## Boundary Value Problems

Impact Factor 1.014

Open Access

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

Boundary Value Problems20132013:7

DOI: 10.1186/1687-2770-2013-7

Accepted: 29 December 2012

Published: 16 January 2013

## 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.

### Keywords

semipositone telegraph equation doubly periodic solution singular cone fixed point theorem

## 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 [19]. In [10], 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 [11] use the continuation type existence principle to investigate the following singular periodic problem:
More recently, using a weak force condition, Wang [12] has built some existence results for the following periodic boundary value problem:

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 [1317] and the references therein. In these references, the nonlinearities are nonnegative.

On the other hand, the authors [18] study the semipositone telegraph system

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
(1)
where is a constant, is a positive parameter, , may change sign and is singular at , namely,

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

Lemma 1.1 [21]

Let E be a Banach space, and be a cone in E. Assume , are open subsets of E with , , and let be a completely continuous operator such that either
1. (i)

, and , ; or

2. (ii)

, and , .

Then T has a fixed point in .

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

## 2 Preliminaries

Let be the torus defined as
Doubly 2π-periodic functions will be identified to be functions defined on . We use the notations

to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space denotes the space of distributions on .

By a doubly periodic solution of Eq. (1) we mean that a satisfies Eq. (1) in the distribution sense, i.e.,
First, we consider the linear equation
(2)

where , , and .

Let be the differential operator
acting on functions on . Following the discussion in [14], we know that if , has the resolvent ,

where is the unique solution of Eq. (2), and the restriction of on () or is compact. In particular, is a completely continuous operator.

For , the Green function of the differential operator is explicitly expressed; see Lemma 5.2 in [14]. From the definition of , we have

For convenience, we assume the following condition holds throughout this paper:

(H1) , on , and .

Finally, if −ξ is replaced by in Eq. (2), the author [13] has proved the following unique existence and positive estimate result.

Lemma 2.1 Let . Then Eq. (2) has a unique solution , is a linear bounded operator with the following properties:
1. (i)

is a completely continuous operator;

2. (ii)
If , a.e , has the positive estimate
(3)

## 3 Main result

Theorem 3.1 Assume (H1) holds. In addition, if satisfies

(H2) , uniformly ,

(H3) is continuous,

(H4) there exists a nonnegative function such that

(H5) , where the limit function ,

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

is a Banach space with the norm . Define a cone by

where . Let , . By Lemma 2.1, it is easy to obtain the following lemmas.

Lemma 3.2 If is a nonnegative function, the linear boundary value problem
has a unique solution . The function satisfies the estimates
Lemma 3.3 If the boundary value problem

has a solution with , then is a positive doubly periodic solution of Eq. (1).

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

We obtain the conclusion that , and is completely continuous.

For any , then , and T is defined. On the other hand, for , the complete continuity is obvious by Lemma 2.1. And we can have

Thus, .

Now we prove that the operator T has one fixed point and for all sufficiently small λ.

Since , there exists such that
Furthermore, we have . It follows that
Let . Then and . Set
For any and , we can verify that
Then we have
On the other hand,
By the Fatou lemma, one has
Hence, there exists a positive number such that
Hence, we have
For any , we have . On the other hand, since , we can get
From above, we can have
Therefore, by Lemma 1.1, the operator T has a fixed point and

So, Eq. (1) has a positive solution .

Step 2. By conditions (H2) and (H3), it is clear to obtain that
Let . For any , we have . Then define the operator A as follows:

It is easy to prove that , and is completely continuous.

And for any , define
Furthermore, for any , we have
Thus, from the above inequality, there exists such that
Since , then there is such that
where μ satisfies . For any , then we have
By Lemma 2.1, it is clear to obtain that

Therefore, by Lemma 1.1, A has a fixed point in and , which is another positive periodic solution of Eq. (1).

Finally, from Step 1 and Step 2, Eq. (1) has two positive doubly periodic solutions and for sufficiently small λ. □

Example

Consider the following problem:

It is clear that satisfies the conditions (H1)-(H5).

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
College of Science, Hohai University
(2)
Department of Mathematics, Nanjing University of Aeronautics and Astronautics

## 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.View Article
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.MATH