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# 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

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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:

( | u | p 2 u ) + h ( u ) u = g ( u ) + c ( t ) .

More recently, using a weak force condition, Wang [12] has built some existence results for the following periodic boundary value problem:

{ u t t u x x + c 1 u t + a 11 ( t , x ) u + a 12 ( t , x ) v = f 1 ( t , x , u , v ) + χ 1 ( t , x ) , v t t v x x + c 2 v t + a 21 ( t , x ) u + a 22 ( t , x ) v = f 2 ( t , x , u , v ) + χ 2 ( t , x ) .

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

{ u t t u x x + c 1 u t + a 1 ( t , x ) u = b 1 ( t , x ) f ( t , x , u , v ) , v t t v x x + c 2 v t + a 2 ( t , x ) v = b 2 ( t , x ) g ( t , x , u , v ) ,

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

{ u t t u x x + c u t + a ( t , x ) u = λ f ( t , x , u ) , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x ) , (1)

where c > 0 is a constant, λ > 0 is a positive parameter, a ( t , x ) C ( R × R , R ) , f ( t , x , u ) may change sign and is singular at u = 0 , namely,

lim u 0 + f ( t , x , u ) = + .

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

Lemma 1.1[21]

LetEbe a Banach space, and K E be a cone inE. Assume Ω 1 , Ω 2 are open subsets ofEwith 0 Ω 1 , Ω ¯ 1 Ω 2 , and let T : K ( Ω ¯ 2 Ω 1 ) K be a completely continuous operator such that either

(i) T u u , u K Ω 1 and T u u , u K Ω 2 ; or

(ii) T u u , u K Ω 1 and T u u , u K Ω 2 .

ThenThas a fixed point in K ( Ω ¯ 2 Ω 1 ) .

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

### 2 Preliminaries

Let 2 be the torus defined as

2 = ( R / 2 π Z ) × ( R / 2 π Z ) .

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

L p ( 2 ) , C ( 2 ) , C α ( 2 ) , D ( 2 ) = C ( 2 ) ,

to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space D ( 2 ) denotes the space of distributions on 2 .

By a doubly periodic solution of Eq. (1) we mean that a u L 1 ( 2 ) satisfies Eq. (1) in the distribution sense, i.e.,

2 u ( φ t t φ x x c φ t + a ( t , x ) φ ) d t d x = λ 2 f ( t , x , u ) φ d t d x .

First, we consider the linear equation

u t t u x x + c u t ξ u = h ( t , x ) , in  D ( 2 ) , (2)

where c > 0 , μ R , and h ( t , x ) L 1 ( 2 ) .

Let £ ξ be the differential operator

£ ξ u = u t t u x x + c u t ξ u ,

acting on functions on 2 . Following the discussion in [14], we know that if ξ < 0 , £ ξ has the resolvent R ξ ,

R ξ : L 1 ( 2 ) C ( 2 ) , h i ( t , x ) u i ( t , x ) ,

where u ( t , x ) is the unique solution of Eq. (2), and the restriction of R ξ on L p ( 2 ) ( 1 < p < ) or C ( 2 ) is compact. In particular, R ξ : C ( 2 ) C ( 2 ) is a completely continuous operator.

For ξ = c 2 / 4 , the Green function G ( t , x ) of the differential operator £ ξ is explicitly expressed; see Lemma 5.2 in [14]. From the definition of G ( t , x ) , we have

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

(H1) a ( t , x ) C ( 2 , R ) , 0 a ( t , x ) c 2 4 on 2 , and 2 a ( t , x ) d t d x > 0 .

