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 \(\mu \in [0,+\infty )\).
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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 2π-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 2π-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.
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Acknowledge my teacher.
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This work is supported by Anhui Province higher discipline top talent academic funding project (No.gxbjZD2022097) and KJ2021A1231
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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
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DOI: https://doi.org/10.1186/s13661-024-01894-8