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On positive periodic solutions of second order singular equations
Boundary Value Problems volume 2018, Article number: 114 (2018)
Abstract
Using the fixed point theorem, we study the existence and multiplicity of positive periodic solutions for the second order differential equations
For given nonnegative constants \(0<\beta_{1}<\beta_{2}<\cdots<\beta_{N}\), the function f may be singular at \(x=\beta_{i}\).
Introduction
We say that the differential equation
is singular if the nonlinear term f is singular, which means that a vectorvalued function f is only defined on \({\mathbb {R}}\times {\mathbb {R}}^{N}\setminus\Omega\) with \(\Omega\subset {\mathbb {R}}^{N}\) being nonempty and for any \(x_{0}\in\Omega\),
During the last two decades, the existence of nontrivial periodic solutions of (1.1) has been studied by many researchers in nonsingular case as well as singular case. Usually, the proof is based on either the method of upper and lower solutions [2, 10, 15], fixed point theorems [6, 7, 17–19], alternative principle of Leray–Schauder [4, 11], or topological degree theory [23, 24]. To our attention, Wang in [22] studied the existence, multiplicity, and nonexistence results for positive solutions of the singular periodic boundary value problem (1.1) in terms of different values of \(\lambda\in {\mathbb {R}}\). But most of these papers are concerned with singularity at \(x=x_{0}\). Later, a similar idea is used to study the singular periodic systems with two parameters,
where \((\lambda,\mu)\in {\mathbb {R}}^{2}_{+}\setminus\{0,0\}\). One nice result proved in [20] is that there exist three nonempty subsets of \({\mathbb {R}}^{2}_{+}\setminus\{0,0\}:\Gamma,\Delta_{1},\Delta_{2}\) such that \({\mathbb {R}}^{2}_{+}\setminus\{0,0\}=\Gamma\cup\Delta_{1}\cup\Delta_{2}\) and (1.2) has at least two positive periodic solutions for \((\lambda,\mu)\in\Delta_{1}\), one positive periodic solution for \((\lambda,\mu)\in\Gamma\), and no positive periodic solutions for \((\lambda,\mu)\in\Delta_{2}\). Note that in this paper the word “singularity” is understood in a more general way because we only need that \(f_{1}\) is singular at the whole line xaxis and \(f_{2}\) is singular at the whole line yaxis. The proof is based on the following vector version of Krasnosel’skii’s fixed point theorem [14]. In addition, there are many papers concerned with the existence and multiplicity of radial solutions of the elliptic equations with the regular or singular nonlinearities. We refer the readers to [1, 3, 5, 8, 12, 13, 16, 21].
Motivated by these recent developments, in this paper, we investigate the existence and multiplicity of Tperiodic solutions of the following problem:
where the function f may be singular at \(x=\beta_{i}\) (\(i=1,\ldots,N\)). For convenience, we give the following notations:
For \(\nu= 0\) or \(\nu= +\infty\), there exist nonnegative constants \(f^{\nu}\) defined as
The main results can be expressed as follows.
Theorem 1.1
Assume that the function f satisfies the following conditions:

(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 \((\beta_{i},\beta_{i+1})\), \(i=1,\ldots,N1\),
$$\begin{gathered} \mathrm{(i)}\quad f(x)a^{*}\beta_{i}\geq0, \quad\textit{for }x\in (\beta_{i},\beta_{i+1}), \\ \mathrm{(ii)}\quad\textit{ there exists }\bar{R_{i}}\in(\beta _{i},\beta_{i+1})\textit{ such that} \\ \hphantom{\mathrm{(ii)}\quad\ }\max_{x\in[\theta\bar{R_{i}}+(1\theta)\beta_{i}, \bar {R_{i}}]}f(x)< \frac{\bar{R_{i}}\beta_{i}}{MT}+a_{*}\beta_{i}. \end{gathered} $$
Then (1.3) has at least \((2N2)\) positive periodic solutions.
Theorem 1.2
Assume that (A) and (B) hold. In addition, the function f satisfies the following condition:

(C)
There exists \(\bar{R}>\beta_{N}\) such that
$$\begin{gathered} \mathrm{(i)}\quad f(x)a^{*}\beta_{N}\geq0, \quad\textit{for }x\in (\beta_{N},\bar{R}), \\ \mathrm{(ii)}\quad\max_{x\in[\theta\bar{R}+(1\theta)\beta_{N}, \bar{R}]}f(x)< \frac{\bar{R}\beta_{N}}{MT}+a_{*} \beta_{n}. \end{gathered} $$
Then (1.3) has at least \((2N1)\) positive periodic solutions.
Theorem 1.3
Assume that (A) and (B) hold. In addition, the function f satisfies the following condition:

