^{\mu}\). *Then there is a constant* \(\gamma _{0}>0\) *such that*

### Proof

Let \(u\in D\), then *u* satisfies

Since \(u\in D\), it is easy to see that there are two points \(t_{1}, t_{2}\in R\) such that \(u(t_{1})=\max _{t\in [0,T]}u(t)\), \(u(t_{2})=\min _{t\in [0,T]}u(t)\), and \(0< t_{1}-t_{2}\le T\). By integrating (2.18) over the interval \([t_{2},t_{1}]\), we get

and then

Assumption (2.16) ensures that there is a constant \(\gamma _{0}> 0\) such that

Combining (2.19) with (2.20), we get that

□

### Lemma 2.6

*Assume* \(\overline{\varphi}>0\) *and* \(\overline{e}=0\) *for a*.*e*. \(t\in [0,T]\) *and suppose that the following assumptions*:

*and*

*hold*. *Then*, *there exists a constant* \(\gamma _{1}> 0\) *such that*

### Proof

Since \(u\in D\), the function *u* satisfies (2.18). Then there are two points \(t_{1}, t_{2}\in R\) such that \(u(t_{1})=\max _{t\in [0,T]}u(t)\), \(u(t_{2})=\min _{t\in [0,T]}u(t)\), and \(0< t_{2}-t_{1}< T\). By integrating over the interval \([t_{1},t_{2}]\), we get

thus, by the assumptions of (2.6), (2.22), and (2.24), according to the proof of Lemma 2.4, we obtain

So, we have

Assumption (2.24) now ensures that there is a constant \(\gamma _{1}> \gamma _{0}> 0\) such that

Therefore, (2.27) and (2.28) imply

□

### Lemma 2.7

*Assume* \(\overline{\varphi}>0\) *and* \(\overline{e}=0\) *for a*.*e*. \(t\in [0,T]\) *and suppose that the following assumptions*:

*and*

*hold*, *where* \(H_{1}(x)=F(x)+T\overline{\varphi _{-}}x^{\mu}\). *Then there is a constant* \(\gamma _{2}>0\) *such that*

### Proof

Since \(u\in D\), it is easy to see that there exist two points \(t_{1}, t_{2}\in R\) such that \(u(t_{1})=\max _{t\in [0,T]}u(t)\), \(u(t_{2})=\min _{t\in [0,T]}u(t)\), and \(0< t_{2}-t_{1}< T\). By integrating over the interval \([t_{1},t_{2}]\), we get

and then

Assumption (2.31) ensures that there is a constant \(\gamma _{2}> 0\) such that

So, it is easy to see from (2.33) that

□

### Lemma 2.8

*Assume* \(\overline{\varphi}>0\) *and* \(\overline{e}=0\) *for a*.*e*. \(t\in [0,T]\) *and suppose that the following assumptions*:

*as well as*

*and*

*hold*. *Then*, *there exists a constant* \(\gamma _{3}> 0\) *such that*

### Proof

Let \(u\in D\), then *u* satisfies (2.18). Let \(t_{1}\) and \(t_{2}\) be defined as in the proof of Lemma 2.6, that is, \(u(t_{1})=\max _{t\in [0,T]}u(t)\), \(u(t_{2})=\min _{t\in [0,T]}u(t)\), and \(0< t_{2}-t_{1}< T\). By integrating over the interval \([t_{1},t_{2}]\), we get

Thus, by the assumptions of (2.6), (2.35), and (2.36), and according to the proof of Lemma 2.6, we have

which together with (2.39) yields

On the other hand, assumption (2.37) gives that there exits a constant \(\gamma _{3}> 0\) such that

Combining (2.41) with (2.42), we get that

□

## 3 Main results

### Theorem 3.1

*Assume* \(\overline{\varphi}>0\) *and* \(\overline{e}=0\) *for a*.*e*. \(t\in [0,T]\) *and suppose that the assumptions of* (2.15) *and* (2.16) *in Lemma *2.4, *as well as the assumption* (2.24) *in Lemma *2.5, *hold*. *Then for each* \(\mu \in [0, +\infty )\), *equation* (1.5) *has at least one positive* *T*-*periodic solution*.

### Proof

Due to assumptions of Lemma 2.4, we see that there are two constants \(\gamma _{0}> 0\), \(\gamma _{1}> 0\) such that \(\min u(t) \geq \gamma _{0}\), \(\max u(t)\leq \gamma _{1}\).

