Abstract
In this paper, we study the existence of periodic solutions of Rayleigh equations with singularities \(x''+f(t, x')+g(x)=p(t)\). By using the limit properties of the time map, we prove that the given equation has at least one 2π periodic solution.
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1 Introduction
In this paper, we are concerned with the existence of periodic solutions of singular Rayleigh equations
where \(g: (0, +\infty)\to\mathbf{R}\) is continuous and has a singularity at the origin, \(f: \mathbf{R^{2}}\to\mathbf{R^{2}}\) is continuous and 2π periodic with respect to the first variable t, \(p: \mathbf{R}\to\mathbf{R}\) is continuous and 2π periodic.
Equation (1.1) can be used to model the oscillations of a clarinet reed [1]. The dynamic behaviors of (1.1) have been widely investigated due to their applications in many fields such as physics, mechanics, and the engineering technique fields (see [2–8] and the references therein). Recently, the periodic problem of equations with singularities has been studied widely because of their background in applied sciences (see [9–13] and the references therein).
When \(f\equiv0\), (1.1) is a conservation system
Assume that g satisfies
and
moreover, the primitive function G of g satisfies
where \(G(x)=\int_{1}^{x}g(u)\,du\). It was proved in [10] that (1.2) has at least one 2π periodic solution.
It is well known that time maps play an important role in studying the existence and multiplicity of periodic solutions of (1.2) [14, 15]. Assume that g satisfies
Condition (h3) implies that there exists a constant \(d>0\) such that
Let us consider the autonomous system
or its equivalent system
The first integral of (1.4) is the curve
where c is an arbitrary constant. From conditions (h i ) (\(i=1,2,3\)) we know that, for \(c>0\) sufficiently large, \(\Gamma_{c}\) is a closed curve. Let \((x(t), y(t))\) be any solution of (1.4) whose orbit is \(\Gamma_{c}\). Clearly, this solution is periodic. Let \(T(c)\) denote the least positive period of this solution. It is not hard to calculate
where \(0< d(c)< c\), \(G(d(c))=G(c)\), \(\lim_{c\to+\infty}d(c)=0\). From [10] we know that, if conditions (h i ) (\(i=1,2,3\)) hold, then
Now, let us set
In this paper, we deal with the existence of periodic solutions of (1.1) by using the asymptotic properties of the time map τ. Assume that the limit
holds uniformly for \(t\in[0, 2\pi]\). We obtain the following result.
Theorem 1.1
Assume that conditions (h i ) (\(i=1,2,3,4\)) hold. Then (1.1) possesses at least one 2π periodic solution provided that the inequality
holds.
Using Theorem 1.1, we can obtain the following corollary.
Corollary 1.2
Assume that conditions (h i ) (\(i=1,2,3,4\)) hold. Then (1.1) possesses at least one 2π periodic solution provided that the inequality
holds.
Throughout this paper, we always use the notations:
for any continuous 2π periodic function \(x(t)\). For a function \(I(c, \cdot)\), the notation \(I=o(1)\) means that, for \(c\to+\infty\), \(I\to0\) holds uniformly with respect to the other variables.
2 A continuation lemma
It is well known that the continuation theorem plays a key role in studying the existence of periodic solutions of ordinary differential equations. Now we shall introduce a continuation lemma for (1.1). To this end, we consider the equivalent system of (1.1),
Now, we embed system (2.1) into a family of equations with one parameter \(\lambda\in[0, 1]\),
Lemma 2.1
Assume that conditions (h i ) (\(i=1,2,3,4\)) hold. Suppose that there exists a constant \(\zeta\geq d\) (d is given in (1.3)) such that, if \((x(t), y(t))\) is a 2π-periodic solution of system (2.2) for some \(\lambda\in(0, 1)\), then
Then system (2.1) has at least one 2π-periodic solution.
We shall use a classical consequence of Mawhin’s continuation theorem [16], Theorem 7.2 to prove Lemma 2.1. For the reader’s convenience, we restate it here.
Lemma 2.2
Let \(\Psi=\Psi(t, z; \lambda):[0, 2\pi]\times\mathbf{R}^{m}\times[0, 1]\to\mathbf{R}^{m}\) be a continuous function and let \(\Omega\subset\mathbf{R}^{m}\) be a (non-empty) open bounded set (with boundary ∂Ω and closure Ω̄). Assume the following conditions:
-
(1)
for any 2π-periodic solution \(z(t)\) of \(z'=\lambda\Psi(t, z; \lambda)\) with \(\lambda\in(0, 1)\), such that \(z(t)\in\bar{\Omega}\), for all \(t\in[0, 2\pi]\), it follows that \(z(t)\in\Omega\), for all \(t\in[0, 2\pi]\);
-
(2)
\(\Psi_{0}(z)\neq0\), for each \(z\in\partial\Omega\) and \(d_{B}(\Psi_{0}, \Omega, 0)\neq0\), where
$$\Psi_{0}(z)=\frac{1}{2\pi}\int_{0}^{2\pi} \Psi(t, z; 0)\,dt,\quad \textit{for } z\in\mathbf{R}^{m}. $$
Then the equation \(z'=\Psi(t, z; 1)\) has at least one 2π-periodic solution and \(z(t)\in\bar{\Omega}\), for all \(t\in[0, 2\pi]\).
