- Research
- Open Access

# Multiplicity of positive solutions to second-order singular differential equations with a parameter

- Shengjun Li
^{1, 2}Email author, - Fang-fang Liao
^{3}and - Hailong Zhu
^{4}

**2014**:115

https://doi.org/10.1186/1687-2770-2014-115

© Li et al.; licensee Springer. 2014

**Received:**15 January 2014**Accepted:**28 April 2014**Published:**14 May 2014

## Abstract

We study the existence and multiplicity of positive periodic solutions for second-order nonlinear damped differential equations by combing the analysis of positiveness of the Green function for a linear damped equation. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on the Guo-Krasnosel’skii fixed point theorem on compression and expansion of cones.

**MSC:**34B16, 34C25.

## Keywords

- positive solutions
- singular
- Guo-Krasnosel’skii fixed point theorem

## 1 Introduction

*T*-periodic solutions for the following second-order singular differential equation:

where *p* is a strictly positive absolutely continuous function. Such equations, even in the case $p\equiv 1$, where they are referred to as being of Schrödinger or Klein-Gordon type, appear in many scientific areas including quantum field theory, gas dynamics, fluid mechanics and chemistry.

Electrostatic or gravitational forces are the most important examples of singular interactions. During the last few decades, the study of the existence of positive solutions for singular differential equations has deserved the attention of many researchers [1–7]. Some strong force conditions introduced by Gordon [8] are standard in the related earlier works [6, 7, 9]. Compared with the case of strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity is more recent, but has also attracted many researchers [2, 3, 10, 11]. In particular, the degree theory [6, 7], the method of upper and lower solutions [11, 12], Schauder’s fixed point theorem [2, 10], some fixed point theorems in cones for completely continuous operators [13–16] and a nonlinear Leray-Schauder alternative principle [17–19] are the most relevant tools.

However, singular differential equation (1.1), in which the nonlinearity is dependent on the derivative and does not require *f* to be nonnegative, has not attracted much attention in the literature. There are not so many existence results for (1.1) even when the nonlinearity is independent of the derivative. In this paper, we try to fill this gap and establish the existence of positive *T*-periodic solutions of (1.1) using the Guo-Krasnosel’skii fixed point theorem on compression and expansion of cones, which has been used to study positive solutions for systems of ordinary, functional differential equations [14–16]. We remark that it is sufficient to prove that $T:K\cap ({\tilde{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\to K$ is continuous and completely continuous in Lemma 3.2 (Section 3). This point is essential and advantageous.

and prove its positiveness. The fact is very crucial to our arguments. Moreover, the one-signed property of the Green function implies that a maximum principle and an anti-maximum principle hold for the corresponding linear differential equations subject to various boundary conditions, which is an important topic in differential equations (see [20, 21]). In Section 3, by employing the Guo-Krasnosel’skii fixed point theorem, we prove the existence of twin positive solutions for (1.1) under the positiveness of the Green function associated with (1.2)-(1.3). To illustrate the new results, some applications are also given.

## 2 The Green function and its positiveness

*T*-periodic solution is the trivial one. When (1.2)-(1.3) is nonresonant, as a consequence of Fredholm’s alternative, equation (2.1) admits a unique

*T*-periodic solution which can be written as

- (H)
$a(t)$, $b(t)$ are continuous functions and $b(t)>0$ for all $t\in [0,T]$.

**Lemma 2.1**

*Let*$K(s,\tau ):[0,T]\times [0,T]\to [0,+\mathrm{\infty})$

*be a continuous function*.

*Then*,

*for any nonnegative continuous function*$\phi (t)$

*defined on*$[0,T]$,

*the integral equation*

*has a unique solution*$x(t)$,

*which is continuous on*$[0,T]$

*and satisfies the following inequality*:

*Proof*We solve equation (2.2) by the method of successive approximations. Let

which implies that the series ${\sum}_{n=0}^{\mathrm{\infty}}{x}_{n}(t)$ converges uniformly with respect to $t\in [0,T]$. Obviously, $x(t)={\sum}_{n=0}^{\mathrm{\infty}}{x}_{n}(t)$ is a continuous solution of (2.2). Moreover, inequality (2.3) holds because $\phi (t)$ and $K(s,\tau )$ are nonnegative functions.

