Abstract
In this paper, by application of the Manasevich-Mawhin continuation theorem, we investigate the existence of a positive periodic solution for a kind of ϕ-Laplacian singular Liénard equation with attractive and repulsive singularities.
Similar content being viewed by others
1 Introduction
Liénard equation [1]
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, Van der Pol oscillator
is a Liénard equation.
From then on, there have been a good amount of work on periodic solutions for Liénard equations (see [2–12] and the references cited therein). Some classical tools have been used to study Liénard equation in the literature, including Mawhin’s coincidence degree theorem [2, 3, 5], topological degree methods [4, 6], Schauder’s fixed point theorem [7], Massera’s theorem [8], the Manasevich-Mawhin continuation theorem [9, 12], generalized polar coordinates [10] and the Poincaré map [11].
At the same time, some authors began to consider Liénard equation with singularity [13–20]. For example, in 1996, Zhang [20] discussed a kind of singular Liénard equation
where g was on the singular case, i.e., when \(g(t,x)\rightarrow+\infty\), as \(x\rightarrow0^{+}\). By application of coincidence degree theory, the author obtained that (1.2) had at least one periodic solution. Afterwards, Jebelean and Mawhin investigated the following quasilinear equation of p-Laplacian type:
where g satisfied slightly strong singularity, i.e.,
The authors proved that the above problem had at least one positive periodic solution through a basic application of the Manasevich-Mawhin continuation theorem. Recently, Xin and Cheng [19] studied the following p-Laplacian Liénard equation with singularity and deviating argument:
By applications of coincidence degree theory and some analysis skills, they obtained that (1.3) had at least one positive periodic solution.
All the aforementioned results concern singular Liénard equation and singular p-Laplacian Liénard equation. There are few results on the ϕ-Laplacian Liénard equation with singularity. Motivated by [13, 19, 20], in this paper, we further consider the following ϕ-Laplacian Liénard equation:
where \(f:\mathbb{R}\to\mathbb{R}\) is an \(L^{2}\)-Carathéodory function, which means, it is measurable in the first variable and continuous in the second variable, and for every \(0< r< s\), there exists \(h_{r,s}\in L^{2}[0,T]\) such that \(\vert g(t,x(t)) \vert \leq h_{r,s}\) for all \(x\in[r,s]\) and a.e. \(t\in[0,T]\); τ is a positive constant; \(e\in L^{2}(\mathbb{R})\) is a T-periodic function; \(g:(0,+\infty)\to \mathbb{R}\) is the \(L^{2}\)-function, the nonlinear term g of (1.4) can be with a singularity at origin, i.e.,
It is said that (1.4) is of attractive type (resp. repulsive type) if \(g(x)\rightarrow+\infty\) (resp. \(g(x)\rightarrow-\infty\)) as \(x\rightarrow0^{+}\).
Moreover, \(\phi:\mathbb{R}\rightarrow\mathbb{R}\) is a continuous function and \(\phi(0)=0\), which satisfies
- \((A_{1})\) :
-
\((\phi(x_{1})-\phi(x_{2}))(x_{1}-x_{2})>0\) for \(\forall x_{1}\neq x_{2}, x_{1}, x_{2}\in\mathbb{R}\);
- \((A_{2})\) :
-
There exists a function \(\alpha:[0,+\infty]\rightarrow[0,+\infty], \alpha(s)\rightarrow+\infty\) as \(s\rightarrow+\infty\), such that \(\phi(x)\cdot x\geq\alpha( \vert x \vert ) \vert x \vert \) for \(\forall x\in\mathbb{R}\).
It is easy to see that ϕ represents a large class of nonlinear operators, including \(\vert u \vert ^{p-2}u: \mathbb{R}\rightarrow\mathbb{R} \) which is a p-Laplacian operator.
The remaining part of the paper is organized as follows. In Section 2, we give some preliminary lemmas. In Section 3, by employing the Manasevich-Mawhin continuation theorem, we state and prove the existence of a positive periodic solution for (1.4) with attractive singularity. In Section 4, we investigate the existence result for (1.4) with repulsive singularity. In Section 5, two numerical examples demonstrate the validity of the method. Our results improve and extend the results in [13, 15, 18–20].
