# Multiple positive doubly periodic solutions for a singular semipositone telegraph equation with a parameter

## Abstract

In this paper, we study the multiplicity of positive doubly periodic solutions for a singular semipositone telegraph equation. The proof is based on a well-known fixed point theorem in a cone.

MSC:34B15, 34B18.

## 1 Introduction

Recently, the existence and multiplicity of positive periodic solutions for a scalar singular equation or singular systems have been studied by using some fixed point theorems; see . In , the authors show that the method of lower and upper solutions is also one of common techniques to study the singular problem. In addition, the authors  use the continuation type existence principle to investigate the following singular periodic problem:

${\left({|{u}^{\prime }|}^{p-2}{u}^{\prime }\right)}^{\prime }+h\left(u\right){u}^{\prime }=g\left(u\right)+c\left(t\right).$

More recently, using a weak force condition, Wang  has built some existence results for the following periodic boundary value problem:

$\left\{\begin{array}{c}{u}_{tt}-{u}_{xx}+{c}_{1}{u}_{t}+{a}_{11}\left(t,x\right)u+{a}_{12}\left(t,x\right)v={f}_{1}\left(t,x,u,v\right)+{\chi }_{1}\left(t,x\right),\hfill \\ {v}_{tt}-{v}_{xx}+{c}_{2}{v}_{t}+{a}_{21}\left(t,x\right)u+{a}_{22}\left(t,x\right)v={f}_{2}\left(t,x,u,v\right)+{\chi }_{2}\left(t,x\right).\hfill \end{array}$

The proof is based on Schauder’s fixed point theorem. For other results concerning the existence and multiplicity of positive doubly periodic solutions for a single regular telegraph equation or regular telegraph system, see, for example, the papers  and the references therein. In these references, the nonlinearities are nonnegative.

On the other hand, the authors  study the semipositone telegraph system

$\left\{\begin{array}{c}{u}_{tt}-{u}_{xx}+{c}_{1}{u}_{t}+{a}_{1}\left(t,x\right)u={b}_{1}\left(t,x\right)f\left(t,x,u,v\right),\hfill \\ {v}_{tt}-{v}_{xx}+{c}_{2}{v}_{t}+{a}_{2}\left(t,x\right)v={b}_{2}\left(t,x\right)g\left(t,x,u,v\right),\hfill \end{array}$

where the nonlinearities f, g may change sign. In addition, there are many authors who have studied the semipositone equations; see [19, 20].

Inspired by the above references, we are concerned with the multiplicity of positive doubly periodic solutions for a general singular semipositone telegraph equation

$\left\{\begin{array}{c}{u}_{tt}-{u}_{xx}+c{u}_{t}+a\left(t,x\right)u=\lambda f\left(t,x,u\right),\hfill \\ u\left(t+2\pi ,x\right)=u\left(t,x+2\pi \right)=u\left(t,x\right),\hfill \end{array}$
(1)

where $c>0$ is a constant, $\lambda >0$ is a positive parameter, $a\left(t,x\right)\in C\left(R×R,R\right)$, $f\left(t,x,u\right)$ may change sign and is singular at $u=0$, namely,

$\underset{u\to {0}^{+}}{lim}f\left(t,x,u\right)=+\mathrm{\infty }.$

The main method used here is the following fixed-point theorem of a cone mapping.

Lemma 1.1 

Let E be a Banach space, and $K\subset E$ be a cone in E. Assume ${\mathrm{\Omega }}_{1}$, ${\mathrm{\Omega }}_{2}$ are open subsets of E with $0\in {\mathrm{\Omega }}_{1}$, ${\overline{\mathrm{\Omega }}}_{1}\subset {\mathrm{\Omega }}_{2}$, and let $T:K\cap \left({\overline{\mathrm{\Omega }}}_{2}\setminus {\mathrm{\Omega }}_{1}\right)\to K$ be a completely continuous operator such that either

1. (i)

$\parallel Tu\parallel \le \parallel u\parallel$, $u\in K\cap \partial {\mathrm{\Omega }}_{1}$ and $\parallel Tu\parallel \ge \parallel u\parallel$, $u\in K\cap \partial {\mathrm{\Omega }}_{2}$; or

2. (ii)

$\parallel Tu\parallel \ge \parallel u\parallel$, $u\in K\cap \partial {\mathrm{\Omega }}_{1}$ and $\parallel Tu\parallel \le \parallel u\parallel$, $u\in K\cap \partial {\mathrm{\Omega }}_{2}$.

