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 wellknown fixed point theorem in a cone.
MSC: 34B15, 34B18.
Keywords:
semipositone telegraph equation; doubly periodic solution; singular; cone; fixed point theorem1 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:
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 [1317] 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
The main method used here is the following fixedpoint theorem of a cone mapping.
Lemma 1.1[21]
LetEbe a Banach space, and
(i)
(ii)
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.
2 Preliminaries
Let
Doubly 2πperiodic functions will be identified to be functions defined on
to denote the spaces of doubly periodic functions with the indicated degree of regularity.
The space
By a doubly periodic solution of Eq. (1) we mean that a
First, we consider the linear equation
where
Let
acting on functions on
where
For
For convenience, we assume the following condition holds throughout this paper:
(H1)
Finally, if −ξ is replaced by
Lemma 2.1Let
(i)
(ii) If
3 Main result
Theorem 3.1Assume (H1) holds. In addition, if
(H2)
(H3)
(H4) there exists a nonnegative function
(H5)
then Eq. (1) has at least two positive doubly periodic solutions for sufficiently smallλ.
where
Lemma 3.2If
has a unique solution
Lemma 3.3If the boundary value problem
has a solution
Proof of Theorem 3.1 Step 1. Define the operator T as follows:
We obtain the conclusion that
For any
Thus,
Now we prove that the operator T has one fixed point
Since
Furthermore, we have
Let
For any
Then we have
On the other hand,
By the Fatou lemma, one has
Hence, there exists a positive number
Hence, we have
For any
From above, we can have
Therefore, by Lemma 1.1, the operator T has a fixed point
So, Eq. (1) has a positive solution
Step 2. By conditions (H2) and (H3), it is clear to obtain that
Let
It is easy to prove that
And for any
Furthermore, for any
Thus, from the above inequality, there exists
Since
where μ satisfies
By Lemma 2.1, it is clear to obtain that
Therefore, by Lemma 1.1, A has a fixed point in
Finally, from Step 1 and Step 2, Eq. (1) has two positive doubly periodic solutions
Example
Consider the following problem:
It is clear that
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.
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