2 Preliminary lemmas
For the T-periodic boundary value problem
here \(\tilde{f}:[0,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) is assumed to be Carathéodory.
Lemma 2.1
Manasevich-Mawhin [21]
Let Ω be an open bounded set in \(C^{1}_{T}:=\{x\in C^{1}(\mathbb{R},\mathbb{R}): x\textit{ is $T$-periodic}\}\). If
-
(i)
for each \(\lambda\in(0,1)\), the problem
$$\bigl(\phi\bigl(x'\bigr)\bigr)'=\lambda \tilde{f} \bigl(t,x,x'\bigr), \qquad x(0)=x(T), \qquad x'(0)=x'(T) $$has no solution on ∂Ω;
-
(ii)
the equation
$$F(a):=\frac{1}{T} \int^{T}_{0}\tilde{f}\bigl(t,x,x' \bigr)\,dt=0 $$has no solution on \(\partial\Omega\cap\mathbb{R}\);
-
(iii)
the Brouwer degree of F
$$\deg\{F,\Omega\cap\mathbb{R},0\}\neq0, $$then the periodic boundary value problem (2.1) has at least one periodic solution on Ω̄.
Next, we embed equation (1.4) into the following equation family with a parameter \(\lambda\in(0,1]\):
By applications of Lemma 2.1, we obtain the following result.
Lemma 2.2
Suppose that \((A_{1})\) and \((A_{2})\) hold. Assume that there exist positive constants \(E_{1}, E_{2}, E_{3}\) and \(E_{1}< E_{2}\) such that the following conditions hold:
-
(1)
Each possible periodic solution x to equation (2.2) such 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 equation
$$g(C)-\frac{1}{T} \int^{T}_{0} e(t)\,dt=0 $$satisfies \(E_{1}< C< E_{2}\).
-
(3)
It holds
$$\biggl(g(E_{1})-\frac{1}{T} \int^{T}_{0} e(t)\,dt \biggr) \biggl(g(E_{2})- \frac {1}{T} \int^{T}_{0} e(t)\,dt \biggr)< 0. $$Then (1.4) has at least one T-periodic solution.
3 Main results (I): periodic solution of (1.4) with attractive singularity
In this section, we investigate the existence of a positive periodic solution for (1.4) with attractive singularity.
Theorem 3.1
Assume that conditions \((A_{1})\) and \((A_{2})\) hold. Suppose that 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 \(x\in(d_{2},+\infty)\).
- \((H_{2})\) :
-
There exist positive constants \(a, b\) and m such that
$$ g(x)\leq a x^{m}+b, \quad \textit{for all } x>0. $$(3.1) - \((H_{3})\) :
-
(Attractive singularity) \(\lim_{x\to 0^{+}}\int^{1}_{x}g(s)\,ds=+\infty\).
- \((H_{4})\) :
-
There exists a constant \(\gamma>0\) such that \(\inf_{x\in\mathbb{R}} \vert f(t,x) \vert \geq\gamma>0\).
Then (1.4) has a positive T-periodic solution.
Proof
Firstly, we claim that there exists a point \(t_{1}\in[0,T]\) such that
Let \(\underline{t}\), t̅ be, respectively, the global minimum point and the global maximum point \(x(t)\) on \([0,T]\); then \(x'(\underline{t})=0\) and \(x'(\overline{t})=0\), and we claim that
In fact, if (3.3) does not hold, then \((\phi(x'(\underline{t})))'<0\) and there exists \(\varepsilon>0\) such that \((\phi(x'(t)))'<0\) for \(t\in(\underline{t}-\varepsilon,\underline{t}+\varepsilon)\). Therefore \(\phi(x'(t))\) is strictly decreasing for \(t\in(\underline{t}-\varepsilon,\underline{t}+\varepsilon)\). From \((A_{1})\), we know that \(x'(t)\) is strictly decreasing for \(t\in(\underline{t}-\varepsilon,\underline{t}+\varepsilon)\). This contradicts the definition of \(\underline{t}\). Thus, (3.3) is true. From (2.2) and (3.3), we have
Similarly, we can get
From \((H_{1})\), (3.4) and (3.5), we have
In view of x being a continuous function, we can get (3.2).
