Abstract
In this paper, the problem of the existence of a periodic solution is studied for the second order differential equation with a singularity of repulsive type
where \(g(x)\) is singular at \(x=0\), φ and h are T-periodic functions. By using the continuation theorem of Manásevich and Mawhin, a new result on the existence of positive periodic solution is obtained. It is interesting that the sign of the function \(\varphi(t)\) is allowed to change for \(t\in[0,T]\).
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1 Introduction
The aim of this paper is to search for positive T-periodic solutions for a second order differential equation with a singularity in the following form:
where \(f:[0,\infty)\rightarrow R\) is an arbitrary continuous function, \(g\in C((0,+\infty), (0,+\infty))\), and \(g(x)\) is singular of repulsive type at \(x=0\), i.e., \(g(x)\rightarrow+\infty\) as \(x\rightarrow 0^{+}\), \(\varphi,h: R\rightarrow R\) are T-periodic functions with \(h\in L^{2}([0,T], R)\) and \(\varphi\in C([0,T], R)\), and the sign of the function φ is allowed to change for \(t\in[0,T]\).
The study of the problem of periodic solutions to scalar equations with a singularity began with work of Forbat and Huaux [1, 2], where the singular term in the equations models the restoring force caused by a compressed perfect gas (see [3–6] and the references therein). In the past years, many works used the methods, such as the approaches of critical point theory [7–12], the techniques of some fixed point theorems [13–15], and the approaches of topological degree theory, in particular, of some continuation theorems of Mawhin (see [6, 16–22]), to study the existence of positive periodic solutions for some second order ordinary differential equations with singularities. For example, in [15], by using a fixed point theorem in cones, the existence of positive periodic solutions to equation (1.1) was investigated for the conservative case, i.e., \(f(x)\equiv0\). But the function \(\varphi(t)\) is required to be \(\varphi(t)\ge0\) for all \(t\in [0,T]\). The method of topological degree theory, together with the technique of upper and lower solutions, was first used by Lazer and Solimini in the pioneering paper [18] for considering the problem of a periodic solution to a second order differential equations with singularities. Jebelean and Mawhin in [6] considered the problem of a p-Laplacian Liénard equation of the form
and
where \(p>1\) is a constant, \(f: [0,+\infty)\rightarrow R\) is an arbitrary continuous function, \(h: R\rightarrow R\) is a T-periodic function with \(h\in L^{\infty}([0,T],R)\), \(g: (0,+\infty)\rightarrow (0,+\infty)\) is continuous, \(g(x)\rightarrow+\infty\) as \(x\rightarrow 0^{+}\). They extended the results of Lazer and Solimini in [16] to equation (1.2) and equation (1.3). For equation (1.3), the crucial condition is that the function \(g(x)\) is bounded, which means that equation (1.3) is not singular at \(x=+\infty\).
By using a continuation theorem of Mawhin, Zhang in [18] studied the problem of periodic solutions of the Liénard equation with a singularity of repulsive type,
where \(f: {R}\rightarrow{R}\) is continuous, \(g: {R}\times(0,+\infty )\rightarrow{R}\) is an \(L^{2}\)-Carathéodory function with T-periodic in the first argument, and it is singular at \(x=0\), i.e., \(g(t,x)\) is unbounded as \(x\rightarrow0^{+}\). Different from the equation studied in [6, 16], which is only singular at \(x=0\), equation (1.4) is provided with both singularities at \(x=+\infty\) and at \(x=0\). In [19], Wang further studied the existence of positive periodic solutions for a delay Liénard equation with a singularity of repulsive type
In [18, 19], the following balance condition between the singular force at the origin and at infinity is needed.
(h1) There exist constants \(0< D_{1}< D_{2}\) such that if x is a positive continuous T-periodic function satisfying
then
From the proof of [18, 19], we see that the balance condition (h1) is crucial for estimating a priori bounds of periodic solutions. Now, the question is how to investigate the existence of positive periodic solutions for the equations like equation (1.4) or equation (1.5) without the balance condition (h1).