Next we prove the uniqueness. To do so, we first show that the solution of (2.2) is unique on $[{t}_{1},{t}_{2}]\subset [0,T]$ with ${t}_{2}<\sqrt{2/{\beta}_{1}}$. Then the uniqueness of the solution on $[0,T]$ is direct using the continuation property.

Hence it follows that ${x}_{1}(t)={x}_{2}(t)$ for all $t\in [{t}_{1},{t}_{2}]$. □

**Lemma 2.2**$u(t)$

*and*$v(t)$

*satisfy the following integral equations*:

*Proof*Since $u(t)$ is a solution of (1.2), we have

*t*and noticing ${u}^{\prime}(0)=0$, we obtain

To obtain (2.4), we only need to integrate the above equality from 0 to *t* and notice $u(0)=1$. In a similar way, we can prove (2.5). □

**Lemma 2.3**

*For the solution*$x(t)$

*of boundary value problem*(2.1)-(1.3),

*the formula*

*holds*,

*where*

*is the Green function*, *the number* *D* *is defined by* $D=u(T)+{v}^{\prime}(T)-2$.

*Proof*It is easy to see that the general solution of equation (2.1) has the form

After not very complicated calculations, we can get (2.7) and (2.8). □

**Remark 2.4**As a direct application of Lemma 2.3, if $a(t)=0$, $b(t)={k}^{2}>0$, then the Green function $G(t,s)$ of boundary value problem (2.1)-(1.3) has the form

**Lemma 2.5** *Assume that* (H) *holds*. *Then the Green function* $G(t,s)$ *associated with* (2.1)-(1.3) *is positive for all* $s,t\in [0,T]$.

*Proof*Since $G(t,s)=G(s,t)$, it is enough to prove that $G(t,s)>0$ for $0\le s\le t<T$. Recall that $u(t)$ and $v(t)$ satisfy integral equations (2.4) and (2.5). By condition (H) and Lemma 2.1, it follows that

Using Lemma 2.1, we get from (2.12) that ${E}_{1}(t,s)>0$ for all $t\in (s,T]$.

Again using Lemma 2.1, we get from (2.13) that ${E}_{2}(t,s)>0$ if $(t,s)\ne (T,0)$, and the proof is completed. □

## 3 Main results

In this section, we state and prove the new existence results for (1.1). The proof is based on the following well-known fixed point theorem on compression and expansion of cones, which we state here for the convenience of the reader, after introducing the definition of a cone.

**Definition 3.1**Let

*X*be a Banach space and let

*K*be a closed, nonempty subset of

*X*.

*K*is a cone if

- (i)
$\alpha u+\beta v\in K$ for all $u,v\in K$ and all $\alpha ,\beta >0$,

- (ii)
$u,-u\in K$ implies $u=0$.

We also recall that a compact operator means an operator which transforms every bounded set into a relatively compact set. Let us define the function $\omega (t)=\lambda {\int}_{0}^{T}G(t,s)\phantom{\rule{0.2em}{0ex}}ds$ and use ${\parallel \cdot \parallel}_{1}$ to denote the usual ${L}^{1}$-norm over $(0,T)$, by $\parallel \cdot \parallel $ we denote the supremum norm of $\mathbb{C}[0,T]$.

**Lemma 3.2** [22]

*Let*

*X*

*be a Banach space and*

*K*(⊂

*X*)

*be a cone*.

*Assume that*${\mathrm{\Omega}}_{1}$, ${\mathrm{\Omega}}_{2}$

*are open subsets of*

*X*

*with*$0\in {\mathrm{\Omega}}_{1}$, ${\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2}$,

*and let*

*be a completely continuous operator such that either*

- (i)
$\parallel \mathcal{A}u\parallel \ge \parallel u\parallel $, $u\in K\cap \partial {\mathrm{\Omega}}_{1}$

*and*$\parallel \mathcal{A}u\parallel \le \parallel u\parallel $, $u\in K\cap \partial {\mathrm{\Omega}}_{2}$;*or* - (ii)
$\parallel \mathcal{A}u\parallel \le \parallel u\parallel $, $u\in K\cap \partial {\mathrm{\Omega}}_{1}$

*and*$\parallel \mathcal{A}u\parallel \ge \parallel u\parallel $, $u\in K\cap \partial {\mathrm{\Omega}}_{2}$.

