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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 [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:

( | 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 [1317] 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]

Let E be a Banach space, and KE be a cone in E. Assume Ω 1 , Ω 2 are open subsets of E with 0 Ω 1 , Ω ¯ 1 Ω 2 , and let T:K( Ω ¯ 2 Ω 1 )K be a completely continuous operator such that either

  1. (i)

    Tuu, uK Ω 1 and Tuu, uK Ω 2 ; or

  2. (ii)

    Tuu, uK Ω 1 and Tuu, uK Ω 2 .

Then T has 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/2πZ)×(R/2πZ).

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 ) dtdx=λ 2 f(t,x,u)φdtdx.

First, we consider the linear equation

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

where c>0, μR, and h(t,x) L 1 ( 2 ).

Let £ ξ be the differential operator

£ ξ u= u t t u x x +c u t ξu,

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), 0a(t,x) c 2 4 on 2 , and 2 a(t,x)dtdx>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 2.1 Let 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:

  1. (i)

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

  2. (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 3.1 Assume (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)dtdx=+, 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 small λ.

C( 2 ) is a Banach space with the norm u= max ( t , x ) 2 |u(t,x)|. Define a cone KC( 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 ={uK:u=r}, [ u ] + =max{u,0}. By Lemma 2.1, it is easy to obtain the following lemmas.

Lemma 3.2 If 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)=λP ( h ( t , x ) ) λ G ¯ G ̲ a L 1 h L 1 .

Lemma 3.3 If 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:

(Tu)(t,x)=λP [ f ( t , x , [ u ( t , x ) ω ( t , x ) ] + ) + h ( t , x ) ] .

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

For any uK{uK: [ u ( t , x ) ω ( t , x ) ] + =0}, then [ u ( t , x ) ω ( t , x ) ] + >0, and T is defined. On the other hand, for uK{uK: [ 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{uK: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)dtdx=+, there exists r 1 2 such that

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

Furthermore, we have 2 f(t,x,δ r 1 )dtdx 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)dtdx>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 dtdx λ 1 δ 1 G ̲ 1 ( 4 π 2 ) 1 ,uδ r 2 .

Hence, we have

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

For any u K r 2 , we have δ r 2 =δuu(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:

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

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

Au<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)μu,for u R +  with 0<u r 3 ,

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

f(t,x,u)μu(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).

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Acknowledgements

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

Author information

Correspondence to Fanglei Wang.

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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.

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Keywords

  • semipositone telegraph equation
  • doubly periodic solution
  • singular
  • cone
  • fixed point theorem