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.

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

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,

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.

Let ⊤ 2 be the torus defined as

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

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.,

First, we consider the linear equation

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

Let £ ξ be the differential operator

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

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

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

(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

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

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

Lemma 3.3If the boundary value problem

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:

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

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

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

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

Then we have

On the other hand,

By the Fatou lemma, one has

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

Hence, we have

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

From above, we can have

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

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:

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

Furthermore, for any u ∈ ∂ K r 4 , we have

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

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

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

By Lemma 2.1, it is clear to obtain that

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:

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

The authors declare that they have no competing interests.

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.

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

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