Proof of Lemma 2.1
We shall use Lemma 2.2 to prove this continuation lemma. Set
Then there exists \(\tilde{t}\in[0, 2\pi]\) such that
From condition (h1) we know that there exists a constant \(0< d_{0}< d\) such that
Therefore, we have
Meanwhile, we have
We claim that there exist constants \(0<\varepsilon<d_{0}\) and \(c>0\) such that, if \((x(t), y(t))\) is a 2π periodic solution of (2.2) with \(x(t)\leq\zeta\), \(t\in[0, 2\pi]\), then
Integrating the second equality of (2.2) on \([0, 2\pi]\) and applying the first equality of (2.2), we get
Then we obtain
where \(I_{1}=\{t\in[0, 2\pi]: 0< x(t)< d_{0}\}\), \(I_{2}=\{t\in[0, 2\pi]: d_{0}\leq x(t)\leq\zeta\}\). Hence, we have
where \(M_{1}=2\pi\cdot\max\{|g(x)|:d_{0}\leq x\leq\zeta\}\).
Let us take a fixed constant δ satisfying \(0<(1+\frac{\pi}{\sqrt{3}})\delta<1\). From (h4) we see that there exists \(R_{\delta}>0\) such that, for any \(|s|\geq R_{\delta}\) and \(t\in[0, 2\pi]\),
Set
Then we see that, for any \((t, s)\in\mathbf{R}^{2}\),
Set \(M=2M_{1}+2\pi M_{2}+\|p\|_{1}\). Then we obtain
From (2.2) we know that \(x(t)\) satisfies the equation as follows:
Multiplying (2.6) by \(x(t)-\bar{x}\) with \(\bar{x} =\frac{1}{2\pi}\int_{0}^{2\pi}x(s)\,ds\), and integrating the equality on \([0, 2\pi]\), we get from (2.5) and (2.6)
Using the Wirtinger inequality and the Sobolev inequality, we have
Then we get
which means that
where \(\gamma=\delta(1+\frac{\pi}{\sqrt{3}})\), \(c_{0}=M_{2}\sqrt{2\pi}+\sqrt{\frac{\pi}{6}}(M+\|p\|_{1})\). Since \(0<\gamma<1\), we have
Integrating the first equation of (2.2) on \([0, 2\pi]\) and noticing \(\lambda\in(0, 1]\), we get
which implies that there exists \(t_{0}\in[0, 2\pi]\) such that \(y(t_{0}) = 0\). Then we get from (2.4), (2.5), and (2.7)
Therefore,
Let \(x(t_{*})\) (\(t_{*}\in[0, 2\pi]\)) be the minimum of \(x(t)\). Then we have \(x'(t_{*})=0\) and \(x''(t_{*})\geq0\). Since \(x(t_{*})\) satisfies
we have
Hence,
which implies
Let \(x(t^{*})\) (\(t^{*}\in[0, 2\pi]\)) be the maximum of \(x(t)\). Then we have \(x'(t^{*})=0\) and \(x''(t^{*})\leq0\). Similarly, we can obtain
From (2.9) and (2.10) we see that there exists \(\bar{t}\in[0, 2\pi]\) such that
In what follows, we shall prove that there exists \(0<\varepsilon<d_{0}\) such that, for any 2π periodic solution \((x(t), y(t))\) of (2.2) with \(x(t)\leq\zeta\), \(t\in[0, 2\pi]\),
The right inequality \(x(t)<\zeta\) (\(t\in[0, 2\pi]\)) follows directly from the condition \(\max\{x(t):t\in[0, 2\pi]\}\neq\zeta\) and \(x(t)\leq\zeta\), \(t\in[0, 2\pi]\). Next, we prove the left inequality. Otherwise, there exist a sequence \(\{\lambda_{n}\}\) with \(\lambda_{n}\in(0, 1]\) and a sequence of 2π periodic solutions of (2.2) \(\{(x_{n}(t), y_{n}(t))\}\) (with \(\lambda=\lambda_{n}\) in (2.2)), satisfying \(x_{n}(t)\leq\zeta\), \(t\in[0, 2\pi]\), and
Without loss of generality, we assume that, for every n,
Set \(\varepsilon_{n}=x_{n}(t_{n})=\min_{t\in[0, 2\pi]}x_{n}(t)\), \(t_{n}\in [0, 2\pi]\). From (2.11) and (2.12) we see that there exists \(\alpha_{n}\in(t_{n}, t_{n}+2\pi)\) such that
Since \((x_{n}(t), y_{n}(t))\) satisfies the equation
we have
Recalling \(x_{n}'(t)=\lambda_{n} y_{n}(t)\), we get
Integrating both sides of (2.13) over the interval \([t_{n}, \alpha_{n}]\) and using the fact \(x_{n}'(t_{n})=\lambda_{n} y_{n}(t_{n})=0\), we obtain
Therefore, we get
From (h2) we have
Next, we shall estimate the right hand side of (2.14). First, it follows from (2.8) that we have
Meanwhile, according to (2.4) and (2.7), we get
Obviously, we have
Hence, the right hand side of (2.14) is bounded. This conclusion contradicts (2.15).