Then T has a fixed point in $K\cap \left({\overline{\mathrm{\Omega }}}_{2}\setminus {\mathrm{\Omega }}_{1}\right)$.

The paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, we give the main result.

## 2 Preliminaries

Let ${\mathrm{\top }}^{2}$ be the torus defined as

${\mathrm{\top }}^{2}=\left(R/2\pi Z\right)×\left(R/2\pi Z\right).$

Doubly 2π-periodic functions will be identified to be functions defined on ${\mathrm{\top }}^{2}$. We use the notations

${L}^{p}\left({\mathrm{\top }}^{2}\right),\phantom{\rule{2em}{0ex}}C\left({\mathrm{\top }}^{2}\right),\phantom{\rule{2em}{0ex}}{C}^{\alpha }\left({\mathrm{\top }}^{2}\right),\phantom{\rule{2em}{0ex}}D\left({\mathrm{\top }}^{2}\right)={C}^{\mathrm{\infty }}\left({\mathrm{\top }}^{2}\right),\dots$

to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space ${D}^{\prime }\left({\mathrm{\top }}^{2}\right)$ denotes the space of distributions on ${\mathrm{\top }}^{2}$.

By a doubly periodic solution of Eq. (1) we mean that a $u\in {L}^{1}\left({\mathrm{\top }}^{2}\right)$ satisfies Eq. (1) in the distribution sense, i.e.,

${\int }_{{\mathrm{\top }}_{2}}u\left({\phi }_{tt}-{\phi }_{xx}-c{\phi }_{t}+a\left(t,x\right)\phi \right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx=\lambda {\int }_{{\mathrm{\top }}^{2}}f\left(t,x,u\right)\phi \phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx.$

First, we consider the linear equation

(2)

where $c>0$, $\mu \in R$, and $h\left(t,x\right)\in {L}^{1}\left({\mathrm{\top }}^{2}\right)$.

Let ${\text{£}}_{\xi }$ be the differential operator

${\text{£}}_{\xi }u={u}_{tt}-{u}_{xx}+c{u}_{t}-\xi u,$

acting on functions on ${\mathrm{\top }}^{2}$. Following the discussion in , we know that if $\xi <0$, ${\text{£}}_{\xi }$ has the resolvent ${R}_{\xi }$,

${R}_{\xi }:{L}^{1}\left({\mathrm{\top }}^{2}\right)\to C\left({\mathrm{\top }}^{2}\right),\phantom{\rule{2em}{0ex}}{h}_{i}\left(t,x\right)↦{u}_{i}\left(t,x\right),$

where $u\left(t,x\right)$ is the unique solution of Eq. (2), and the restriction of ${R}_{\xi }$ on ${L}^{p}\left({\mathrm{\top }}^{2}\right)$ ($1) or $C\left({\mathrm{\top }}^{2}\right)$ is compact. In particular, ${R}_{\xi }:C\left({\mathrm{\top }}^{2}\right)\to C\left({\mathrm{\top }}^{2}\right)$ is a completely continuous operator.

For $\xi =-{c}^{2}/4$, the Green function $G\left(t,x\right)$ of the differential operator ${\text{£}}_{\xi }$ is explicitly expressed; see Lemma 5.2 in . From the definition of $G\left(t,x\right)$, we have For convenience, we assume the following condition holds throughout this paper:

(H1) $a\left(t,x\right)\in C\left({\mathrm{\top }}^{2},R\right)$, $0\le a\left(t,x\right)\le \frac{{c}^{2}}{4}$ on ${\mathrm{\top }}^{2}$, and ${\int }_{{\mathrm{\top }}^{2}}a\left(t,x\right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx>0$.