Motivated by this, in this paper, we study the existence of positive T-periodic solutions for equation (1.1) under the condition that the sign of the function φ is allowed to change for \(t\in [0,T]\). For this case, the balance condition (h1) may not be satisfied. By using the continuation theorem of Manásevich and Mawhin, a new result on the existence of positive periodic solutions is obtained.
2 Preliminary lemmas
Throughout this paper, let \(C_{T}=\{x\in C(R,R): x(t+T)= x(t) \mbox{ for all } t\in R\}\) with the norm defined by \(|x|_{\infty}=\max_{t\in [0,T]}|x(t)|\). For any T-periodic solution \(y(t)\) with \(y\in L^{1}([0,T], R)\), \(y_{+}(t)\) and \(y_{-}(t)\) denote \(\max\{y(t),0\}\) and \(-\min\{y(t),0\}\), respectively, and \(\bar{y}=\frac{1}{T}\int _{0}^{T}y(s)\,ds\). Clearly, \(y(t)=y_{+}(t)-y_{-}(t)\) for all \(t\in R\), and \(\bar {y}=\overline{y_{+}}-\overline{y_{-}}\).
The following lemma is a consequence of Theorem 3.1 in [23].
Lemma 1
Assume that there exist positive constants \(M_{0}\), \(M_{1}\), and \(M_{2}\) 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 x to the equation
$$u''+\lambda f(u)u'-\lambda g(u)+\lambda \varphi(t)u=\lambda h(t) $$satisfies the inequalities \(M_{0}< x(t)< M_{1}\) and \(|x'(t)|< M_{2}\) for all \(t\in[0,T]\).
-
2.
Each possible solution c to the equation
$$g(c)-c\bar{\varphi}+\bar{h}=0 $$satisfies the inequality \(M_{0}< c< M_{1}\).
-
3.
We have
$$\bigl(g(M_{0})-\bar{\varphi}M_{0}+\bar{h} \bigr) \bigl(g(M_{1})-\bar{\varphi }M_{1}+\bar{h} \bigr)< 0. $$Then equation (1.1) has at least one T-periodic solution u such that \(M_{0}< u(t)< M_{1}\) for all \(t\in[0,T]\).
Lemma 2
[19]
Let x be a continuous T-periodic continuously differential function. Then, for any \(\tau\in(0,T]\),
In order to study the existence of positive periodic solutions to equation (1.1), we list the following assumptions.
[H1] The function \(\varphi(t)\) satisfies the following conditions:
and
[H2] there are constants \(A>0\) and \(M>0\) such that \(g(x)\in(0,A)\) for all \(x>M\);
[H3] \(\int_{0}^{1}g(s)\,ds=+\infty\).
Remark 1
If assumptions [H1]-[H2] hold, then there are constants \(D_{1}\) and \(D_{2}\) with \(0< D_{1}< D_{2}\) such that
and
Furthermore, assumption \(\sigma_{1}\in(0,1)\) in [H1] is different from the corresponding condition \(\int_{0}^{T}\varphi_{+}(s)\,ds<\frac{4}{T}\) in [20].