*Then* $\mathcal{A}$ *has a fixed point in* $K\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})$.

*Let*$X=\mathbb{C}[0,T]$

*and define*

*where* *σ* *is as in* (2.14).

*One may readily verify that*

*K*

*is a cone in*

*X*.

*Now*,

*suppose that*$F:[0,T]\times \mathbb{R}\times \mathbb{R}\to [0,\mathrm{\infty})$

*is a continuous function*.

*Define an operator*

*for* $x\in X$ *and* $t\in [0,T]$.

**Lemma 3.3** $\mathcal{A}:X\to K$ *is well defined*.

*Proof*Let $x\in X$, then we have

This implies that $\mathcal{A}(X)\subset K$ and the proof is completed. □

It is easy to prove.

**Lemma 3.4** $\mathcal{A}$ *is continuous and completely continuous*.

Now we present our main result.

**Theorem 3.5** *Suppose that* (1.1) *satisfies* (H). *Furthermore*, *assume that*

_{1}) $f:[0,T]\times {\mathbb{R}}^{+}\times \mathbb{R}\to \mathbb{R}$

*is continuous and there exists a constant*$M>0$

*such that*

(H_{2}) ${lim}_{x\to {0}^{+}}f(t,x,y)=+\mathrm{\infty}$ *and* ${lim}_{x\to +\mathrm{\infty}}f(t,x,y)/x=+\mathrm{\infty}$ *uniformly* $(t,y)\in {\mathbb{R}}^{2}$.

*Then* (1.1) *has at least two positive* *T*-*periodic solutions for sufficiently small* *λ*.

*Proof*To show that (1.1) has a positive solution, we should only show that

*x*satisfying (1.3) and $x(t)>M\omega (t)$ for $t\in [0,T]$. If it is right, then $\varphi (t)=x(t)-M\omega (t)$ is a solution of (1.1) since

where $-{\omega}^{\u2033}(t)+a(t){\omega}^{\prime}(t)+b(t)\omega (t)=\lambda $ is used.

For $r>0$, let ${\mathrm{\Omega}}_{r}=\{x\in K:\parallel x\parallel <r\}$ and note that $\partial {\mathrm{\Omega}}_{r}=\{x\in K:\parallel x\parallel =r\}$.

This implies $\parallel \mathcal{A}x\parallel \le \parallel x\parallel $.

So, equation (1.1) has a positive solution ${x}_{1}(t)={\tilde{x}}_{1}(t)-M\omega (t)\ge \sigma {r}_{1}-\frac{\sigma MT}{2}\ge \frac{\sigma MT}{2}$.

where *γ* satisfies $\lambda \gamma \sigma AT>1$.

we can conclude that ${x}_{1}$ and ${x}_{2}$ are the desired distinct positive periodic solutions of (1.1) for $\lambda <min\{{\lambda}^{\ast},{\lambda}^{\ast \ast}\}$. □

**Example**Let the nonlinearity in (1.1) be

where $\alpha >0$, $\beta >1$, $\gamma \ge 0$, $c(t),d(t),e(x)\in \mathbb{C}[0,T]$. It is clear that $f(t,x,y)$ satisfies conditions (H_{1}), (H_{2}). Then (1.1) has at least two positive *T*-periodic solutions for sufficiently small *λ*.

## Declarations

### Acknowledgements

The authors express their thanks to the referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 11161017, No. 11301001, No. 11301139), Hainan Natural Science Foundation (Grant No.113001), Excellent Youth Scholars Foundation and the Natural Science Foundation of Anhui Province of PR China (No. 2013SQRL030ZD).