Finally, if −ξ is replaced by $a\left(t,x\right)$ in Eq. (2), the author  has proved the following unique existence and positive estimate result.

Lemma 2.1 Let $h\left(t,x\right)\in {L}^{1}\left({\mathrm{\top }}^{2}\right)$. Then Eq. (2) has a unique solution $u\left(t,x\right)=P\left[h\left(t,x\right)\right]$, $P:{L}^{1}\left({\mathrm{\top }}^{2}\right)\to C\left({\mathrm{\top }}^{2}\right)$ is a linear bounded operator with the following properties:

1. (i)

$P:C\left({\mathrm{\top }}^{2}\right)\to C\left({\mathrm{\top }}^{2}\right)$ is a completely continuous operator;

2. (ii)

If $h\left(t,x\right)>0$, a.e $\left(t,x\right)\in {\mathrm{\top }}^{2}$, $P\left[h\left(t,x\right)\right]$ has the positive estimate

$\underline{G}{\parallel h\parallel }_{{L}^{1}}\le P\left[h\left(t,x\right)\right]\le \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\parallel h\parallel }_{{L}^{1}}.$
(3)

## 3 Main result

Theorem 3.1 Assume (H1) holds. In addition, if $f\left(t,x,u\right)$ satisfies

(H2) ${lim}_{u\to {0}^{+}}f\left(t,x,u\right)=+\mathrm{\infty }$, uniformly $\left(t,x\right)\in {\mathrm{\top }}^{2}$,

(H3) $f:{\mathrm{\top }}^{2}×\left(0,+\mathrm{\infty }\right)\to \left(-\mathrm{\infty },+\mathrm{\infty }\right)$ is continuous,

(H4) there exists a nonnegative function $h\left(t,x\right)\in C\left({\mathrm{\top }}^{2}\right)$ such that

$f\left(t,x,u\right)+h\left(t,x\right)\ge 0,\phantom{\rule{1em}{0ex}}\left(t,x\right)\in {\mathrm{\top }}^{2},u>0,$

(H5) ${\int }_{{\mathrm{\top }}^{2}}{F}_{\mathrm{\infty }}\left(t,x\right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx=+\mathrm{\infty }$, where the limit function ${F}_{\mathrm{\infty }}\left(t,x\right)={lim inf}_{u\to +\mathrm{\infty }}\frac{f\left(t,x,u\right)}{u}$,

then Eq. (1) has at least two positive doubly periodic solutions for sufficiently small λ.

$C\left({\mathrm{\top }}^{2}\right)$ is a Banach space with the norm $\parallel u\parallel ={max}_{\left(t,x\right)\in {\mathrm{\top }}^{2}}|u\left(t,x\right)|$. Define a cone $K\subset C\left({\mathrm{\top }}^{2}\right)$ by

$K=\left\{u\in C\left({\mathrm{\top }}^{2}\right):u\ge 0,u\left(t,x\right)\ge \delta \parallel u\parallel \right\},$

where $\delta =\frac{{\underline{G}}^{2}{\parallel a\parallel }_{{L}^{1}}}{\overline{G}}\in \left(0,1\right)$. Let $\partial {K}_{r}=\left\{u\in K:\parallel u\parallel =r\right\}$, ${\left[u\right]}^{+}=max\left\{u,0\right\}$. By Lemma 2.1, it is easy to obtain the following lemmas.

