 Research
 Open Access
On positive periodic solutions of second order singular equations
 Yunhai Wang^{1} and
 Yuanfang Ru^{2}Email author
 Received: 3 April 2018
 Accepted: 11 July 2018
 Published: 20 July 2018
Abstract
Keywords
 Periodic solutions
 Second order differential equations
 Singular
 Fixed point theorem
1 Introduction
The main results can be expressed as follows.
Theorem 1.1
 (A)f may be singular at \(x=\beta_{i}\) (\(i=1,\ldots,N\)) and continuous on$$(0,\beta_{1},), (\beta_{1},\beta_{2}),\ldots, ( \beta_{N1},\beta _{N}),(\beta_{N},+\infty); $$
 (B)For any interval \((\beta_{i},\beta_{i+1})\), \(i=1,\ldots,N1\),$$\begin{gathered} \mathrm{(i)}\quad f(x)a^{*}\beta_{i}\geq0, \quad\textit{for }x\in (\beta_{i},\beta_{i+1}), \\ \mathrm{(ii)}\quad\textit{ there exists }\bar{R_{i}}\in(\beta _{i},\beta_{i+1})\textit{ such that} \\ \hphantom{\mathrm{(ii)}\quad\ }\max_{x\in[\theta\bar{R_{i}}+(1\theta)\beta_{i}, \bar {R_{i}}]}f(x)< \frac{\bar{R_{i}}\beta_{i}}{MT}+a_{*}\beta_{i}. \end{gathered} $$
Theorem 1.2
 (C)There exists \(\bar{R}>\beta_{N}\) such that$$\begin{gathered} \mathrm{(i)}\quad f(x)a^{*}\beta_{N}\geq0, \quad\textit{for }x\in (\beta_{N},\bar{R}), \\ \mathrm{(ii)}\quad\max_{x\in[\theta\bar{R}+(1\theta)\beta_{N}, \bar{R}]}f(x)< \frac{\bar{R}\beta_{N}}{MT}+a_{*} \beta_{n}. \end{gathered} $$
Theorem 1.3
 (D)
There exists \(\bar{r}\in(0,\beta_{1})\) such that \(f^{*}(\bar{r})<\frac{\bar{r}}{MT}\).
Corollary 1.4
Assume that (A) and (B) hold. If \(f^{0}=0\), then (1.3) has at least \((2N1)\) positive periodic solutions.
Theorem 1.5
Assume that (A), (B), and (D) hold. If \(f^{0}=+\infty\), then (1.3) has at least \((2N)\) positive periodic solutions.
Theorem 1.6
Assume that (A), (B), (C), and (D) hold. Then (1.3) has at least \((2N)\) positive periodic solutions.
Theorem 1.7
 (E)There exists \(\hat{R}>\frac{\beta_{N}}{\theta}\) such that$$\begin{gathered} \mathrm{(i)}\quad f(x)a^{*}\beta_{N}\geq0, \quad \textit{for }x\in(\beta_{N},\hat {R}), \\ \mathrm{(ii)}\quad\max_{x\in[\theta\hat{R}, \hat{R}]}f(x)< \frac {\bar{R}}{MT}+\min\biggl\{ a_{*}\frac{1}{MT},0\biggr\} \beta_{N}. \end{gathered} $$
Theorem 1.8
Assume that (A), (B), (C), and (D) hold. If \(f^{0}=+\infty\), then (1.3) has at least \((2N+1)\) positive periodic solutions.
Theorem 1.9
Assume that (A), (B), (D), and (E) hold. If \(f^{\infty}=+\infty\), then (1.3) has at least \((2N+1)\) positive periodic solutions.
Theorem 1.10
Assume that (A), (B), (D), and (E) hold. If \(f^{0}=+\infty\) and \(f^{\infty}=+\infty\), then (1.3) has at least \((2N+2)\) positive periodic solutions.
2 Preliminary
 (H)The function a is continuous, positive, Tperiodic and the linear equation \(\ddot{x}+a(t)x=0\) has a positive Green’s function \(G(t,s)\), i.e.,$$\begin{gathered} G(t,s)>0 \quad\mbox{for all } (t,s)\in[0,T] \times[0,T], \\ m=\min_{0\leq s,t\leq T}G(t,s)>0, \qquad M=\max_{0\leq s,t\leq T}G(t,s), \qquad\theta =m/M\in(0,1). \end{gathered} $$