Multiplying both sides of (2.2) by \(x'(t)\) and integrating over the interval \([0,T]\), we have
Moreover, we have
and
since \(dx(t)=\frac{dx(t-\tau)}{d(t-\tau)}\,dt=dx(t-\tau)\).
Substituting (3.7) and (3.8) into (3.6), we have
From (3.9), we have
From \((H_{4})\), we know
Therefore, we can get
where \(\Vert e \Vert _{2}= (\int^{T}_{0} \vert e(t) \vert ^{2}\,dt )^{\frac{1}{2}}\). It is easy to see that there exists a positive constant \(M_{1}'\) (independent of λ) such that
From (3.2) and (3.10), we have
On the other hand, integrating both sides of (2.2) over \([0,T]\), we have
Therefore, from (3.10), (3.12) and \((H_{2})\), we have
where \(f_{M_{1}}:=\max_{0\leq x(t)\leq M_{1}} \vert f(t,x) \vert , \Vert f_{M_{1}} \Vert _{2}:= (\int^{T}_{0} \vert f(t,x(t)) \vert ^{2}\,dt )^{\frac {1}{2}}\). As \(x(0)=x(T)\), there exists a point \(t_{2}\in[0,T]\) such that \(x'(t_{2})=0\), while \(\phi(0)=0\), from (3.10), (3.12) and (3.13), we have
We claim that there exists a positive constant \(M_{2}>M_{2}'+1\) such that, for all \(t\in\mathbb{R}\),
In fact, if \(x'\) is not bounded, then from the definition of α, there exists a positive constant \(M_{2}''\) such that \(\alpha( \vert x' \vert )>M_{2}''\) for some \(x'\in\mathbb{R}\). However, from \((A_{2})\), we have
Then we can get
which is a contradiction. So, (3.15) holds.
From (2.2), we have
Multiplying both sides of (3.16) by \(x'(t)\) and integrating on \([\xi,t]\), here \(\xi\in[0,T]\), we get
By (3.14) and (3.15), we can get
Moreover, from (3.15), we have
From (3.17), we have
From \((H_{3})\), we know that there exists a constant \(M_{3}>0\) such that
Similarly, we can consider \(t\in[0,\xi]\).
Let \(E_{1}<\min\{d_{1},M_{3}\}\), \(E_{2}>\max\{d_{2}, M_{1}\}\), \(E_{3}>M_{2}\) be constants, from (3.11), (3.15) and (3.19), we can get that the periodic solution x to (2.2) satisfies
Then condition (1) of Lemma 2.1 is satisfied. For a possible solution C to equation
it satisfies \(E_{1}< C<E_{2}\). Therefore, condition (2) of Lemma 2.2 holds. Finally, we consider condition (3) of Lemma 2.2 is also satisfied. In fact, from \((H_{1})\), we have
and
So condition (3) is also satisfied. By application of Lemma 2.2, we get that (1.4) has at least one positive periodic solution. □
4 Main results (II): periodic solution of (1.4) with repulsive singularity
In this section, we consider (1.4) in the case that \(f(t,x)\equiv f(x)\). Then (1.4) can be written as
We will discuss the existence of a positive periodic solution for (4.1) with repulsive singularity.
Theorem 4.1
Assume that conditions \((A_{1})\) and \((A_{2})\) hold. Suppose that the following conditions hold:
- \((H_{1}^{*})\) :
-
There exist constants \(0< d_{1}^{*}< d_{2}^{*}\) such that \(g(x)<0\) for \(x\in(0,d_{1}^{*})\) and \(g(x)>0\) for \(x\in(d_{2}^{*},+\infty)\).
- \((H_{2}^{*})\) :
-
\(\int^{T}_{0}e(t)\,dt=0\).
- \((H_{3}^{*})\) :
-
(Repulsive singularity) \(\lim_{x\to 0^{+}}\int^{1}_{x}g(s)\,ds=-\infty\).