## Authors’ Affiliations

## References

- Bravo JL, Torres PJ: Periodic solutions of a singular equation with indefinite weight.
*Adv. Nonlinear Stud.*2010, 10: 927-938.MathSciNetGoogle Scholar - Chu J, Torres PJ: Applications of Schauder’s fixed point theorem to singular differential equations.
*Bull. Lond. Math. Soc.*2007, 39: 653-660. 10.1112/blms/bdm040MathSciNetView ArticleGoogle Scholar - Chu J, Lin X, Jiang D, O’Regan D, Agarwal RP: Multiplicity of positive solutions to second order differential equations.
*Bull. Aust. Math. Soc.*2006, 73: 175-182. 10.1017/S0004972700038764MathSciNetView ArticleGoogle Scholar - Lazer AC, Solimini S: On periodic solutions of nonlinear differential equations with singularities.
*Proc. Am. Math. Soc.*1987, 99: 109-114. 10.1090/S0002-9939-1987-0866438-7MathSciNetView ArticleGoogle Scholar - Wang F, Zhang F, Ya Y: Existence of positive solutions of Neumann boundary value problem via a convex functional compression-expansion fixed point theorem.
*Fixed Point Theory*2010, 11: 395-400.MathSciNetGoogle Scholar - Yan P, Zhang M: Higher order nonresonance for differential equations with singularities.
*Math. Methods Appl. Sci.*2003, 26: 1067-1074. 10.1002/mma.413MathSciNetView ArticleGoogle Scholar - Zhang M: Periodic solutions of equations of Ermakov-Pinney type.
*Adv. Nonlinear Stud.*2006, 6: 57-67.MathSciNetGoogle Scholar - Gordon WB: Conservative dynamical systems involving strong forces.
*Trans. Am. Math. Soc.*1975, 204: 113-135.View ArticleGoogle Scholar - del Pino MA, Manásevich RF: Infinitely many
*T*-periodic solutions for a problem arising in nonlinear elasticity.*J. Differ. Equ.*1993, 103: 260-277. 10.1006/jdeq.1993.1050View ArticleGoogle Scholar - Franco D, Torres PJ: Periodic solutions of singular systems without the strong force condition.
*Proc. Am. Math. Soc.*2008, 136: 1229-1236.MathSciNetView ArticleGoogle Scholar - Rachunková I, Tvrdý M, Vrkoc̆ I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems.
*J. Differ. Equ.*2001, 176: 445-469. 10.1006/jdeq.2000.3995View ArticleGoogle Scholar - Bonheure D, De Coster C: Forced singular oscillators and the method of lower and upper solutions.
*Topol. Methods Nonlinear Anal.*2003, 22: 297-317.MathSciNetGoogle Scholar - Cabada A, Cid JA, Infante G: New criteria for the existence of non-trivial fixed points in cones.
*Fixed Point Theory Appl.*2013., 2013: Article ID 125Google Scholar - Wang H: Positive periodic solutions of singular systems with a parameter.
*J. Differ. Equ.*2010, 249: 2986-3002. 10.1016/j.jde.2010.08.027View ArticleGoogle Scholar - Wang H: On the number of positive solutions of nonlinear systems.
*J. Math. Anal. Appl.*2003, 281: 287-306. 10.1016/S0022-247X(03)00100-8MathSciNetView ArticleGoogle Scholar - Wang F, An Y: Multiple positive doubly periodic solutions for a singular semipositone telegraph equation with a parameter.
*Bound. Value Probl.*2013., 2013: Article ID 7Google Scholar - Jiang D, Chu J, Zhang M: Multiplicity of positive periodic solutions to superlinear repulsive singular equations.
*J. Differ. Equ.*2005, 211: 282-302. 10.1016/j.jde.2004.10.031MathSciNetView ArticleGoogle Scholar - Li S, Liang L, Xiu Z: Positive solutions for nonlinear differential equations with periodic boundary condition.
*J. Appl. Math.*2012. 10.1155/2012/528719Google Scholar - Zhu H, Li S: Existence and multiplicity results for nonlinear differential equations depending on a parameter in semipositone case.
*Abstr. Appl. Anal.*2012. 10.1155/2012/215617Google Scholar - Cabada A, Cid JA, Tvrdy M: A generalized anti-maximum principle for the periodic one-dimensional
*p*-Laplacian with sign-changing potential.*Nonlinear Anal.*2010, 72: 3436-3446. 10.1016/j.na.2009.12.028MathSciNetView ArticleGoogle Scholar - Campos J, Mawhin J, Ortega R: Maximum principles around an eigenvalue with constant eigenfunctions.
*Commun. Contemp. Math.*2008, 10: 1243-1259. 10.1142/S021919970800323XMathSciNetView ArticleGoogle Scholar - Guo D, Lakshmikanantham V:
*Nonlinear Problems in Abstract Cones*. Academic Press, New York; 1988.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.