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
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 , 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  use the continuation type existence principle to investigate the following singular periodic problem:
More recently, using a weak force condition, Wang  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  study the semipositone telegraph system
Inspired by the above references, we are concerned with the multiplicity of positive doubly periodic solutions for a general singular semipositone telegraph equation
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.
LetEbe a Banach space, and be a cone inE. Assume , are open subsets ofEwith , , and let be 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.
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
where , , and .
Let be the differential operator
acting on functions on . Following the discussion in , 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 . 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  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 Main result
Theorem 3.1Assume (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 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 3.2If is a nonnegative function, the linear boundary value problem
has a unique solution . The function satisfies the estimates
Lemma 3.3If 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
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. □
Consider the following problem:
It is clear that 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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