- \((H_{4}^{*})\) :
-
There exists a constant \(\gamma^{*}>0\) such that \(\inf_{x\in\mathbb{R}} \vert f(x) \vert \geq\gamma^{*}>0\).
Then (4.1) has at least one positive T-periodic solution.
Proof
Firstly, we embed equation (4.1) into the following equation family with a parameter \(\lambda\in(0,1]\):
Integrating both sides of (4.2) over \([0,T]\), from \((H_{2}^{*})\), we have
From the continuity of g, we know there exists \(t_{1}^{*}\in[0,T]\) such that
Let \(t_{3}=t_{1}^{*}-\tau\), from assumption \((H_{1})\) we can get
We follow the same strategy and notation as in the proof of Theorem 3.1. We know that there exists \(M_{1}^{*}>0\) such that
Next, we prove that there exists a positive constant \(M_{2}^{*}\) such that \(\Vert x' \Vert \leq M_{2}^{*}\).
In fact, we get from (4.3) that
where \(g^{+}(x)=\max\{g(x),0\}\). Since \(g^{+}(x(t-\tau))\geq0\), from \((H_{1}^{*})\) we know \(x(t-\tau)\geq d_{2}^{*}\). Then we have
where \(\Vert g^{+}_{M_{1}} \Vert =\max_{d_{2}^{*}\leq x\leq M_{1}}g^{+}(x)\).
As \(x(0)=x(T)\), there exists a point \(t_{4}\in[0,T]\) such that \(x'(t_{4})=0\), which \(\phi(0)=0\), we have
Thus, from (3.15) we know that there exists some positive constant \(M_{2}^{*}\) such that
The proof left is the same as that of Theorem 3.1. □
5 Examples
Example 5.1
Consider the following ϕ-Laplacian Liénard equation with attractive singularity
where \(\phi(u)=ue^{ \vert u \vert ^{2}}\), τ is a positive constant and \(0<\tau <T\), \(m\geq0\) and \(\kappa\geq1\).
Comparing (5.1) to (1.4), it is easy to see that \(g(x)=-3x^{m}(t-\tau)+\frac{1}{x^{\kappa}(t-\tau)}\), \(f(t,x)=5x^{6}\cos^{t}+10\), \(e(t)=e^{\cos^{2}t}, T=\pi\). Obviously, we get
and
So, conditions \((A_{1})\) and \((A_{2})\) hold. Moreover, it is easy to see that there exist constants \(d_{1}=0.1\) and \(d_{2}=1\) such that \((H_{1})\) holds. \(g(x)\leq3x^{m}+1\), here \(a=3, b=1\), condition \((H_{2})\) holds. Consider \(\vert f(t,x) \vert = \vert 5x^{6} \cos^{2}t+10 \vert \geq10:=\gamma\), so, condition \((H_{4})\) is satisfied. Next, we consider that condition \((H_{3})\) is also satisfied. In fact, \(\lim_{x\to 0^{+}}\int^{1}_{x}g(s)\,ds=\lim_{x\to 0^{+}}\int^{1}_{x}(-x^{m}+\frac{1}{x^{\kappa}})\,ds=+\infty\), thus, condition \((H_{3})\) holds. Therefore, by Theorem 3.1, we know that (5.1) has at least one positive π-periodic solution.
Example 5.2
Consider the ϕ-Laplacian Liénard equation with repulsive singularity:
where relativistic operator \(\phi(u)=\frac{u}{\sqrt{1- (\frac{ \vert u \vert }{c} )^{2}}}\), here c is the speed of light in the vacuum and \(c>0\), τ is a constant and \(0\leq\tau< T\), \(m\geq0\).
It is clear that \(T=2\pi\), \(g(x)=5x^{m}(t-\tau)-\frac{1}{x^{\kappa}(t-\tau)}\), \(f(x)=x^{4}+3\), \(e(t)=\sin t\). It is obvious that \((H_{1}^{*})\)-\((H_{4}^{*})\) hold. Now we consider conditions \((A_{1})\) and \((A_{2})\).
and
Then conditions \((A_{1})\) and \((A_{2})\) hold. Therefore, by Theorem 4.1, we know that (5.2) has at least one positive periodic solution.
