- Research
- Open Access
Periodic solutions for a kind of prescribed mean curvature Liénard equation with a singularity and a deviating argument
- Shiping Lu1 and
- Fanchao Kong2Email author
https://doi.org/10.1186/s13662-015-0474-y
© Kong and Lu; licensee Springer. 2015
- Received: 25 February 2015
- Accepted: 14 April 2015
- Published: 9 May 2015
Abstract
Keywords
- periodic solution
- continuation theorem
- prescribed mean curvature Liénard equation
- deviating argument
- singularity
1 Introduction
Theorem 1.1
- (h1):
-
(Balance condition) There exist constants \(0< D_{1}<D_{2}\) such that if x is a positive continuous T-periodic function satisfyingthen$$\int_{0}^{T} g\bigl(t,x(t)\bigr)\, dt=0, $$$$D_{1}\leq x(\tau)\leq D_{2}\quad \textit{for some } \tau \in[0,T]. $$
- (h2):
-
(Degree condition) \(\overline{g}(x)<0\) for all \(x\in(0,D_{1})\), and \(\overline{g}(x)>0\) for all \(x>D_{2}\).
- (h3):
-
(Decomposition condition) \(g(t,x)=g_{0}(x)+g_{1}(t,x)\), where \(g_{0}\in C((0,+\infty), {R})\) and \(g_{1}:[0,T]\times[0,+\infty)\rightarrow {R}\) is an \(L^{2}\)-Carathéodory function, i.e., \(g_{1}\) is measurable with respect to the first variable, continuous with respect to the second one, and for any \(b>0\) there is \(h_{b}\in L^{2}((0,T);[0,+\infty))\) such that \(|g_{1}(t,x)|\leq h_{b}(t)\) for a.e. \(t\in [0,T]\) and all \(x\in[0,b]\).
- (h4):
-
(Strong force condition at \(x=0\)) \(\int_{0}^{1} g_{0}(x)\, dx=-\infty\).
- (h5):
-
(Small force condition at \(x=\infty\))$$\bigl\Vert \varphi^{+}\bigr\Vert _{1}< \frac{\sqrt{3}}{T} \quad \bigl(\varphi^{+}=\max\bigl\{ \varphi ^{+},0\bigr\} \bigr). $$
Theorem 1.2
- (h1):
-
(Balance condition) There exist constants \(0< D_{1}<D_{2}\) such that if x is a positive continuous T-periodic function satisfyingthen$$\int_{0}^{T} g\bigl(t,x(t)\bigr)\, dt=0, $$$$D_{1}\leq x(\tau)\leq D_{2}\quad \textit{for some } \tau \in[0,T]. $$
- (h2):
-
(Degree condition) \(\overline{g}(x)<0\) for all \(x\in(0,D_{1})\), and \(\overline{g}(x)>0\) for all \(x>D_{2}\).
- (h3):
-
(Decomposition condition) \(g(t,x)=g_{0}(x)+g_{1}(t,x)\), where \(g_{0}\in C((0,+\infty), {R})\) and \(g_{1}:[0,T]\times[0,+\infty )\rightarrow {R}\) is an \(L^{2}\)-Carathéodory function, i.e., \(g_{1}\) is measurable with respect to the first variable, continuous with respect to the second one, and for any \(b>0\) there is \(h_{b}\in L^{2}((0,T);[0,+\infty))\) such that \(|g_{1}(t,x)|\leq h_{b}(t)\) for a.e. \(t\in[0,T]\) and all \(x\in[0,b]\).
- (h4):
-
(Strong force condition at \(x=0\)) \(\int_{0}^{1} g_{0}(x)\, dx=-\infty\).
- (\(\mathrm{h}'_{5}\)):
-
(Small force condition at \(x=\infty\))$$\| \varphi\|_{\infty}< \biggl(\frac{\sqrt{\pi}}{T}\biggr)^{2}. $$
However, to the best of our knowledge, the studying of periodic solutions for the prescribed mean curvature equation with a singularity is relatively infrequent, and the method of finding a priori bounds is different from the other prescribed mean curvature equations which have no singularities. So, it is worthwhile and interesting to explore this topic.
The structure of the rest of this paper is as follows. In Section 2, we state some necessary definitions and lemmas. In Section 3, we prove the main result. Finally, we give an example of an application in Section 4.
2 Preliminary
In order to use Mawhin’s continuation theorem, we first recall it.
- (a)
ImL is a closed subset of Y,
- (b)
\(\operatorname{dim} {\operatorname{Ker}L}=\operatorname{codim} {\operatorname{Im}L}<\infty\).
- (c)
\({K_{p}(I-Q)N}(\overline{\Omega})\) is a relative compact set of X,
- (d)
\({QN}(\overline{\Omega})\) is a bounded set of Y,
Lemma 2.1
[19]
- (1)
\({Lx\neq\lambda Nx}\), \(\forall(x,\lambda)\in\partial\Omega\times (0,1)\);
- (2)
\({QN}x\neq0\), \(\forall x\in \operatorname{Ker}L\cap\partial\Omega\);
- (3)
\(\operatorname{deg}\{{JQN},\Omega\cap{\operatorname{Ker}L},0\}\neq0\), where \({J: \operatorname{Im}Q}\rightarrow{\operatorname{Ker}L}\) is a homeomorphism,
- (H1):
-
There exist positive constants \(D_{1}\) and \(D_{2}\) with \(D_{1}< D_{2}\) such that
- (1)
for each positive continuous T-periodic function \(x(t)\) satisfying \(\int_{0}^{T} g( x(t))\, dt=0\), there exists a positive point \(\tau\in [0,T]\) such that \(D_{1}\leq x(\tau)\leq D_{2}\);
- (2)
\(g(x)<0\) for all \(x\in(0,D_{1})\) and \(g(x)>0\) for all \(x>D_{2}\).
- (1)
- (H2):
-
\(g(x(t))=g_{1}(x(t))+g_{0}(x(t))\), where \(g_{1}: (0,+\infty )\rightarrow {R}\) is a continuous function and
- (1)
there exist positive constants \(m_{0}\) and \(m_{1}\) such that \(g(x)\leq m_{0}x+m_{1}\) for all x in \((0,+\infty)\);
- (2)
\(\int_{0}^{1} g_{0}(x)\, dx=-\infty\).
- (1)
- (H3):
-
There exist positive constants γ, \(c_{0}\), \(c_{1} \) such that \(\gamma< f(x)\leq c_{0}|x|+c_{1}\) for all x in \((0,+\infty)\).
Throughout this paper, define \(A:=( \int_{0}^{T}|e(t)|^{2}\, dt)^{\frac {1}{2}}+\sup_{t\in[0,T]}|e(t)|<+\infty\).
3 Existence of periodic solutions
Theorem 3.1
Proof
Theorem 3.2
Assume that all the conditions in Theorem 3.1 hold, then Eq. (1.5) has at least one positive T-periodic solution.
Proof
In the following, we prove that the condition (3) of Lemma 2.1 is also satisfied.
Therefore, by applying Lemma 2.1, we can conclude that Eq. (1.5) has at least one positive T-periodic solution. □
4 Example
Declarations
Acknowledgements
The work was supported by the National Natural Science Foundation of China (Grant No. 11271197) and the Ministry of Education Key Project of Natural Science (207047).
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.
Authors’ Affiliations
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