Lemma 3.2 If $h\left(t,x\right)\in C\left({\mathrm{\top }}^{2}\right)$ is a nonnegative function, the linear boundary value problem

$\left\{\begin{array}{c}{u}_{tt}-{u}_{xx}+c{u}_{t}+a\left(t,x\right)u=\lambda h\left(t,x\right),\hfill \\ u\left(t+2\pi ,x\right)=u\left(t,x+2\pi \right)=u\left(t,x\right)\hfill \end{array}$

has a unique solution $\omega \left(t,x\right)$. The function $\omega \left(t,x\right)$ satisfies the estimates

$\lambda \underline{G}{\parallel h\parallel }_{{L}^{1}}\le \omega \left(t,x\right)=\lambda P\left(h\left(t,x\right)\right)\le \lambda \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\parallel h\parallel }_{{L}^{1}}.$

Lemma 3.3 If the boundary value problem

$\left\{\begin{array}{c}{u}_{tt}-{u}_{xx}+c{u}_{t}+a\left(t,x\right)u=\lambda \left[f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\right],\hfill \\ u\left(t+2\pi ,x\right)=u\left(t,x+2\pi \right)=u\left(t,x\right)\hfill \end{array}$

has a solution $\stackrel{˜}{u}\left(t,x\right)$ with $\parallel \stackrel{˜}{u}\parallel >\lambda \frac{{\overline{G}}^{2}}{{\underline{G}}^{3}{\parallel a\parallel }_{{L}^{1}}^{2}}{\parallel h\parallel }_{{L}^{1}}$, then ${u}^{\ast }\left(t,x\right)=\stackrel{˜}{u}\left(t,x\right)-\omega \left(t,x\right)$ is a positive doubly periodic solution of Eq. (1).

Proof of Theorem 3.1 Step 1. Define the operator T as follows:

$\left(Tu\right)\left(t,x\right)=\lambda P\left[f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\right].$

We obtain the conclusion that $T\left(K\mathrm{\setminus }\left\{u\in K:{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}=0\right\}\right)\subseteq K$, and $T:K\mathrm{\setminus }\left\{u\in K:{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}=0\right\}\to K$ is completely continuous.

For any $u\in K\mathrm{\setminus }\left\{u\in K:{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}=0\right\}$, then ${\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}>0$, and T is defined. On the other hand, for $u\in K\mathrm{\setminus }\left\{u\in K:{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}=0\right\}$, the complete continuity is obvious by Lemma 2.1. And we can have

$\begin{array}{rcl}\left(Tu\right)\left(t,x\right)& =& \lambda P\left[f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\right]\\ \ge & \lambda \underline{G}{\parallel f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\parallel }_{{L}^{1}}\\ \ge & \underline{G}\frac{\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\overline{G}}\parallel T\left(u\right)\parallel \\ \ge & \delta \parallel Tu\parallel .\end{array}$

Thus, $T\left(K\mathrm{\setminus }\left\{u\in K:u\left(t,x\right)\le \omega \left(t,x\right)\right\}\right)\subseteq K$.

Now we prove that the operator T has one fixed point $\stackrel{˜}{u}\in K$ and $\parallel \stackrel{˜}{u}\parallel >\lambda \frac{{\overline{G}}^{2}}{{\underline{G}}^{3}{\parallel a\parallel }_{{L}^{1}}^{2}}{\parallel h\parallel }_{{L}^{1}}$ for all sufficiently small λ.

Since ${\int }_{{\mathrm{\top }}^{2}}{F}_{\mathrm{\infty }}\left(t,x\right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx=+\mathrm{\infty }$, there exists ${r}_{1}\ge 2$ such that

${\int }_{{\mathrm{\top }}^{2}}\frac{f\left(t,x,u\right)}{u}\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx\ge \frac{1}{\delta },\phantom{\rule{1em}{0ex}}u\ge \delta {r}_{1}.$

Furthermore, we have ${\int }_{{\mathrm{\top }}^{2}}f\left(t,x,\delta {r}_{1}\right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx\ge {r}_{1}\ge 2$. It follows that Let $\mathrm{\Phi }\left(t,x\right)=max\left\{f\left(t,x,u\right):\frac{\delta }{2}{r}_{1}\le u\le {r}_{1}\right\}+h\left(t,x\right)$. Then $\mathrm{\Phi }\in {L}^{1}\left({\mathrm{\top }}^{2}\right)$ and ${\int }_{{\mathrm{\top }}^{2}}\mathrm{\Phi }\left(t,x\right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx>0$. Set