To use Lemma 2.2, we define an open bounded set \(\Omega=\{(x, y): \varepsilon< x<\zeta, -c-1<y<c+1\}\), and a map \(S: (0, +\infty)\times \mathbf{R}\to\mathbf{R}^{2}\), \(S(x, y)=(y, -g(x)-\bar{f}+\bar{p})\). Then, for any 2π-periodic solution \((x(t), y(t))\) of system (2.2), such that \((x(t), y(t))\in\bar{\Omega}\), for all \(t\in[0, 2\pi]\), we have \((x(t), y(t))\in\Omega\), for all \(t\in[0, 2\pi]\). Therefore, the first condition of Lemma 2.2 is satisfied. Obviously, S does not vanish outside the rectangle Ω. Furthermore, the Brouwer degree of S, \(d_{B}(S, \Omega, 0)\), is defined and \(d_{B}(S, \Omega, 0)=d_{B}(g, (\varepsilon, \zeta), \bar{p}-\bar{f})=1\) because g is continuous and \(g(\varepsilon)<\bar{p}-\bar{f}\), \(g(\zeta)>\bar{p}-\bar{f}\). According to Lemma 2.2, system (2.1) has at least one 2π periodic solution. □
Lemma 2.3
[14]
Assume that \(g: \mathbf{R}\to \mathbf{R}\) is continuous and \(\lim_{|x|\to+\infty}\operatorname{sgn}(x)g(x)=+\infty\). Then, for any constant \(\nu\in\mathbf{R}\),
where
with \(\tilde{G}(x)=\int_{0}^{x}g(s)\,ds\).
Remark 2.4
When \(g: [0, +\infty)\to\mathbf{R}\) is continuous and satisfies \(\lim_{x\to+\infty}g(x)=+\infty\), we can also define \(\tau_{g}(c)\) and \(\tau_{g}(\nu, c)\) for \(c>0\) large enough. In this case, we know from Lemma 2.3 that, for any constant ν,
When \(g: (0, +\infty)\to\mathbf{R}\) is continuous and \(\lim_{x\to+\infty}g(x)=+\infty\), we can get a similar estimate. Under this condition, it is noted that g may have a singularity at the origin, \(x=0\), namely, \(\lim_{x\to0^{+}}g(x)=-\infty\). For any constant \(\nu\in\mathbf{R}\) and sufficiently large \(c\geq1\), let us set
where \(G(x)=\int_{1}^{x}g(s)\,ds\). Then we have
where τ is defined by (1.5).
In fact, let us consider a function \(g_{0}: [0, +\infty)\to\mathbf{R}\), \(g_{0}(x)=g(x+1)\), \(x\geq0\). Obviously, \(g_{0}\) is continuous on the interval \([0, +\infty)\) and satisfies \(\lim_{x\to+\infty}g_{0}(x)=+\infty\). Then we have, for \(x\geq0\),
According to Lemma 2.3, we get
When \(c>0\) is large enough, we have
Similarly, we have
Consequently, we get
Therefore, the conclusion (2.16) holds.
3 Proof of Theorem 1.1
In this section, we shall use the continuation Lemma 2.1 given in Section 2 to prove Theorem 1.1.
Proof of Theorem 1.1
Let us set
Then there exist \(0<\varepsilon_{0}<\frac{1}{3}(\tau-2\pi)\) and a sequence \(\{c_{n}\}\) with \(\lim_{n\to\infty}c_{n}=+\infty\) such that, for every n,
We shall prove that the condition of Lemma 2.1 is satisfied for \(\zeta=c_{n}\) with n sufficiently large.
