Abstract
Based on a variational approach, we prove that a second-order singular damped differential equation has at least one periodic solution when some reasonable assumptions are satisfied.
Boundary Value Problems volume 2015, Article number: 5 (2015)
Based on a variational approach, we prove that a second-order singular damped differential equation has at least one periodic solution when some reasonable assumptions are satisfied.
The purpose of this paper is to study the existence of T-periodic solutions for second-order singular damped differential equation
where \(q, g\in C(\mathbb{R}/{T\mathbb{Z},\mathbb{R}})\) with \(\int_{0}^{T} q(t)\,dt=0\), and the nonlinearity \(f\in C((0,\infty),\mathbb{R})\) admits a repulsive singularity at \(u=0\), which means that
Second-order singular differential equations have attracted many researchers’ attention because of the wide applications in applied sciences. For example, they can describe the dynamics of particles under the action of Newtonian-type forces caused by compressed gases [1]. If \(q(t)\equiv0\), then Eq. (1.1) reduces to the following singular differential equation:
The existence of periodic solutions for Eq. (1.2) has attracted the attention of many researchers, and some classical tools have been used in the literature, including the method of upper and lower solutions [2, 3], degree theory [4], some fixed point theorems in cones for completely continuous operators [5, 6], Schauder’s fixed point theorem [7], a nonlinear Leray-Schauder alternative principle [7, 8] and variational methods [9–11].
Recently, Eq. (1.1) has also been investigated by several authors; see, for instance, [12, 13] (application of Leray-Schauder alternative principle) and [14] (using Schauder’s fixed point theorem). In general cases, it is very difficult or impossible to apply variational methods to Eq. (1.1) when \(\int_{0}^{T}q(t)\,dt>0\). In this paper, we consider the case \(\int_{0}^{T}q(t)\,dt=0\) and under some reasonable assumptions, we establish the corresponding variational framework of T-periodic solutions for Eq. (1.1) on an appropriate Sobolev space and give a new criterion to guarantee the existence of at least one nontrivial T-periodic solution of Eq. (1.1) using a variant of the mountain pass theorem. We refer the reader to [15–17] for the details about variational methods.
In order to state our main result, we need the following assumptions:
\(q, g\in C(\mathbb{R}/{T\mathbb{Z}})\) with \(\int_{0}^{T} q(t)\,dt=0\);
\(f\in C((0,\infty),\mathbb{R})\) has a repulsive singularity at \(u=0\), i.e.,
\(\lim_{u\rightarrow0^{+}}F(u)=+\infty\), where \(F(u)= \int_{1}^{u} f(s)\,ds\);
\(M=\sup\{f(s): 0< s<+\infty\}\) is bounded;
\(\lim_{u\rightarrow+\infty} (F(u)- \bar{g} u) =+\infty\), where \(\bar{g}\) is defined by
Assume that (H1)-(H5) are satisfied. Then Eq. (1.1) has at least one nontrivial T-periodic solution.
The existence of T-periodic solutions for the following singular damped differential equation
was discussed in [12–14] by using the Leray-Schauder alternative principle or Schauder’s fixed point theorem. However, all of them required that the Green function associated to the linear equation problem
is positive for all \((t,s)\in[0,T]\times[0,T]\). For example, in [13] and [14] it is supposed that
In [12], two criteria to make the Green function positive were given. In particular, one criterion was proved when \(\int_{0}^{T} q(t)\,dt=0\) and
Note that in Theorem 1.1, conditions (1.4) and (1.5) do not hold because \(p(t)\equiv0\) and \(\int_{0}^{T} q(t)\,dt=0\) in our case.
From (H1), it is obvious that
where \(Q(t)= \int_{0}^{t} q(s)\,ds\). In addition, it is easy to find the functions \(f(u)\) and \(g(t)\) which satisfy assumptions (H2)-(H5). For example, if we take
where \(e>0\) and \(\gamma\geq1\) are constants and choose \(g\in C(\mathbb{R}/{T\mathbb{Z}},\mathbb{R})\) such that
then (H2)-(H5) are satisfied.
If we take \(q(t)\equiv0\) in (1.1) and \(e=1\) in (1.7), then (1.1) reduces to the following repulsive-type equation:
It was proved in [18] that Eq. (1.9) (with \(\gamma\geq1\)) has a positive T-periodic solution if and only if (1.8) holds. One open problem is whether we can obtain the sufficient and necessary conditions to guarantee the existence of positive T-periodic solutions for the following special form of Eq. (1.1) with \(\gamma\geq1\):
The remaining part of this paper is organized as follows. Some preliminaries are presented in Section 2. In Section 3, the proof of Theorem 1.1 is given.