${\lambda }^{\ast }=min\left\{\frac{{\delta }^{2}}{2\underline{G}{\parallel h\parallel }_{{L}^{1}}},\frac{2\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\overline{G}{\parallel \mathrm{\Phi }\parallel }_{{L}^{1}}}\right\}.$

For any $u\in \partial {K}_{{r}_{1}}$ and $0<\lambda <{\lambda }^{\ast }$, we can verify that

$\begin{array}{rcl}u\left(t,x\right)-\omega \left(t,x\right)& \ge & \delta \parallel u\parallel -\omega \left(t,x\right)\\ =& \delta {r}_{1}-\omega \left(t,x\right)\\ \ge & \delta {r}_{1}-\lambda \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\parallel h\parallel }_{{L}^{1}}\\ \ge & \delta {r}_{1}-\frac{\delta {r}_{1}}{2}\\ =& \frac{\delta {r}_{1}}{2}.\end{array}$

Then we have

$\begin{array}{rcl}\parallel Tu\parallel & =& \lambda \parallel P\left[f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\right]\parallel \\ \le & \lambda \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\parallel f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\parallel }_{{L}^{1}}\\ \le & \lambda \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\parallel \mathrm{\Phi }\left(t,x\right)\parallel }_{{L}^{1}}\\ <& 2\le {r}_{1}=\parallel u\parallel .\end{array}$

On the other hand,

$\underset{u\to +\mathrm{\infty }}{lim inf}\frac{f\left(t,x,u-\omega \left(t,x\right)\right)}{u}=\underset{u\to +\mathrm{\infty }}{lim inf}\frac{f\left(t,x,u\right)}{u}={F}_{\mathrm{\infty }}\left(t,x\right).$

By the Fatou lemma, one has Hence, there exists a positive number ${r}_{2}>\delta {r}_{2}>{r}_{1}$ such that

${\int }_{{\mathrm{\top }}^{2}}\frac{f\left(t,x,u-\omega \left(t,x\right)\right)+h\left(t,x\right)}{u}\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx\ge {\lambda }^{-1}{\delta }^{-1}{\underline{G}}^{-1}{\left(4{\pi }^{2}\right)}^{-1},\phantom{\rule{1em}{0ex}}u\ge \delta {r}_{2}.$

Hence, we have

${\int }_{{\mathrm{\top }}^{2}}f\left(t,x,u-\omega \left(t,x\right)\right)+h\left(t,x\right)\phantom{\rule{0.2em}{0ex}}dt\phantom{\rule{0.2em}{0ex}}dx\ge {\lambda }^{-1}{\underline{G}}^{-1}{\left(4{\pi }^{2}\right)}^{-1}{r}_{2},\phantom{\rule{1em}{0ex}}u\ge \delta {r}_{2}.$

For any $u\in \partial {K}_{{r}_{2}}$, we have $\delta {r}_{2}=\delta \parallel u\parallel \le u\left(t,x\right)\le \parallel u\parallel ={r}_{2}$. On the other hand, since $0<\lambda <{\lambda }^{\ast }$, we can get

$\begin{array}{rcl}u\left(t,x\right)-\omega \left(t,x\right)& \ge & \delta {r}_{2}-\omega \left(t,x\right)\\ \ge & \delta \frac{{r}_{2}}{\delta }-\lambda \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}\\ \ge & \delta {r}_{2}-\delta \\ >& 0.\end{array}$

From above, we can have

$\begin{array}{rcl}\parallel Tu\parallel & \ge & \lambda P\left[f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\right]\\ \ge & \lambda \underline{G}{\parallel f\left(t,x,{\left[u\left(t,x\right)-\omega \left(t,x\right)\right]}^{+}\right)+h\left(t,x\right)\parallel }_{{L}^{1}}\\ \ge & \lambda \underline{G}4{\pi }^{2}{\lambda }^{-1}{\underline{G}}^{-1}{\left(4{\pi }^{2}\right)}^{-1}{r}_{2}\\ =& {r}_{2}.\end{array}$

Therefore, by Lemma 1.1, the operator T has a fixed point $\stackrel{˜}{u}\left(t,x\right)\in K$ and So, Eq. (1) has a positive solution $\stackrel{ˆ}{u}\left(t,x\right)=\stackrel{˜}{u}\left(t,x\right)-\omega \left(t,x\right)\ge \delta$.