(D)
There exists \(\bar{r}\in(0,\beta_{1})\) such that \(f^{*}(\bar{r})<\frac{\bar{r}}{MT}\).
Then (1.3) has at least \((2N1)\) positive periodic solutions.
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
Assume that (A) and (B) hold. In addition, the function f satisfies the following condition:

(E)
There exists \(\hat{R}>\frac{\beta_{N}}{\theta}\) such that
$$\begin{gathered} \mathrm{(i)}\quad f(x)a^{*}\beta_{N}\geq0, \quad \textit{for }x\in(\beta_{N},\hat {R}), \\ \mathrm{(ii)}\quad\max_{x\in[\theta\hat{R}, \hat{R}]}f(x)< \frac {\bar{R}}{MT}+\min\biggl\{ a_{*}\frac{1}{MT},0\biggr\} \beta_{N}. \end{gathered} $$
If \(f^{\infty}=+\infty\), then (1.3) has at least \((2N)\) positive periodic solutions.
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.
Preliminary
The function a satisfies the following assumption:

(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.,
$$\begin{gathered} G(t,s)>0 \quad\mbox{for all } (t,s)\in[0,T] \times[0,T], \\ m=\min_{0\leq s,t\leq T}G(t,s)>0, \qquad M=\max_{0\leq s,t\leq T}G(t,s), \qquad\theta =m/M\in(0,1). \end{gathered} $$
From (H), it is clear that a function \(x(t)\) is a Tperiodic solution of (1.3) if and only if
Define the operator \(A:K\rightarrow E\) by
Let E denote the Banach space \(C[0,T]\) with the usual maxnorm and define a subcone K by
For \(r>0\), let
Therefore, the solution of (1.3) is equivalent to the fixed point of the operator A. The discussion is based on the following wellknown fixed point theorem.
Lemma 2.1
([9, Theorem 2.3.4])
Let E be a Banach space and \(K\subset E\) be a cone in E. Assume that \(\Omega_{1}\), \(\Omega_{2}\) are open subsets of E with \(0\in\Omega_{1}\), \(\overline{\Omega}_{1}\subset\Omega_{2}\), and let \(T:K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\rightarrow K\) be a completely continuous operator such that either

(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 \(K\cap(\overline{\Omega}_{2}\setminus\Omega_{1})\).
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
Since \(f(\cdot)\) is singular at \(x=\beta_{1}^{}\), there exists sufficiently small \(\delta<(1\theta)\beta_{1}\) such that \(f(x)\geq\rho x\), for \(0<x\beta_{1}< \delta_{1}\), where ρ satisfies \(\rho m \Omega_{2}\theta\geq1\), \(\Omega_{2}\) is given as follows. Choose \(R\in(\beta_{1}\delta,\beta_{1})\). Then, for \(x\in \partial K_{R}\), one has \(x(t)\geq\theta\Vert x \Vert\geq\theta R\). If \(\theta R\geq\beta _{1}\delta\), then \(\frac{\beta_{1}\delta}{\theta}\leq R\leq\beta_{1}\). Further, we have \(\delta>(1\theta)\beta_{1}\), which yields a contradiction. Now we only consider the case \(\theta R<\beta_{1}\delta\). Let \([0,T]=\Omega_{1}\cup\Omega_{2}\), where
Since \(\Vert x \Vert=R\) and x is continuous, \(\Omega_{2}\) is nonempty and \(\Omega_{2}>0\). Therefore, we have
For any \(x\in\partial K_{\bar{r}}\), we have
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
Since \(f^{0}=0\), there exists sufficiently small \(\bar{r}>0\) such that \(f(x)\leq\epsilon x\) for \(x\in[0,\bar{r}]\), where \(\epsilon M T<1\). Then, for any \(x\in\partial K_{\bar{r}}\), we have
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 \(x\in \partial K_{R}\)
Since \(f^{0}=+\infty\), there exists sufficiently small \(r<\bar{r}\) such that \(f(x)\geq\varrho x\) for \(x\in[0,r]\), where \(\varrho m\theta T>1\). Then, for any \(x\in\partial K_{r}\), we have
For any \(x\in\partial K_{\bar{r}}\), we have
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
Let \(\omega=u\beta_{N}\). Then (1.3) is equivalent to the problem
From (A) and (C), it follows that
 \((a_{1})\) :