In this section, we present some auxiliary results, which will be used in the proof of our main result. First, we define the truncation function \(f_{\lambda}: \mathbb{R}\rightarrow \mathbb{R}\), \(0<\lambda<1\), by
Note that condition (H2) implies that \(f_{\lambda}\) is continuous with respect to \(u\in \mathbb{R}\).
In what follows, for \(\lambda\in(0,1)\), we consider the following modified equation:
Let
Then the problem of the existence of T-periodic solutions for Eq. (2.1) has a variational structure with corresponding functional \(\Phi_{\lambda}\) given by
and defined on the Hilbert space
equipped with the norm
for \(u\in H_{T}^{1}\).
[15, Proposition 1.3] (Wirtinger’s inequality)
If \(u\in H_{T}^{1}\) and \(\int_{0}^{T} u(t)\,dt=0\), then
Under the conditions of Theorem 1.1, similar to [19, Theorems 2.1 and 2.2], it is easy to verify that \(\Phi _{\lambda}\) is continuously differentiable, weakly lower semicontinuous on \(H_{T}^{1}\) and
Moreover, critical points of \(\Phi_{\lambda}\) on \(H_{T}^{1}\) are T-periodic solutions of Eq. (2.1).
In order to obtain the existence of T-periodic solutions of Eq. (2.1), the following version of the mountain pass theorem will be used in our argument.
[15, Theorem 4.10]
Let X be a Banach space, and let \(\varphi\in C^{1}(X,\mathbb{R})\). Assume that there exist \(x_{0}, x_{1}\in X\) and a bounded open neighborhood Ω of \(x_{0}\) such that \(x_{1} \in X\backslash\overline{\Omega}\) and
Let
and
If φ satisfies the (PS)-condition (that is, a sequence \(\{u_{n}\}\) in X satisfying \(\varphi(u_{n})\) is bounded and \(\varphi'(u_{n})\rightarrow0\) as \(n\rightarrow+\infty\) has a convergent subsequence), then c is a critical value of φ and \(c>\max\{\varphi(x_{0}),\varphi (x_{1})\}\).
In this section, we give the proof of Theorem 1.1.
The proof will be divided into four steps.
Step 1. \(\Phi_{\lambda}\) satisfies the (PS)-condition.
Let \(\{u_{n}\}_{n\in \mathbb{N}}\) be a sequence in \(H_{T}^{1}\) such that \(\{\Phi _{\lambda}'(u_{n})\}_{n\in \mathbb{N}}\) is bounded and \(\Phi_{\lambda }'(u_{n})\rightarrow0\) as \(n\rightarrow+\infty\). Then there exist a constant \(c_{1}>0\) and a sequence \(\{\epsilon_{n}\} _{n\in \mathbb{N}}\subset \mathbb{R}^{+}\) with \(\epsilon_{n}\rightarrow0\) as \(n\rightarrow +\infty\) such that, for all n,
and for every \(v\in H_{T}^{1}\),
Using a standard argument, it is sufficient to show that \(\{u_{n}\}_{n\in \mathbb{N}}\) is bounded in \(H_{T}^{1}\), and this will be enough to derive the (PS)-condition.
Taking \(v(t)\equiv-1\) in (3.2), we obtain that
So that
Let
and
It follows from (3.3) that
where M is defined in (H4). Hence, there exists \(c_{3}>0\) such that
On the other hand, if we take, in (3.2), \(v(t)\equiv w_{n}(t):=u_{n}(t)-\bar{u}_{n}\), where \(\bar{u}_{n}\) is the average of \(u_{n}\) over the interval \([0,T]\), we get (taking into account (3.4))
Using the Poincaré-Wirtinger inequality for zero mean functions in the Sobolev space \(H_{T}^{1}\), we know that there exists \(c_{6}>0\) such that
Now suppose that
Since (3.5) holds, we have, passing to a subsequence if necessary, that either
(i) Assume that the second possibility occurs. We have
Thus, using the Sobolev and Poincaré inequalities to \(M_{n}-u_{n}(\cdot )\), we have, from (3.5),
In view of (3.1), we see that
is bounded, which contradicts (H5).
(ii) Assume that the first possibility occurs, i.e., \(m_{n}\rightarrow-\infty\) as \(n\rightarrow+\infty\). We replace \(M_{n}\) by \(-m_{n}\) in the preceding arguments, and we also get a contradiction.
Therefore, \(\Phi_{\lambda}\) satisfies the (PS) condition. This completes the proof of the claim.
Step 2. In what follows, let
and
We show that there exists \(m>0\) such that \(\inf_{u\in \partial\Omega}\Phi_{\lambda}(u)\geq-m\) whenever \(\lambda\in(0,1)\).