Now, we will show that there exists a positive constant \(M > 0\) such that \(\max _{t\in [0,T]} |u'(t)| \leq M\), uniformly for \(u\in D\). If \(u(t_{1})=\max _{t\in [0,T]}\), \(t_{1}\in [0,T]\), then \(u'(t_{1})=0\). Letting \(t\in [0, T]\), we integrate (2.8) over the interval \([t_{1},t]\) and get

which yields

and then we obtain

So, we have

Let \(m_{1}=\min \{\gamma _{0}, D_{1}\}\) and \(m_{2}= \{\gamma _{1}, D_{2}\}\) be two constants, where \(D_{1}\) and \(D_{2}\) are the constants determined in Remark 2.3. Then we get that every possible positive *T*-periodic solution \(x(t)\) to equation (1.5) satisfies

Furthermore, we have

by using Lemma 2.1, thus equation (1.5) has at least one positive *T*-periodic solution.

On the other hand, by Lemmas 2.6 and 2.7, we get the same conclusion as in Theorem 3.1, which can be proved similarly. Thus, the proofs are omitted. □

### Theorem 3.2

*Assume* \(\overline{\varphi}>0\) *and* \(\overline{e}=0\) *for a*.*e*. \(t\in [0,T]\) *and suppose that the assumptions of* (2.30) *and* (2.31) *in Lemma *2.6, *as well as the assumption* (2.37) *in Lemma *2.7, *hold*. *Then for each* \(\mu \in [0, +\infty )\), *equation* (1.5) *has at least one positive* *T*-*periodic solution*.

## 4 Example

In this section, we present two examples to demonstrate the main results.

### Example 4.1

Considering the following equation:

Corresponding to equation (1.5), in (4.1), \(e(t)=\sin (t)\), \(\varphi (t)=1+2\cos{t}\), \(T=2\pi \). Obviously, \(\overline{\varphi}=1 > 0\), and \(\overline{e}=0\) for all \(t \in [0,T]\) with \(\overline{\varphi _{+}}= \frac{5}{6} +\frac{1}{\pi}\) and \(\overline{\varphi _{-}}=\frac{1}{\pi}- \frac{1}{6} \). Since \(F(x)= -\frac{1}{x^{3}} + (\frac{5\pi}{3} + 2)x^{\frac{5}{2}}\), we can easily verify that equation (4.1) satisfies

and

Obviously, (4.2), (4.3), and (4.4) imply that assumptions (2.15), (2.16), and (2.24) hold. Thus, by using Theorem 3.1, equation (4.1) has at least one positive 2*π*-periodic solution.

### Example 4.2

Now consider

Corresponding to equation (1.5), here, \(e(t)=\sin{t}\), \(\varphi (t)=1+2\cos{t}\), \(T=2\pi \). Clearly, \(\overline{\varphi}=1 > 0\), and \(\overline{e}=0\) for all \(t \in [0,T]\) with \(\overline{\varphi _{+}}= \frac{5}{6} +\frac{1}{\pi}\) and \(\overline{\varphi _{-}}=\frac{1}{\pi}- \frac{1}{6} \). Since \(F(x)= \frac{1}{x^{3}} - (2-\frac{\pi}{3})x^{\frac{5}{2}}\), we can easily verify that (4.1) satisfies

and

Obviously, (4.6), (4.7), (4.8) imply that assumptions (2.30), (2.31), and (2.37) hold. Thus, by using Theorem 3.2, equation (4.5) has at least one positive 2*π*-periodic solution.

### Remark 4.3

In (4.5), since \(\mu =\frac{3}{2}>1\) and \(\varphi (t)=1+2\cos t\) is a sign-changing function, the result of Example 4.2 can be obtained neither by using the main results of [23], nor by using the theorems of [23]. In this sense, the theorems of the present paper are new results on the existence of positive periodic solutions for singular Liénard equations.

## Data Availability

No datasets were generated or analysed during the current study.

## References

Torres, P.J.: Mathematical Models with Singularities-a Zoo of Singular Creatures. Atlantis Press, Amsterdam (2015). ISBN:978-94-6239-105-5

Plesset, M.S.: The dynamics of cavitation bubbles. J. Appl. Mech.

