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Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type
- Shiping Lu^{1}Email author and
- Xingchen Yu^{1}
- Received: 8 February 2018
- Accepted: 19 June 2018
- Published: 28 June 2018
Abstract
Keywords
- Boundary value problem
- Periodic solution
- Singularity
- Upper and lower functions
1 Introduction
Theorem 1.1
Remark 1.1
In this paper, a function \(u:[0,T]\rightarrow (0,+\infty )\) is said to be a positive solution to problem (1.7)–(1.8) if \(u:[0,T]\rightarrow R_{+}\) is absolutely continuous together with its first derivative on \([0,T]\) and satisfies (1.8) together with (1.7) almost everywhere on \([0,T]\). From this definition, r, \(\varphi : R\rightarrow R\) being T-periodic, we can easily find that if \(u_{0}: [0,T]\rightarrow R\) is a positive solution to boundary value problem (1.7)–(1.8), then \(\tilde{u}: R\rightarrow R\), which is a T-periodic extension of \(u_{0}(t)\), is a positive T-periodic solution to (1.7). Thus, the existence of positive periodic solutions to (1.7) is equivalent to the existence of positive solutions to boundary value problem (1.7)–(1.8).
For convenience, in the end of this introduction, we give some notations used throughout the paper:
\(R_{+}=(0,+\infty )\), \(R_{+}^{0}=[0,+\infty )\), \([x]_{+}= \max \{x,0 \}\), \([x]_{-}=\max \{-x,0\}\); \(AC^{1}([0,T];R)\) is the set of functions \(u:[0,T]\rightarrow R\) such that u and \(u'\) are absolutely continuous; \(\overline{p}=\frac{1}{T}\int_{0}^{T}\vert p(s) \vert \,ds\) for \(p\in L([0,T];R)\).
2 Preliminary lemmas
The method of lower and upper functions is one of the most widely used methods in nonlinear analysis. Its main idea goes back at least to Picard. Many mathematical researchers have obtained rich results by using this method. For a complete historical review of the method, we refer to the monograph [32]. Now, we give the definitions of upper and lower functions.
Definition 2.1
Definition 2.2
The following proposition can be found in [32] (or a more general case in [33]).
Proposition 2.1
We further show some auxiliary results obtained by Hakl, Torres and Zamora [16].
Lemma 2.1
([16])
Lemma 2.2
([16])
Lemma 2.3
Proof
3 Main results
3.1 Construction of lower function
Theorem 3.1
Let \(\varphi ,r\in L([0,T],R)\) be such that \(\operatorname{ess} \inf \{r(t):t \in [0,T]\}>0\). Then there exists a lower function α to problem (1.7)–(1.8) satisfying \(0<\alpha (t)<1\).
Proof
Theorem 3.2
Let φ, \(r\in L([0,T],R)\), \(r(t)\ge 0\) a.e. \(t\in [0,T]\) and \(\bar{r}>0\). Suppose \(\frac{T}{4} \int_{0}^{T}[\varphi (s)]_{-}(s)\,ds<1\). Then there exists a lower function α to problem (1.7)–(1.8) such that \(0<\alpha (t)<1\) for \(t\in [0,T]\).
Proof
By Lemma 2.2 there exists a solution u to (2.2)–(2.3) such that (2.6) and (2.7) hold.
3.2 Construction of upper function
Theorem 3.3
Proof
Theorem 3.4
Proof
3.3 Existence theorems
Theorem 3.5
Proof
By Theorems 3.1 and 3.3 there exist a lower function \(0<\alpha (t)<1\) and an upper function \(\beta (t)>1\). Therefore, the result can be obtained directly from Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.1
Theorem 3.6
Proof
By Theorems 3.1 and 3.4 there exist a lower function \(0<\alpha (t)<1\) and an upper function \(\beta (t)>1\). Therefore, the result is a direct consequence of Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.2
Theorem 3.7
- (1)
\(r(t)\ge 0\) for a.e. \(t\in [0,T]\), \(\bar{r}>0\), and \(\operatorname{ess} \sup \{r(t)\}<+\infty \);
- (2)
\(\frac{T}{4}\int_{0}^{T}[\varphi (s)]_{-}\,ds<1\);
- (3)
\(\frac{T}{4}\int_{0}^{T}[\varphi (t)]_{+}\,ds\int_{0}^{T}[\varphi (t)]_{-}\,ds \leq \int_{0}^{T}[\varphi (t)]_{-}\,ds-\int_{0}^{T}[\varphi (t)]_{+}\,ds\).
Proof
By Theorems 3.2 and 3.3 there exist a lower function \(0<\alpha (t)<1\) and an upper function \(\beta (t)>1\). Therefore, the result is a direct consequence of Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.3
Theorem 3.8
- (1)
\(r(t)\ge 0\) for a.e. \(t\in [0,T]\);
- (2)
\(\frac{T}{4}\int_{0}^{T}[\varphi (s)]_{-}(s)\,ds<1\);
- (3)
\(\frac{T}{4}\int_{0}^{T}[\varphi (s)]_{+}\,ds\int_{0}^{T}[\varphi (s)]_{-}\,ds< \int_{0}^{T}[\varphi (s)]_{-}\,ds-\int_{0}^{T}[\varphi (s)]_{+}\,ds\).
Proof
By Theorems 3.2 and 3.4 there exist a lower function \(0<\alpha (t)<1\) and an upper function \(\beta (t)>1\). Therefore, the result follows immediately from Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.4
Remark 3.1
Obviously, the conclusions associated with Examples 3.1–3.4 can be obtained neither by using the results of [7] nor by using the results of [16] (see Theorem 1.1), since the singularity term \(\frac{r(t)}{x^{\mu }}\) is nonautonomous. Furthermore, even if \(f(x)\equiv 0\) for \(x\in (0,+\infty )\), the above conclusion associated with Example 3.4 cannot be deduced from the main theorem of [17] (Theorem 1 of [17]). This is due to the fact that the condition of \(\delta \in [0,1)\) is required in [17].
4 Conclusions
In this paper, we study the periodic problem for Liénard equations with a singularity of attractive type in the case of \(r(t)\ge 0\) for a.e. \(t\in [0,T]\). The proofs of main results are based on the method of upper and lower functions. It is interesting that the singularity term \(\frac{r(t)}{x^{\mu }}\) in (1.7) is nonautonomous, which generalizes the corresponding results in the known literature where \(r(t)\) is a constant function. In the next research, we will continue to study the periodic problem to the singular equation like (1.7) where \(r(t)\) is a changing-sign function.
Declarations
Acknowledgements
The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
Availability of data and materials
Not applicable.
Funding
The authors are gratefully acknowledge support from NSF of China (No. 11271197).
Authors’ contributions
Both authors contributed equally to this work. Both authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that there is no conflict of interest regarding the publication of this manuscript. The authors declare that they have no competing interests.
Consent for publication
Not applicable.
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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