For any \(u\in\partial\Omega\), we have \(\min u=u(t_{u})=1\) for some \(t_{u}\). By (2.2), we obtain that
The Hölder inequality and the fact that \(u'(t)=(u(\cdot)-1)'(t)\) imply that
Applying the Poincaré inequality to \(u(\cdot)-1\), we get
The above inequality shows that
Since \(\min u=1\), we have that \(\|u(\cdot)-1\|_{H_{T}^{1}}\rightarrow +\infty\) is equivalent to \(\|u'\|_{L^{2}}\rightarrow+\infty\). Hence
which yields that \(\Phi_{\lambda}\) is coercive. Thus it has a minimizing sequence. The weak lower semicontinuity of \(\Phi_{\lambda}\) implies that
It follows that there exists \(m>0\) such that \(\inf_{u\in \partial\Omega}\Phi_{\lambda}(u)\geq-m\) for all \(\lambda\in(0,1)\).
Step 3. We show that there exists \(\lambda_{0}\in(0,1)\) with the property that, for every \(\lambda\in(0,\lambda_{0})\), any solution u of Eq. (2.1) satisfying \(\Phi_{\lambda}(u)\geq-m\) is such that \(\min u\geq\lambda_{0}\), and hence u is a solution of Eq. (1.1).
On the contrary, assume that there are sequences \(\{\lambda_{n}\}_{n\in \mathbb{N}}\) and \(\{u_{n}\}_{n\in \mathbb{N}}\) such that
\(\lambda_{n}\leq\frac{1}{n}\);
\(u_{n}\) is a solution of Eq. (2.1) with \(\lambda =\lambda_{n}\);
\(\Phi_{\lambda_{n}}(u_{n})\geq-m\);
\(\min u_{n}<\frac{1}{n}\).
Since
we have
On the other hand, since \(u_{n}(0)=u_{n}(T)\), there exists \(\tau_{n} \in (0,T)\) such that
Therefore, we obtain that
which, from (3.6), yields that
Since \(\Phi_{\lambda_{n}}(u_{n})>-m\), it follows that there exist two constants \(R_{1}\) and \(R_{2}\) with \(0< R_{1}<R_{2}\) such that
If not, \(u_{n}\) would tend uniformly to 0 or +∞. In both cases, in view of (H3), (H5) and (3.7), we have
which contradicts the fact that \(\Phi_{\lambda_{n}}(u_{n})\geq-m\).
Let \(\tau_{n}^{1}\), \(\tau_{n}^{2}\) be such that, for n large enough,
Multiplying Eq. (2.1) by \(u_{n}'\) and integrating the resulting equation on \([\tau_{n}^{1},\tau_{n}^{2}]\) (or \([\tau_{n}^{2},\tau_{n}^{1}]\)), we get
It is clear that
where
Since q is bounded, g is integrable and \(\|u_{n}'\|_{L^{\infty}}\leq c_{8}\) (see (3.7)), it follows that J is bounded, and consequently, \(J_{1}\) is bounded. On the other hand, we have
which yields that
However, due to (H3), it follows that \(J_{1}\) is unbounded. This is a contradiction.
Step 4. We prove that \(\Phi_{\lambda}\) has a mountain pass geometry for \(\lambda\leq\lambda_{0}\).
Fix \(\lambda\in(0,\lambda_{0}]\) such that \(f(\lambda)<0\). It is possible because of (H2). Therefore, we have
This implies that
Hence
By (H3), choose \(\lambda\in(0,\lambda_{0}]\) such that
It follows from (3.8) that \(\Phi_{\lambda}(0)<-m\).
Also, using (H5), we can find R large enough such that \(R>1\) and
which implies that
Since Ω is a neighborhood of R, \(0\notin\Omega\) and
Step 1 and Step 4 imply that \(\Phi_{\lambda}\) has a critical point \(u_{\lambda}\) such that
where \(\Gamma=\{\eta\in C([0,1],H_{T}^{1}):\eta(0)=0, \eta(1)=R\}\).
Since \(\inf_{u\in\partial\Omega}\Phi_{\lambda }(u)\geq -m\), it follows from Step 3 that \(u_{\lambda}\) is a solution of Eq. (1.1). Now the proof is finished. □
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The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the correctness and presentation of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 11101304).
The authors declare that they have no competing interests.
All the authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Li, J., Li, S. & Zhang, Z. Periodic solutions for a singular damped differential equation. Bound Value Probl 2015, 5 (2015). https://doi.org/10.1186/s13661-014-0269-1
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DOI: https://doi.org/10.1186/s13661-014-0269-1