Lemma 2.1
([9, Theorem 2.3.4])
 (i)
\(\Vert Tu \Vert\leq\Vert u \Vert\), \(u\in K\cap\partial\Omega_{1}\) and \(\Vert Tu \Vert\geq\Vert u \Vert\), \(u\in K\cap\partial\Omega_{2}\); or
 (ii)
\(\Vert Tu \Vert\geq\Vert u \Vert\), \(u\in K\cap\partial\Omega_{1}\) and \(\Vert Tu \Vert\leq\Vert u \Vert\), \(u\in K\cap\partial\Omega_{2}\).
Then T has a fixed point in \(K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).
3 Proof of the main results
First, we consider the existence of solutions of (1.3) on \((0,\beta_{1})\).
Lemma 3.1
Assume that (H), (A), and (D) hold. Then (1.3) has at least one positive periodic solution.
Proof
From [22], it is clear that \(A:\overline{K_{R}}\backslash K_{\bar{r}}\rightarrow K\), and A is completely continuous on \(\overline{K_{R}}\backslash K_{\bar{r}}\). Therefore, by Lemma 2.1, (1.3) has at least a positive solution \(x(t)\in\overline{K_{R}}\backslash K_{\bar{r}}\). □
Lemma 3.2
Assume that (H) and (A) hold. If \(f^{0}=0\), then (1.3) has at least one positive periodic solution.
Proof
The remainder is similar to the proof of Lemma 3.1, so we omit it. □
Lemma 3.3
Assume that (H), (A), and (D) hold. If \(f^{0}=+\infty\), then (1.3) has at least two positive periodic solutions.
Remark
\(f^{0}=+\infty\) implies that f may be singular at \(x=0\).
Proof
Since \(f(\cdot)\) is singular at \(x=\beta_{1}^{}\), from the proof of Lemma 3.1, there exists \(R\in (\beta_{1}\delta,\beta_{1})>\bar{r}\) such that \(\Vert Ax \Vert> \Vert x \Vert\) for \(x\in \partial K_{R}\)
Further, from (A) and the Arzela–Ascoli theorem it follows that now we verify that \(A:\overline{K_{R}}\backslash K_{r}\rightarrow K\), and A is completely continuous.
Therefore, by Lemma 2.1, (1.3) has at least two positive solutions \(x_{1}(t)\in\overline{K_{R}}\backslash K_{\bar{r}}\) and \(x_{2}(t)\in \overline{K_{\bar{r}}}\backslash K_{r}\). □
Second, we give some existence results of solutions of (1.3) on \((\beta_{N},+\infty)\).
Lemma 3.4
Assume that (H), (A), and (C) hold. Then (1.3) has at least one positive periodic solution.
Remark
Since \(f(\cdot)\) is singular at \(x=\beta_{N}^{+}\), there exists sufficiently small \(\delta<\frac{1\theta}{\theta}\) such that \(f(x)\geq\rho x\) for \(0< x\beta_{N}< \delta\). Choose \(r\in(\beta_{N}, \beta_{N}+\delta)\). Then, for \(x\in \partial K_{r}\), one has \(x(t)\geq\theta\Vert x \Vert\geq\theta r\). If \(\theta r\geq\beta _{n}\), then \(\frac{\beta_{N}}{\theta}\leq r\leq\beta_{N}+\delta\), which yields a contradiction. Therefore \(\theta r<\beta_{N}\), which implies that A is not continuous on \(\overline{K_{R}}\backslash K_{r}\), where \(R>\frac{\beta_{N}}{\theta}\).
Proof
 \((a_{1})\) :