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

Lemma 2.1Let h ( t , x ) L 1 ( 2 ) . Then Eq. (2) has a unique solution u ( t , x ) = P [ h ( t , x ) ] , P : L 1 ( 2 ) C ( 2 ) is a linear bounded operator with the following properties:

(i) P : C ( 2 ) C ( 2 ) is a completely continuous operator;

(ii) If h ( t , x ) > 0 , a.e ( t , x ) 2 , P [ h ( t , x ) ] has the positive estimate

G ̲ h L 1 P [ h ( t , x ) ] G ¯ G ̲ a L 1 h L 1 . (3)

### 3 Main result

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

(H2) lim u 0 + f ( t , x , u ) = + , uniformly ( t , x ) 2 ,

(H3) f : 2 × ( 0 , + ) ( , + ) is continuous,

(H4) there exists a nonnegative function h ( t , x ) C ( 2 ) such that

f ( t , x , u ) + h ( t , x ) 0 , ( t , x ) 2 , u > 0 ,

(H5) 2 F ( t , x ) d t d x = + , where the limit function F ( t , x ) = lim inf u + f ( t , x , u ) u ,

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

C ( 2 ) is a Banach space with the norm u = max ( t , x ) 2 | u ( t , x ) | . Define a cone K C ( 2 ) by

K = { u C ( 2 ) : u 0 , u ( t , x ) δ u } ,

where δ = G ̲ 2 a L 1 G ¯ ( 0 , 1 ) . Let K r = { u K : u = r } , [ u ] + = max { u , 0 } . By Lemma 2.1, it is easy to obtain the following lemmas.

Lemma 3.2If h ( t , x ) C ( 2 ) is a nonnegative function, the linear boundary value problem

{ u t t u x x + c u t + a ( t , x ) u = λ h ( t , x ) , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x )

has a unique solution ω ( t , x ) . The function ω ( t , x ) satisfies the estimates

λ G ̲ h L 1 ω ( t , x ) = λ P ( h ( t , x ) ) λ G ¯ G ̲ a L 1 h L 1 .

Lemma 3.3If the boundary value problem

{ u t t u x x + c u t + a ( t , x ) u = λ [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x )

has a solution u ˜ ( t , x ) with u ˜ > λ G ¯ 2 G ̲ 3 a L 1 2 h L 1 , then u ( t , x ) = u ˜ ( t , x ) ω ( t , x ) is a positive doubly periodic solution of Eq. (1).

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

( T u ) ( t , x ) = λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] .

We obtain the conclusion that T ( K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } ) K , and T : K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } K is completely continuous.

For any u K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } , then [ u ( t , x ) ω ( t , x ) ] + > 0 , and T is defined. On the other hand, for u K { u K : [ u ( t , x ) ω ( t , x ) ] + = 0 } , the complete continuity is obvious by Lemma 2.1. And we can have

( T u ) ( t , x ) = λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] λ G ̲ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) L 1 G ̲ G ̲ a L 1 G ¯ T ( u ) δ T u .

Thus, T ( K { u K : u ( t , x ) ω ( t , x ) } ) K .

Now we prove that the operator T has one fixed point u ˜ K and u ˜ > λ G ¯ 2 G ̲ 3 a L 1 2 h L 1 for all sufficiently small λ.

Since 2 F ( t , x ) d t d x = + , there exists r 1 2 such that

2 f ( t , x , u ) u d t d x 1 δ , u δ r 1 .

Furthermore, we have 2 f ( t , x , δ r 1 ) d t d x r 1 2 . It follows that

Let Φ ( t , x ) = max { f ( t , x , u ) : δ 2 r 1 u r 1 } + h ( t , x ) . Then Φ L 1 ( 2 ) and 2 Φ ( t , x ) d t d x > 0 . Set

λ = min { δ 2 2 G ̲ h L 1 , 2 G ̲ a L 1 G ¯ Φ L 1 } .

For any u K r 1 and 0 < λ < λ , we can verify that

u ( t , x ) ω ( t , x ) δ u ω ( t , x ) = δ r 1 ω ( t , x ) δ r 1 λ G ¯ G ̲ a L 1 h L 1 δ r 1 δ r 1 2 = δ r 1 2 .