Now, we suppose that assumptions [H1] and [H2] hold, and we embed equation (1.1) into the following equation family with a parameter \(\lambda\in(0,1]\):
Let
and
where
\(A>0\) is a constant determined by assumption [H2]. Clearly, \(M_{0}\) and \(A_{0}\) are all independent of \((\lambda,x)\in(0,1]\times \varOmega \). Let \(M>0\) be determined by assumption [H2], then there is a positive integer \(k_{0}\) such that
Lemma 3
Assume that assumptions [H1]-[H2] hold, then there is an integer \(k^{*}>k_{0}\) such that, for each function \(u\in \varOmega \), there is a point \(t_{0}\in[0,T]\) satisfying
Proof
If the conclusion does not hold, then, for each \(k>k_{0}\), there is a function \(u_{k}\in \varOmega \) satisfying
From the definition of Ω, we see
and by using assumption [H2],
By integrating equation (2.5) over the interval \([0,T]\), we have
i.e.,
Since \(\varphi_{+}(t)\ge0\) and \(\varphi_{-}(t)\ge0\) for all \(t\in[0,T]\), it follows from the integral mean value theorem that there is a point \(\xi\in[0,T]\) such that
which together with (2.6) yields
It follows from \(|u_{k}|_{\infty}\le u_{k}(\xi)+T^{\frac{1}{2}} (\int _{0}^{T}|u_{k}'(s)|^{2}\,ds )^{\frac{1}{2}}\) that
On the other hand, by multiplying equation (2.5) with \(u_{k}(t)\), and integrating it over the interval \([0,T]\), we obtain
which together with the fact of \(g(x)>0\) for all \(x>0\) gives
i.e.,
By using Lemma 2, we have
Substituting (2.7) and (2.9) into the above formula,
where
which is determined by assumption [H1]. This gives
i.e.,
where
It follows from (2.9) that
Substituting (2.11)-(2.12) into (2.8), we have
which together with (2.2) yields
By the definition of \(k_{0}\), we see from (2.3) that (2.13) contradicts (2.4). This contradiction implies that the conclusion of Lemma 3 is true. □
3 Main results
Theorem 1
Assume that [H1]-[H3] hold. Then equation (1.1) has at leat one positive T-periodic solution.
Proof
Firstly, we will show that there exist \(M_{1},M_{2}\) with \(M_{1}>k^{*}M\) and \(M_{2}>0\) such that each positive T-periodic solution \(u(t)\) of equation (2.1) satisfies the inequalities
In fact, if u is an arbitrary positive T-periodic solution of equation (2.1), then
This implies \(u\in \varOmega \). So by using Lemma 3 that there is a point \(t_{0}\in[0,T]\) such that
and then
Integrating (3.2) over the interval \([0,T]\), we have
Since \(g(x)\rightarrow+\infty\) as \(x\rightarrow0^{+}\), we see from (3.5) that there is a point \(t_{1}\in[0,T]\) such that
where \(\gamma< k^{*}M\) is a positive constant, which is independent of \(\lambda\in(0,1]\). Similar to the proof of (2.9), we have
By using Lemma 2, we have
where \(t_{0}\) is determined in (3.3). Substituting (3.7) into (3.8), we have
which results in
Since \(\frac{T}{\pi}|\varphi_{+}|_{\infty}^{1/2}<\sigma_{1}\in(0,1)\), it follows from (3.9) that there is a constant \(\rho>0\), which is independent of \(\lambda\in(0,1]\), such that
and then by (3.7), we have
It follows from (3.4) that
i.e.,
Now, if u attains its maximum over \([0,T]\) at \(t_{2}\in[0,T]\), then \(u'(t_{2})=0\) and we deduce from (3.2) that
for all \(t\in[t_{2},t_{2}+T]\). Thus, if \(F'=f\), then
From (3.2), we see that
It follows from (3.10) and (3.11) that
and then
Equations (3.10) and (3.13) imply that (3.1) holds.