Let \((x(t), y(t))\) be any 2π periodic solution of (2.2) for some \(\lambda\in(0, 1]\) and suppose that, for n large enough,
where d is given in (1.3). Assume that \(x(t_{*})\) (\(t_{*}\in[0, 2\pi]\)) is a local minimum of \(x(t)\). From the proof of Lemma 2.1
Then there exists an interval \([\alpha, \beta]\subset[0, 2\pi]\) containing \(t^{*}\), with \(\alpha=\alpha(x, \lambda)\), \(\beta=\beta(x, \lambda)\) such that
and
From (2.2) we have
Integrating both sides of (3.1) on the interval \([t, t^{*}]\) with \(\alpha\leq t\leq t^{*}\), we have
From (h4) we know that, for any sufficiently small \(\varepsilon>0\), there is a constant \(M_{\varepsilon}>0\) such that, for any \((t, y)\in R^{2}\),
Since \(y(t)>0\), \(t\in[\alpha, t^{*}]\), it follows from (3.2) and (3.3) that, for \(t\in[\alpha, t^{*}]\),
where \(M_{\varepsilon}'=M_{\varepsilon}+\|p\|_{\infty}\). Let us set
Then we have
Hence,
Multiplying both sides of (3.4) by \(e^{2\varepsilon t}\) and integrating over the interval \([t, t^{*}]\) yields
Since \(\phi(t^{*})=0\), we have
From \(x'(t)=\lambda y(t)\geq0\), \(t\in[\alpha, t^{*}]\) we know that \(x(t)\) is increasing on the interval \([\alpha, t^{*}]\). Therefore, we get, for \(t\in[\alpha, t^{*}]\),
Furthermore,
Consequently, we can get, for \(t\in[\alpha, t^{*}]\),
where \(\kappa(\varepsilon)=4\pi\varepsilon e^{4\pi\varepsilon}\). Recalling \(x'(t)=\lambda y(t)\) and \(y(t)>0\) for \(t\in[\alpha, t^{*}]\), we have
Hence,
Integrating both sides of (3.5) over interval \([\alpha, t^{*}]\) yields
Similarly, we can get
Therefore, we obtain
Using (h3) we can easily derive that, for \(n\to\infty\),
Then we have
It follows from Remark 2.4 that
Consequently, we have
Furthermore,
Since \(\lim_{\varepsilon\to0^{+}}\sqrt{1+\kappa(\varepsilon)}=1\), there exist a sufficiently small \(\varepsilon>0\) and a sufficiently large n such that, if \(\max_{[0, 2\pi]}x(t)=c_{n}\), then
which contradicts with the inequality \(\beta-\alpha<2\pi\). Then we find \(\zeta=c_{n}\) for n sufficiently large. Consequently, from the continuation Lemma 2.1, we know that (2.1) has at least one 2π periodic solution. □
Proof of Corollary 1.2
Let us denote \(\rho=\liminf_{x\to+\infty}\frac{2G(x)}{x^{2}}<\frac{1}{4}\). Then there exists \(\varepsilon>0\) such that \(\rho_{\varepsilon}=\rho+\varepsilon\in(\rho, \frac{1}{4})\). Define
Therefore, we have
It follows that there exists a sequence \(\{c_{n}\}\) with \(\lim_{n\to+\infty}c_{n}=+\infty\) such that
Consequently,
Hence, we have
As a result, we get
which implies that \(\limsup_{c\to+\infty}\tau(c)>2\pi\). According to Theorem 1.1, (1.1) has at least one 2π periodic solution. □
Remark 3.1
In [12], the existence of periodic solutions of the Hamiltonian systems of the type
was studied. A similar result was obtained (see [12], Corollary 3.13) for system (3.6). However, this corollary cannot be applied directly to obtain the main results of this paper because the asymptotic behavior of the primitive G of the nonlinearity g is treated in present paper.
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Acknowledgements
The authors are grateful to the referees for many valuable suggestions to make the paper more readable. Research supported by National Nature Science Foundation of China, No. 11501381 and the Grant of Beijing Education Committee Key Project, No. KZ201310028031.
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Authors’ contributions
ZW proved a continuation lemma for Rayleigh equations. TM participated in obtaining a prior estimate and helped to draft the manuscript. All authors read and approved the final manuscript.
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Wang, Z., Ma, T. Periodic solutions of Rayleigh equations with singularities. Bound Value Probl 2015, 154 (2015). https://doi.org/10.1186/s13661-015-0427-0
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DOI: https://doi.org/10.1186/s13661-015-0427-0