Then we have

T u = λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] λ G ¯ G ̲ a L 1 f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) L 1 λ G ¯ G ̲ a L 1 Φ ( t , x ) L 1 < 2 r 1 = u .

On the other hand,

lim inf u + f ( t , x , u ω ( t , x ) ) u = lim inf u + f ( t , x , u ) u = F ( t , x ) .

By the Fatou lemma, one has

Hence, there exists a positive number r 2 > δ r 2 > r 1 such that

2 f ( t , x , u ω ( t , x ) ) + h ( t , x ) u d t d x λ 1 δ 1 G ̲ 1 ( 4 π 2 ) 1 , u δ r 2 .

Hence, we have

2 f ( t , x , u ω ( t , x ) ) + h ( t , x ) d t d x λ 1 G ̲ 1 ( 4 π 2 ) 1 r 2 , u δ r 2 .

For any u K r 2 , we have δ r 2 = δ u u ( t , x ) u = r 2 . On the other hand, since 0 < λ < λ , we can get

u ( t , x ) ω ( t , x ) δ r 2 ω ( t , x ) δ r 2 δ λ G ¯ G ̲ a L 1 δ r 2 δ > 0 .

From above, we can have

T u λ P [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] λ G ̲ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) L 1 λ G ̲ 4 π 2 λ 1 G ̲ 1 ( 4 π 2 ) 1 r 2 = r 2 .

Therefore, by Lemma 1.1, the operator T has a fixed point u ˜ ( t , x ) K and

So, Eq. (1) has a positive solution u ˆ ( t , x ) = u ˜ ( t , x ) ω ( t , x ) δ .

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

u 0 = inf { u K : f ( t , x , u ) 0 , ( t , x ) 2 } > 0 .

Let r 4 = min { δ 2 , δ u 0 2 } . For any u ( 0 , r 4 ] , we have f ( t , x , u ) > 0 . Then define the operator A as follows:

( A u ) ( t , x ) = λ P ˆ [ f ( t , x , u ( t , x ) ) ] .

It is easy to prove that A ( K { u C ( 2 ) : 0 < u < r 4 } ) K , and A : K { u C ( 2 ) : 0 < u < r 4 } K is completely continuous.

And for any ρ > 0 , define

M ( ρ ) = max { f ( t , x , u ) : u R + , δ ρ u ρ , ( t , x ) 2 } > 0 .

Furthermore, for any u K r 4 , we have

A u = λ P ˆ [ f ( t , x , u ( t , x ) ) ] λ G ¯ G ̲ a L 1 f ( t , x , u ( t , x ) ) L 1 λ G ¯ G ̲ a L 1 M ( r 4 ) 4 π 2 .

Thus, from the above inequality, there exists λ ¯ such that

A u < u , for  u K r 4 , 0 < λ < λ ¯ .

Since lim u 0 + f ( t , x , u ) = , then there is 0 < r 3 < r 4 2 such that

f ( t , x , u ) μ u , for  u R +  with  0 < u r 3 ,

where μ satisfies λ G ̲ μ δ > 1 . For any u K r 3 , then we have

f ( t , x , u ) μ u ( t , x ) , for  ( t , x ) 2 .

By Lemma 2.1, it is clear to obtain that

A u = λ P ˆ [ f ( t , x , u ( t , x ) ) ] λ G ̲ f ( t , x , u ( t , x ) ) L 1 λ G ̲ μ δ r 3 > r 3 = u .

Therefore, by Lemma 1.1, A has a fixed point in u ¯ ( t , x ) K and u ¯ r 4 δ 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 u ˆ ( t , x ) and u ¯ ( t , x ) for sufficiently small λ. □

Example

Consider the following problem:

{ u t t u x x + 2 u t + sin 2 ( t + x ) u = λ [ 1 u + min { u 2 , u | 1 t π | | 1 x π | } 10 ] , u ( t + 2 π , x ) = u ( t , x + 2 π ) = u ( t , x ) .

It is clear that f ( t , x , u ) 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)