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Periodic solutions for p-Laplacian Rayleigh equations with singularities
- Shipin Lu^{1}Email author,
- Tao Zhong^{1} and
- Lijuan Chen^{1}
- Received: 23 January 2016
- Accepted: 6 May 2016
- Published: 11 May 2016
Abstract
In this paper, the problem of existence of periodic solutions is studied for p-Laplacian Rayleigh equations with a singularity at \(x=0\). By using the topological degree theory, some new results are obtained.
Keywords
- Rayleigh equation
- topological degree
- singularity
- periodic solution
1 Introduction
2 Preliminary lemmas
The following two lemmas (Lemma 2.1 and Lemma 2.2) are all consequences of Theorem 3.1 in [19].
Lemma 2.1
- (1)For each \(\lambda\in(0,1]\), each possible positive T-periodic solution x to the equationsatisfies the inequalities \(\eta_{0}< x(t)<\eta_{1}\) and \(|x'(t)|< M_{2}\) for all \(t\in[0,T]\).$$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'+ \lambda f \bigl(u' \bigr)-\lambda g_{1}(u)+\lambda g_{2}(t,u)=\lambda h(t) $$
- (2)Each possible solution c to the equationsatisfies the inequality \(\eta_{0}< c<\eta_{1}\).$$ g_{1}(c)-g_{2}(c)+\bar{h}=0 $$
- (3)It holds$$ \bigl(g_{1}(\eta_{0})-g_{2}( \eta_{0})+\bar{h} \bigr) \bigl(g_{1}(\eta _{1})-g_{2}(\eta_{1})+\bar{h} \bigr)< 0. $$
Lemma 2.2
- (1)For each \(\lambda\in(0,1]\), each possible positive T-periodic solution x to the equationsatisfies the inequalities \(\eta_{0}< x(t)<\eta_{1}\) and \(|x'(t)|< M_{2}\) for all \(t\in[0,T]\).$$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'+ \lambda f \bigl(u' \bigr)+\lambda g_{1}(u)-\lambda g_{2}(u)=\lambda h(t) $$
- (2)Each possible solution c to the equationsatisfies the inequality \(\eta_{0}< c<\eta_{1}\).$$ g_{1}(c)-g_{2}(c)-\bar{h}=0 $$
- (3)It holds$$ \bigl(g_{1}(\eta_{0})-g_{2}( \eta_{0})-\bar{h} \bigr) \bigl(g_{1}(\eta _{1})-g_{2}(\eta_{1})-\bar{h} \bigr)< 0. $$
In order to study the existence of positive periodic solutions to equation (1.5) and equation (1.6), we list the following assumptions:
(H_{2}) \(g_{1}(x)\ge0\) for all \(x\in(0,+\infty)\), and \(\int _{0}^{1}g_{1}(s)\,ds=+\infty\);
(H_{3}) \(f(0)=0\);
Lemma 2.3
- (1)for each possible positive T-periodic solution \(u(t)\) of equation (2.3) there exists \(\tau\in[0,T]\) such that$$ m_{0}< u(\tau)< m_{1}; $$(2.5)
- (2)each possible solution c to the equationsatisfies the inequality \(m_{0}< c< m_{1}\);$$ g_{1}(c)-g_{2}(c)+\bar{h}=0 $$
- (3)
\(g_{1}(u)-g_{2}(u)+\bar{h}>0\) for all \(u\in(0,m_{0}]\), and \(g_{1}(u)-g_{2}(u)+\bar{h}<0\) for all \(u\in[m_{1},+\infty)\).
Proof
(2) Conclusion (2), as well as conclusion (3), follows directly from assumption (H_{1}). □
Similar to the proof of Lemma 2.3, we obtain the following result.
Lemma 2.4
- (1)each possible positive T-periodic solution \(u(t)\) to equation (2.4) satisfies$$ m_{0}< u(t)< m_{1} \quad\textit{for all } t\in[0,T]; $$
- (2)each possible solution c to the equationsatisfies the inequality \(m_{0}< c< m_{1}\);$$ g_{1}(c)-g_{2}(c)-\bar{h}=0 $$
- (3)
\(g_{1}(u)-g_{2}(u)-\bar{h}>0\) for all \(u\in(0,m_{0}]\), and \(g_{1}(u)-g_{2}(u)-\bar{h}<0\) for all \(u\in[m_{1},+\infty)\).
3 Main results
Theorem 3.1
Assume that assumptions (H_{1}), (H_{2}), and (H_{4}) hold, then equation (1.5) has at least one positive T-periodic solution.
Proof
Theorem 3.2
Assume that assumptions (H_{1}) and (H_{3}) hold, then equation (1.6) has at least one positive T-periodic solution.
Proof
Corollary 3.1
Assume that (3.20), (3.21), and assumptions (H_{2}) and (H_{4}) hold, then equation (1.5) has at least one positive T-periodic solution.
Corollary 3.2
Assume that (3.20), (3.21), and assumption (H_{3}) hold, then equation (1.6) has at least one positive T-periodic solution.
Example 3.1
Example 3.2
Corresponding to equation (1.6), \(f(u)=u^{4}\), \(g_{1}(u)=\frac {1}{u^{\frac{1}{2}}}-\frac{1}{u^{\frac{1}{4}}}\), \(g_{2}(u)=u^{3}\), and \(h(t)=\cos t\). It is easy to see that conditions (3.20) and (3.21), and assumption (H_{3}) are all satisfied. By using Corollary 3.2, we see that equation (3.23) has at least one positive 2π-periodic solution.
Declarations
Acknowledgements
The work is sponsored by the National Natural Science Foundation of China (11271197). The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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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