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
The main method used here is the following fixed-point theorem of a cone mapping.
The paper is organized as follows. In Section 2, some preliminaries are given. In Section 3, we give the main result.
First, we consider the linear equation
acting on functions on . Following the discussion in , we know that if , has the resolvent ,
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:
Finally, if −ξ is replaced by in Eq. (2), the author  has proved the following unique existence and positive estimate result.
3 Main result
then Eq. (1) has at least two positive doubly periodic solutions for sufficiently smallλ.
Lemma 3.3If the boundary value problem
Proof of Theorem 3.1 Step 1. Define the operator T as follows:
Then we have
On the other hand,
By the Fatou lemma, one has
Hence, we have
From above, we can have
Step 2. By conditions (H2) and (H3), it is clear to obtain that
By Lemma 2.1, it is clear to obtain that
Consider the following problem:
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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