Step 2. By conditions (H2) and (H3), it is clear to obtain that

${u}_{0}=inf\left\{u\in K:f\left(t,x,u\right)\le 0,\left(t,x\right)\in {\mathrm{\top }}^{2}\right\}>0.$

Let ${r}_{4}=min\left\{\frac{\delta }{2},\frac{\delta \parallel {u}_{0}\parallel }{2}\right\}$. For any $u\in \left(0,{r}_{4}\right]$, we have $f\left(t,x,u\right)>0$. Then define the operator A as follows:

$\left(Au\right)\left(t,x\right)=\lambda \stackrel{ˆ}{P}\left[f\left(t,x,u\left(t,x\right)\right)\right].$

It is easy to prove that $A\left(K\cap \left\{u\in C\left({\mathrm{\top }}^{2}\right):0<\parallel u\parallel <{r}_{4}\right\}\right)\subseteq K$, and $A:K\cap \left\{u\in C\left({\mathrm{\top }}^{2}\right):0<\parallel u\parallel <{r}_{4}\right\}\to K$ is completely continuous.

And for any $\rho >0$, define

$M\left(\rho \right)=max\left\{f\left(t,x,u\right):u\in {R}^{+},\delta \rho \le u\le \rho ,\left(t,x\right)\in {\mathrm{\top }}^{2}\right\}>0.$

Furthermore, for any $u\in \partial {K}_{{r}_{4}}$, we have

$\begin{array}{rcl}\parallel Au\parallel & =& \lambda \parallel \stackrel{ˆ}{P}\left[f\left(t,x,u\left(t,x\right)\right)\right]\parallel \\ \le & \lambda \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}{\parallel f\left(t,x,u\left(t,x\right)\right)\parallel }_{{L}^{1}}\\ \le & \lambda \frac{\overline{G}}{\underline{G}{\parallel a\parallel }_{{L}^{1}}}M\left({r}_{4}\right)4{\pi }^{2}.\end{array}$

Thus, from the above inequality, there exists $\overline{\lambda }$ such that

Since ${lim}_{u\to {0}^{+}}f\left(t,x,u\right)=\mathrm{\infty }$, then there is $0<{r}_{3}<\frac{{r}_{4}}{2}$ such that

where μ satisfies $\lambda \underline{G}\mu \delta >1$. For any $u\in \partial {K}_{{r}_{3}}$, then we have

By Lemma 2.1, it is clear to obtain that

$\begin{array}{rcl}\parallel Au\parallel & =& \lambda \parallel \stackrel{ˆ}{P}\left[f\left(t,x,u\left(t,x\right)\right)\right]\parallel \\ \ge & \lambda \underline{G}{\parallel f\left(t,x,u\left(t,x\right)\right)\parallel }_{{L}^{1}}\\ \ge & \lambda \underline{G}\mu \delta {r}_{3}\\ >& {r}_{3}=\parallel u\parallel .\end{array}$

Therefore, by Lemma 1.1, A has a fixed point in $\overline{u}\left(t,x\right)\in K$ and $\parallel \overline{u}\parallel \le {r}_{4}\le \frac{\delta }{2}$, which is another positive periodic solution of Eq. (1).