\(\lim_{\omega\rightarrow0^{+}}F(\omega) =+\infty\);
 \((a_{2})\) :

let \(\widetilde{r}=\bar{R}\beta_{N}\), \(F(\omega)\geq0\), for \(\omega\in(0,\widetilde{r})\);
 \((a_{3})\) :

there exists sufficiently small \(\overline {r}<\widetilde{r}\) such thatwhere \(LmT>1\).$$F(\omega)>L \Vert \omega \Vert ,\quad \mbox{for }\omega\in (0,\overline{r}), $$
From \((f_{5})\) and the Arzela–Ascoli theorem it follows that T is completely continuous on \(\overline{K_{\widetilde{r}}}\backslash K_{\bar{r}}\). Therefore, by Lemma 2.1, (3.1) has at least a positive solution \(\omega(t)\in\overline{K_{\widetilde{r}}}\backslash K_{\bar{r}}\). Namely, (1.3) has at least a positive solution \(x(t)=\omega(t)+\beta_{N}\) satisfying \(\Vert x(t) \Vert<\beta_{N}+\widetilde{r}=\bar{R}\). □
Corollary 3.5
 \((f_{1})\) :

\(f(\cdot)\) is continuous on \((\beta_{N},+\infty)\) and \(f(x)\geq a^{*}\beta_{N}\) on \((\beta_{N},+\infty)\).
Proof
Combining Lemma 3.4 and Lemma 2.1, we have that (1.3) has at least one positive periodic solution. □
Lemma 3.6
Assume that (H), (A), and (E) hold. If \(f^{\infty}=+\infty\), then (1.3) has at least two positive periodic solutions.
Proof
It is easy to see that (E) implies (C). Then, from Lemma 3.3, we can obtain a solution \(x_{1}(t)\) satisfying \(\Vert x_{1}(t) \Vert<\bar{R}\).
By the definition of \(f^{\infty}=\infty\), there exists \(\widetilde{R}>\hat{R}\) such that \(f(x)\geq\mu x\) for \(x\geq \widetilde{R}\), where μ satisfies \(\mu m T \theta>1\).
From (A) and the Arzela–Ascoli theorem it follows that A is completely continuous on \(\overline{K_{R}}\backslash K_{\hat{R}}\). Therefore, by Lemma 2.1, (1.3) has another positive solution \(x_{2}(t)\in\overline{K_{R}}\backslash K_{\hat{R}}\). □
Finally, we are studying the existence of solutions of (1.3) on \((\beta_{i},\beta_{i+1})\), \(i=1,2\ldots,N1\).
Lemma 3.7
Assume that (H), (A), and (B) hold. Then (1.3) has at least two positive periodic solutions.
Proof
 \((F_{1})\) :

\(F_{i}(\cdot)\) is nonnegative and continuous on \((0,\beta_{i+1}\beta_{i})\);
 \((F_{2})\) :

\(F_{i}(\cdot)\) is singular at \(x=0^{+}\), \((\beta_{i+1}\beta_{i})^{}\);
 \((F_{3})\) :

Let \(\bar{r_{i}}=\tilde{ R_{i}}\beta_{i}\in (0,\beta_{i+1}\beta_{i})\) such that \(F_{i}^{*}(\bar{r_{i}})<\frac{\bar{r_{i}}}{MT}\).
Now we give the proof of the main results.
Proof of Theorem 1.1
The number of the interval \((\beta_{i},\beta_{i+1})\) is \(N1\), then we can obtain the result from Lemma 3.7. □
Now we just give an example to illustrate Theorem 1.10.
Example
 (I)
\(C_{ij}\) are positive constants; (\(i=0,1,2,3\), \(j=1,2\));
 (II)
\(a^{*}=a_{*}=\mu\);
 (III)
\(\nu_{01}<1\), \(\nu_{11}, \nu_{21}, \nu_{31}>0\), \(\nu_{32}>1\), \(\nu_{02}\), \(\nu_{12}\), \(\nu_{22}\) are even.
Second, if \(C_{0j}\) (\(j=1,2\)) is sufficiently small, there exists \(\bar{r}\in(0,\beta_{1})\) such that \(f^{*}(\bar{r})<\frac{\bar{r}}{MT}\), namely (D) holds.
Therefore, by Theorem 1.10, (3.3) has at least \((2N+2)\) positive periodic solutions.
4 Conclusion
Declarations
Acknowledgements
We would like to thank the referee for his/her valuable comments and helpful suggestions which have led to an improvement of the presentation of this paper.
Availability of data and materials
Not applicable.
Funding
The authors were supported by NNSF of China (No.11501165), the projects of Guizhou Provincial Science and Technology Fund (QKHJICHU [2017], Grant no. 1408, QKHLH [2014], Grant no. 7366).
Authors’ contributions
All the authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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