**16**, 228–231 (1949)Jebelean, P., Mawhin, J.: Periodic solutions of singular nonlinear differential perturbations of the ordinary

*p*-Laplacian. Adv. Nonlinear Stud.**2**(3), 299–312 (2002)Lei, J., Zhang, M.: Twist property of periodic motion of an atom near a charged wire. Lett. Math. Phys.

**60**(1), 9–17 (2002)Huang, J., Ruan, S., Song, J.: Bifurcations in a predator–pry system of Leslite type with deneralized Holling type functional response. J. Differ. Equ.

**257**(6), 1721–1751 (2014)Yu, X., Lu, S.: A multiplicity result for periodic solutions of Liénard equations with an attractive singularity. Appl. Math. Comput.

**346**, 183–192 (2019)Nagumo, M.: On the periodic solution of an ordinary differential equation of second order. In: Zenkokou Shijou Suudaku Danwakai, pp. 54–61 (1994). (in Japanese). English translation in Mitio Nagumo Collected papers, Springer, Berlin (1993)

Torres, P.J.: Weak singularities may help periodic solutions to exist. J. Differ. Equ.

**232**, 277–284 (2007)Li, X., Zhang, Z.: Periodic solutions for second order differential equations with a singular nonlinearity. Nonlinear Anal.

**69**, 3866–3876 (2008)Jiang, D., Chu, J., Zhang, M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ.

**211**, 282–302 (2005)Chu, J., Torres, P.J., Zhang, M.: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ.

**239**, 196–212 (2007)Cheng, Z., Ren, J.: Multiplicity results of positive solutions foe fourth-order nonlinear differential equation with singularity. Math. Methods Appl. Sci.

**38**, 5284–5304 (2016)Cheng, Z., Ren, J.: Positive solutions for fourth-order singular nonlinear differential equation with variable-coefficient. Math. Methods Appl. Sci.

**39**, 2251–2274 (2016)Hakl, R., Torres, P.J.: On periodic solutions of second-order differential equations with attractive–repulsive singularities. J. Differ. Equ.

**248**, 111–126 (2010)Habets, P., Sanchez, L.: Periodic solutions of some Liénard equations with singularities. Proc. Am. Math. Soc.

**109**, 1135–1144 (1990)Zhang, M.: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl.

**203**(1), 254–269 (1996)Kong, F., Lu, S., Luo, Z.: Positive periodic solutions of singular higher order delay differential equations. Results Math.

**72**(1), 1–16 (2017)Lu, S., Zhong, T., Gao, Y.: Periodic solutions of

*p*-Laplacian equations with singularities. Adv. Differ. Equ.**2016**, 146 (2016). https://doi.org/10.1186/s13662-016-0875-6Martins, R.: Existence of periodic solutions for second-order differential equations with singularities and the strong force condition. J. Math. Anal. Appl.

**317**, 1–13 (2006)Yu, X., Lu, S.: A multiplicity result for periodic solutions of Liénard equations with an attractive singularity. Appl. Math. Comput.

**346**, 183–192 (2019)Wang, Z.: Periodic solutions of Lińard equation with a singularity and a deviating argument. Nonlinear Anal., Real World Appl.

**16**(1), 227–234 (2014)Hakl, R., Torres, P.J., Zamora, M.: Periodic solutions of singular second order differential equations: the repulsive case. Topol. Methods Nonlinear Anal.

**39**, 199–220 (2012)Lu, S., Guo, Y., Chen, L.: Periodic solutions for Liénard equation with an indefinite singularity. Nonlinear Anal., Real World Appl.

**45**, 542–556 (2019)Manásevich, R., Mawhin, J.: Periodic solutions for nonlinear systems with

*p*-Laplacian-like operator. J. Differ. Equ.**145**, 367–393 (1998)

## Acknowledgements

Acknowledge my teacher.

## Funding

This work is supported by Anhui Province higher discipline top talent academic funding project (No.gxbjZD2022097) and KJ2021A1231

## Author information

### Authors and Affiliations

### Contributions

Yu Zhu have equally contributed to obtaining new results in this article and also read and approved the final manuscript.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Additional information

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Zhu, Y. Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type.
*Bound Value Probl* **2024**, 85 (2024). https://doi.org/10.1186/s13661-024-01894-8

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/s13661-024-01894-8