References
Liénard, A: Etude des oscillations entretenues, vol. 23 pp. 901-912 (1928)
Cheng, ZB, Ren, JL: Existence of periodic solution for fourth-order Liénard type p-Laplacian generalized neutral differential equation with variable parameter. J. Appl. Anal. Comput. 5, 704-720 (2015)
Cheung, WS, Ren, JL: Periodic solutions for p-Laplacian Liénard equation with a deviating argument. Nonlinear Anal. 59, 107-120 (2004)
Liu, WB, Feng, ZS: Periodic solutions for p-Laplacian systems of Liénard-type. Commun. Pure Appl. Anal. 10, 1393-1400 (2011)
Liu, BW, Huang, LH: Existence and uniqueness of periodic solutions for a kind of Liénard equation with a deviating argument. Appl. Math. Lett. 21, 56-62 (2008)
Ma, TT, Wang, ZH: Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete Contin. Dyn. Syst. 33, 1563-1581 (2013)
Torres, J: Nondegeneracy of the periodically forced Liénard differential equation with ϕ-Laplacian. Commun. Contemp. Math. 13, 283-292 (2011)
Villari, G: An improvement of Massera’s theorem for the existence and uniqueness of a periodic solution for the Liénard equation. Rend. Ist. Mat. Univ. Trieste 44, 187-195 (2012)
Wang, Y, Dai, XZ, Xia, XX: On the existence of a unique periodic solution to a Liénard type p-Laplacian non-autonomous equation. Nonlinear Anal. 71, 275-280 (2009)
Wang, YY, Cheng, SS, Ge, WG: Periodic solutions of generalized Liénard equations with a p-Laplacian-like operator. Bull. Braz. Math. Soc. 39, 21-43 (2008)
Yang, LJ, Zeng, XW: The period function of Liénard systems. Proc. R. Soc. Edinb. A 143, 205-221 (2013)
Xin, Y, Han, XF, Cheng, ZB: Existence and uniqueness of positive periodic solution for ϕ-Laplacian Liénard equation. Bound. Value Probl. 2014, 244 (2014)
Jebelean, P, Mawhin, J: Periodic solutions of forced dissipative p-Liénard equations with singularities. Vietnam J. Math. 32, 97-103 (2004)
Kong, FC, Lu, SP, Liang, ZT: Existence of positive periodic solutions for neutral Liénard differential equations with a singularity. Electron. J. Differ. Equ. 2015 242, (2015)
Mawhin, J: Periodic solutions for quasilinear complex-valued differential systems involving singular ϕ-Laplacians. Rend. Ist. Mat. Univ. Trieste 44, 75-87 (2012)
Li, SJ, Liao, FF, Xing, WY: Periodic solutions for Liénard differential equations with singularities. Electron. J. Differ. Equ. 2015 151, (2015)
Lu, SP, Kong, FC: Periodic solutions for a kind of prescribed mean curvature Liénard equation with a singularity and a deviating argument. Adv. Differ. Equ. 2015, 151 (2015)
Wang, ZH: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal., Real World Appl. 16, 227-234 (2014)
Xin, Y, Cheng, ZB: Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument. Adv. Differ. Equ. 2016, 41 (2016)
Zhang, MR: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203, 254-269 (1996)
Manásevich, R, Mawhin, J: Periodic solutions for nonlinear systems with p-Laplacian-like operator. J. Differ. Equ. 145, 367-393 (1998)
Acknowledgements
YX and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by the National Natural Science Foundation of China (No. 11501170), China Postdoctoral Science Foundation funded project (No. 2016M590886).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YX and ZBC worked together in the derivation of the mathematical results. Both authors read and approved the final manuscript.
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 distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Xin, Y., Cheng, Z. Study on a kind of ϕ-Laplacian Liénard equation with attractive and repulsive singularities. J Inequal Appl 2017, 180 (2017). https://doi.org/10.1186/s13660-017-1410-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1410-3