Below, we will show that there exists a constant \(\gamma_{0}\in (0,\gamma)\), such that each positive T-periodic solution of equation (2.1) satisfies
Suppose that \(u(t)\) is an arbitrary positive T-periodic solution of equation (2.1), then
Let \(t_{1}\) be determined in (3.6). Multiplying (3.15) by \(u'(t)\) and integrating it over the interval \([t_{1},t]\) (or \([t,t_{1}]\)), we get
which yields the estimate
From (3.10) and (3.12), we get
which gives
with
From [H3] there exists \(\gamma_{0}\in(0,\gamma)\) such that
Therefore, if there is a \(t^{*}\in[t_{1},t_{1}+T]\) such that \(u(t^{*})\le\gamma _{0}\), then from (3.17) we get
which contradicts (3.16). This contradiction gives that \(u(t)>\gamma_{0}\) for all \(t\in[0,T]\). So (3.14) holds. Let \(m_{0}=\min\{D_{1},\gamma_{0}\}\) and \(m_{1}\in(M_{1}+D_{2}, +\infty)\) be two constants, then from (3.1) and (3.14), we see that each possible positive T-periodic solution u to equation (2.1) satisfies
This implies that condition 1 and condition 2 of Lemma 1 are satisfied. Also, we can deduce from Remark 1 that
and
which results in
So condition 3 of Lemma 1 holds. By using Lemma 1, we see that equation (1.1) has at least one positive T-periodic solution. The proof is complete. □
Let us consider the equation
where \(f : [0,+\infty)\rightarrow R\) is an arbitrary continuous function, \(\varphi,h: R\rightarrow R\) are T-periodic functions with \(h\in L^{1}([0,T], R)\) and \(\varphi\in C([0,T], R)\), and the sign of the function φ is allowed to change for \(t\in[0,T]\), \(\gamma \ge1\) is a constant. Corresponding to equation (1.1), \(g(x)=\frac{1}{x^{\gamma}}\). For this case, \(g(x)\rightarrow+\infty\) as \(x\rightarrow0^{+}\), and assumptions [H2]-[H3] are satisfied. Thus, by using Theorem 1, we have the following results.
Corollary 1
Assume that the function \(\varphi(t)\) satisfies the following conditions:
and
Then, equation (3.18) possesses at least one positive T-periodic solution.
Remark 2
Corresponding to equation (1.4) and equation (1.5), the function \(g(t,x)\) associated to equation (3.18) can be regarded as
For the case of \(\varphi(t)\ge0\) for all \(t\in[0,T]\), we see that if x is a positive T-periodic continuous function satisfying \(\int _{0}^{T} g(t,x(t))\,dt=0\), then
By applying the integral mean value theorem to the term \(\int _{0}^{T}\varphi(t)x(t)\,dt\) in equation (3.20), one can easily verify that \(g(t,u)\) determined in (3.19) satisfies the balance condition (h1). However, if the sign of the function \(\varphi(t)\) is changeable for \(t\in[0,T]\), then it is unclear from (3.20) whether the balance condition (h1) is satisfied. For this case, the main results of [18, 19] cannot be applied to equation (3.18).
Corollary 2
Assume that the function \(\varphi(t)\) satisfies \(\varphi(t)\ge0\) for all \(t\in[0,T]\) with \(\int_{0}^{T}\varphi(s)\,ds>0\), and
Then, equation (3.18) possesses at least one positive T-periodic solution.
Example 1
Consider the following equation:
where f is an arbitrary continuous function, \(a\in(0,+\infty)\) is a constant. Corresponding to equation (3.18), we have \(\gamma=2\), \(\varphi(t)=a(1+2\sin2t)\) and \(h(t)=\cos2t\), \(T=\pi\). By simply calculating, we can verify that
and then
and
Thus, if \(0< a<\frac{1}{3} (\frac{4\pi+9-3\pi\sqrt{6}}{4\pi+9} )^{2}\), then \(\sigma_{1}\in(0,1)\). By using Corollary 1, we see that equation (3.21) has at least one positive π-periodic solution.
Remark 3
Since the sign of \(\varphi(t)=1+2\sin t\) is changed for \(t\in[0,T]\), whether the right inequality of (1.6) in the balance condition (h1) is satisfied remains unclear. So the conclusion of the example cannot be obtained by using the main results in [18, 19].
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The work is sponsored by the National Natural Science Foundation of China (No. 11271197).
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Lu, S. A new result on the existence of periodic solutions for Liénard equations with a singularity of repulsive type. J Inequal Appl 2017, 37 (2017). https://doi.org/10.1186/s13660-016-1285-8
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DOI: https://doi.org/10.1186/s13660-016-1285-8