Multiple positive doubly periodic solutions for a singular semipositone telegraph equation with a parameter
© Wang and An; licensee Springer. 2013
Received: 26 July 2012
Accepted: 29 December 2012
Published: 16 January 2013
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.
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.
The main method used here is the following fixed-point theorem of a cone mapping.
Lemma 1.1 
, and , ; or
, and , .
Then T has 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.
to denote the spaces of doubly periodic functions with the indicated degree of regularity. The space denotes the space of distributions on .
where , , and .
where is the unique solution of Eq. (2), and the restriction of on () or is compact. In particular, is a completely continuous operator.
For convenience, we assume the following condition holds throughout this paper:
(H1) , on , and .
is a completely continuous operator;
- (ii)If , a.e , has the positive estimate(3)
3 Main result
Theorem 3.1 Assume (H1) holds. In addition, if satisfies
(H2) , uniformly ,
(H3) is continuous,
(H5) , where the limit function ,
then Eq. (1) has at least two positive doubly periodic solutions for sufficiently small λ.
where . Let , . By Lemma 2.1, it is easy to obtain the following lemmas.
has a solution with , then is a positive doubly periodic solution of Eq. (1).
We obtain the conclusion that , and is completely continuous.
Now we prove that the operator T has one fixed point and for all sufficiently small λ.
So, Eq. (1) has a positive solution .
It is easy to prove that , and is completely continuous.
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. □
It is clear that satisfies the conditions (H1)-(H5).
The authors would like to thank the referees for valuable comments and suggestions for improving this paper.
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