Finally, from Step 1 and Step 2, Eq. (1) has two positive doubly periodic solutions $\stackrel{ˆ}{u}\left(t,x\right)$ and $\overline{u}\left(t,x\right)$ for sufficiently small λ. □

Example

Consider the following problem:

$\left\{\begin{array}{c}{u}_{tt}-{u}_{xx}+2{u}_{t}+{sin}^{2}\left(t+x\right)u=\lambda \left[\frac{1}{u}+min\left\{{u}^{2},\frac{u}{|1-\frac{t}{\pi }||1-\frac{x}{\pi }|}\right\}-10\right],\hfill \\ u\left(t+2\pi ,x\right)=u\left(t,x+2\pi \right)=u\left(t,x\right).\hfill \end{array}$

It is clear that $f\left(t,x,u\right)$ satisfies the conditions (H1)-(H5).

## References

1. Chu J, Torres PJ, Zhang M: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ. 2007, 239: 196-212. 10.1016/j.jde.2007.05.007

2. Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665-675. 10.1016/j.jmaa.2011.09.061

3. Chu J, Zhang Z: Periodic solutions of second order superlinear singular dynamical systems. Acta Appl. Math. 2010, 111: 179-187. 10.1007/s10440-009-9539-9

4. Chu J, Li M: Positive periodic solutions of Hill’s equations with singular nonlinear perturbations. Nonlinear Anal. 2008, 69: 276-286. 10.1016/j.na.2007.05.016

5. 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/bdm040

6. 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.2004.10.031

7. Torres PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643-662. 10.1016/S0022-0396(02)00152-3

8. Torres PJ: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 2007, 232: 277-284. 10.1016/j.jde.2006.08.006

9. Wang H: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ. 2010, 249: 2986-3002. 10.1016/j.jde.2010.08.027

10. DeCoster C, Habets P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. 371. In Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations CISM-ICMS. Edited by: Zanolin F. Springer, New York; 1996:1-78.

11. Jebelean P, Mawhin J: Periodic solutions of forced dissipative p -Liénard equations with singularities. Vietnam J. Math. 2004, 32: 97-103.

12. Wang F: Doubly periodic solutions of a coupled nonlinear telegraph system with weak singularities. Nonlinear Anal., Real World Appl. 2011, 12: 254-261. 10.1016/j.nonrwa.2010.06.012

13. Li Y: Positive doubly periodic solutions of nonlinear telegraph equations. Nonlinear Anal. 2003, 55: 245-254. 10.1016/S0362-546X(03)00227-X

14. Ortega R, Robles-Perez AM: A maximum principle for periodic solutions of the telegraph equations. J. Math. Anal. Appl. 1998, 221: 625-651. 10.1006/jmaa.1998.5921

15. Wang F, An Y: Nonnegative doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 2008, 338: 91-100. 10.1016/j.jmaa.2007.05.008

16. Wang F, An Y: Existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 2009, 349: 30-42. 10.1016/j.jmaa.2008.08.003

17. Wang F, An Y: Nonnegative doubly periodic solutions for nonlinear telegraph system with twin-parameters. Appl. Math. Comput. 2009, 214: 310-317. 10.1016/j.amc.2009.03.069

18. Wang F, An Y: On positive solutions of nonlinear telegraph semipositone system. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2009, 16: 209-219.

19. Xu X: Positive solutions for singular semi-positone three-point systems. Nonlinear Anal. 2007, 66: 791-805. 10.1016/j.na.2005.12.019

20. Yao Q: An existence theorem of a positive solution to a semipositone Sturm-Liouville boundary value problem. Appl. Math. Lett. 2010, 23: 1401-1406. 10.1016/j.aml.2010.06.025

21. Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.

## Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions for improving this paper.

## Author information

Authors

### Corresponding author

Correspondence to Fanglei Wang.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

This paper is concerned with a singular semipositone telegraph equation with a parameter and represents a somewhat interesting contribution in the investigation of the existence and multiplicity of doubly periodic solutions of the telegraph equation. All authors typed, read and approved the final manuscript.

## Rights and permissions

Reprints and Permissions

Wang, F., An, Y. Multiple positive doubly periodic solutions for a singular semipositone telegraph equation with a parameter. Bound Value Probl 2013, 7 (2013). https://doi.org/10.1186/1687-2770-2013-7 