Research

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

Fanglei Wang1* and Yukun An2

### Author affiliations

1 College of Science, Hohai University, Nanjing, 210098, P.R. China

2 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:7  doi:10.1186/1687-2770-2013-7

 Received: 26 July 2012 Accepted: 29 December 2012 Published: 16 January 2013

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 cited.

### 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 [1-9]. 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 [13-17] 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]

LetEbe a Banach space, andbe a cone inE. Assume, are open subsets ofEwith, , and letbe a completely continuous operator such that either

(i) , and, ; or

(ii) , and, .

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

(i) is a completely continuous operator;

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

(3)

### 3 Main result

Theorem 3.1Assume (H1) holds. In addition, ifsatisfies

(H2) , uniformly,

(H3) is continuous,

(H4) there exists a nonnegative functionsuch that

(H5) , where the limit function,

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

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 3.2Ifis a nonnegative function, the linear boundary value problem

has a unique solution. The functionsatisfies the estimates

Lemma 3.3If the boundary value problem

has a solutionwith, thenis 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).

### 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.

### Acknowledgements

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

### References

1. Chu, J, Torres, PJ, Zhang, M: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ.. 239, 196–212 (2007). Publisher Full Text

2. Chu, J, Fan, N, Torres, PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl.. 388, 665–675 (2012). Publisher Full Text

3. Chu, J, Zhang, Z: Periodic solutions of second order superlinear singular dynamical systems. Acta Appl. Math.. 111, 179–187 (2010). Publisher Full Text

4. Chu, J, Li, M: Positive periodic solutions of Hill’s equations with singular nonlinear perturbations. Nonlinear Anal.. 69, 276–286 (2008). Publisher Full Text

5. Chu, J, Torres, PJ: Applications of Schauder’s fixed point theorem to singular differential equations. Bull. Lond. Math. Soc.. 39, 653–660 (2007). Publisher Full Text

6. Jiang, D, Chu, J, Zhang, M: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ.. 211, 282–302 (2005). PubMed Abstract | Publisher Full Text

7. Torres, PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ.. 190, 643–662 (2003). Publisher Full Text

8. Torres, PJ: Weak singularities may help periodic solutions to exist. J. Differ. Equ.. 232, 277–284 (2007). Publisher Full Text

9. Wang, H: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ.. 249, 2986–3002 (2010). Publisher Full Text

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

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

12. Wang, F: Doubly periodic solutions of a coupled nonlinear telegraph system with weak singularities. Nonlinear Anal., Real World Appl.. 12, 254–261 (2011). Publisher Full Text

13. Li, Y: Positive doubly periodic solutions of nonlinear telegraph equations. Nonlinear Anal.. 55, 245–254 (2003). Publisher Full Text

14. Ortega, R, Robles-Perez, AM: A maximum principle for periodic solutions of the telegraph equations. J. Math. Anal. Appl.. 221, 625–651 (1998). Publisher Full Text

15. Wang, F, An, Y: Nonnegative doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl.. 338, 91–100 (2008). Publisher Full Text

16. Wang, F, An, Y: Existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl.. 349, 30–42 (2009). Publisher Full Text

17. Wang, F, An, Y: Nonnegative doubly periodic solutions for nonlinear telegraph system with twin-parameters. Appl. Math. Comput.. 214, 310–317 (2009). Publisher Full Text

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

19. Xu, X: Positive solutions for singular semi-positone three-point systems. Nonlinear Anal.. 66, 791–805 (2007). Publisher Full Text

20. Yao, Q: An existence theorem of a positive solution to a semipositone Sturm-Liouville boundary value problem. Appl. Math. Lett.. 23, 1401–1406 (2010). Publisher Full Text

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