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

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 that
$$F(\omega)>L \Vert \omega \Vert ,\quad \mbox{for }\omega\in (0,\overline{r}), $$where \(LmT>1\).
Define the operator \(A:K\rightarrow E\) by
Then, for any \(\omega\in\partial K_{\widetilde{r}}\), from (ii) of (C), we have
Further, we have
For \(\omega\in \partial K_{\bar{r}}\), we have
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
Assume that (H) and (A) hold. In addition,
 \((f_{1})\) :

\(f(\cdot)\) is continuous on \((\beta_{N},+\infty)\) and \(f(x)\geq a^{*}\beta_{N}\) on \((\beta_{N},+\infty)\).
Proof
If \((f_{1})\) holds, then \(F(\omega)\geq\) and (i) of (C) hold. From \((f_{1})\) and \(f^{\infty}=0\) it follows that \(\lim_{\omega\rightarrow+\infty}\frac{F(\omega)}{\omega}=0\). Then there exists sufficiently large \(R>0\) such that \(F(\omega)\leq\epsilon\omega\) for \(\omega\geq R\), where \(\epsilon M T<1\). Choose \(\tilde{r}>\max\{{\frac{R}{\theta},\bar{r}}\}\), where r̄ is given in \((a_{3})\). Then, for any \(\omega\in \partial K_{\tilde{r}}\), one has \(\omega(t)\geq\theta\Vert\omega \Vert>R\), and further we have
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 \(\Vert x_{1}(t) \Vert<\bar{R}\).
For any \(x(t)\in\partial K_{\hat{R}} \), we get
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 \(\mu m T \theta>1\).
Choose \(R=\frac{\widetilde{R}}{\theta}\). Then, for \(x\in \partial K_{R}\), one has \(x(t)\geq\theta\Vert x \Vert\geq\widetilde{R}\), and
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
Let \(\omega=u\beta_{i}\). Then (1.3) is equivalent to the problem
From (A) and (B), it follows that
 \((F_{1})\) :

\(F_{i}(\cdot)\) 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. □
Proof of Theorem 1.2
From Lemmas 3.4 and 3.7 the result follows. □
Proof of Theorem 1.3
From Lemmas 3.1 and 3.7 the result follows. □
Proof of Corollary 1.4
From Lemmas 3.2 and 3.7 the result follows. □
Proof of Theorem 1.5
From Lemmas 3.3 and 3.7 the result follows. □
Proof of Theorem 1.6
From Lemmas 3.1, 3.4, and 3.7 the result follows. □
Proof of Theorem 1.7
. From Lemmas 3.6 and 3.7 the result follows. □
Proof of Theorem 1.8
From Lemmas 3.3, 3.4, and 3.7 the result follows. □
Proof of Theorem 1.9
From Lemmas 3.1, 3.6, and 3.7 the result follows. □
Proof of Theorem 1.10
From Lemmas 3.3, 3.6, and 3.7 the result follows. □
Now we just give an example to illustrate Theorem 1.10.
Example
For convenience, we consider the following periodic boundary value problem:
where \(a(t)=\mu\) is a constant such that (H) holds, and \(f(x)\) is expressed by
where

(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.
It is obvious that (A) holds. First we show that assumption (B) holds.
If \(C_{ij}\) (\(i=1,2\), \(j=1,2\)) is sufficiently small, there exists \(\tilde{R_{i}}\in(\beta_{i},\beta_{i+1})\) such that the above inequality holds.
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.
Third, it is clear that \(f(x)a^{*}\beta_{N}\geq 0\) for \(x\in(\beta_{N},\hat{R})\). In addition, if \(C_{3j}\) (\(j=1,2\)) is sufficiently small, there exists \(\hat{R}>\frac{\beta_{N}}{\theta}\) such that
namely (E) holds.
Finally, we can verify that
Therefore, by Theorem 1.10, (3.3) has at least \((2N+2)\) positive periodic solutions.
Conclusion
Some sufficient conditions are given to illustrate the existence and multiplicity of positive periodic solutions for the second order differential equations
For given nonnegative constants \(0<\beta_{1}<\beta_{2}<\cdots<\beta_{N}\), the function f may be singular at \(x=\beta_{i}\).
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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.
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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).
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Wang, Y., Ru, Y. On positive periodic solutions of second order singular equations. Bound Value Probl 2018, 114 (2018). https://doi.org/10.1186/s1366101810365
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Keywords
 Periodic solutions
 Second order differential equations
 Singular
 Fixed point theorem