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.
Keywordssemipositone telegraph equation doubly periodic solution singular cone fixed point theorem
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.
- Chu J, Torres PJ, Zhang M: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ. 2007, 239: 196-212. 10.1016/j.jde.2007.05.007MATHMathSciNetView ArticleGoogle Scholar
- Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665-675. 10.1016/j.jmaa.2011.09.061MATHMathSciNetView ArticleGoogle Scholar
- Chu J, Zhang Z: Periodic solutions of second order superlinear singular dynamical systems. Acta Appl. Math. 2010, 111: 179-187. 10.1007/s10440-009-9539-9MATHMathSciNetView ArticleGoogle Scholar
- Chu J, Li M: Positive periodic solutions of Hill’s equations with singular nonlinear perturbations. Nonlinear Anal. 2008, 69: 276-286. 10.1016/j.na.2007.05.016MATHMathSciNetView ArticleGoogle Scholar
- Chu J, Torres PJ: Applications of Schauder’s fixed point theorem to singular differential equations. Bull. Lond. Math. Soc. 2007, 39: 653-660. 10.1112/blms/bdm040MATHMathSciNetView ArticleGoogle Scholar
- Jiang D, Chu J, Zhang M: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 2005, 211: 282-302. 10.1016/j.jde.2004.10.031MATHMathSciNetView ArticleGoogle Scholar
- Torres PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643-662. 10.1016/S0022-0396(02)00152-3MATHView ArticleGoogle Scholar
- Torres PJ: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 2007, 232: 277-284. 10.1016/j.jde.2006.08.006MATHView ArticleGoogle Scholar
- Wang H: Positive periodic solutions of singular systems with a parameter. J. Differ. Equ. 2010, 249: 2986-3002. 10.1016/j.jde.2010.08.027MATHView ArticleGoogle Scholar
- DeCoster C, Habets P: Upper and lower solutions in the theory of ODE boundary value problems: classical and recent results. 371. In Nonlinear Analysis and Boundary Value Problems for Ordinary Differential Equations CISM-ICMS. Edited by: Zanolin F. Springer, New York; 1996:1-78.View ArticleGoogle Scholar
- Jebelean P, Mawhin J: Periodic solutions of forced dissipative p -Liénard equations with singularities. Vietnam J. Math. 2004, 32: 97-103.MATHMathSciNetGoogle Scholar
- Wang F: Doubly periodic solutions of a coupled nonlinear telegraph system with weak singularities. Nonlinear Anal., Real World Appl. 2011, 12: 254-261. 10.1016/j.nonrwa.2010.06.012MATHMathSciNetView ArticleGoogle Scholar
- Li Y: Positive doubly periodic solutions of nonlinear telegraph equations. Nonlinear Anal. 2003, 55: 245-254. 10.1016/S0362-546X(03)00227-XMATHMathSciNetView ArticleGoogle Scholar
- Ortega R, Robles-Perez AM: A maximum principle for periodic solutions of the telegraph equations. J. Math. Anal. Appl. 1998, 221: 625-651. 10.1006/jmaa.1998.5921MATHMathSciNetView ArticleGoogle Scholar
- Wang F, An Y: Nonnegative doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 2008, 338: 91-100. 10.1016/j.jmaa.2007.05.008MATHMathSciNetView ArticleGoogle Scholar
- Wang F, An Y: Existence and multiplicity results of positive doubly periodic solutions for nonlinear telegraph system. J. Math. Anal. Appl. 2009, 349: 30-42. 10.1016/j.jmaa.2008.08.003MATHMathSciNetView ArticleGoogle Scholar
- Wang F, An Y: Nonnegative doubly periodic solutions for nonlinear telegraph system with twin-parameters. Appl. Math. Comput. 2009, 214: 310-317. 10.1016/j.amc.2009.03.069MATHMathSciNetView ArticleGoogle Scholar
- Wang F, An Y: On positive solutions of nonlinear telegraph semipositone system. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2009, 16: 209-219.MATHMathSciNetGoogle Scholar
- Xu X: Positive solutions for singular semi-positone three-point systems. Nonlinear Anal. 2007, 66: 791-805. 10.1016/j.na.2005.12.019MATHMathSciNetView ArticleGoogle Scholar
- Yao Q: An existence theorem of a positive solution to a semipositone Sturm-Liouville boundary value problem. Appl. Math. Lett. 2010, 23: 1401-1406. 10.1016/j.aml.2010.06.025MATHMathSciNetView ArticleGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.MATHGoogle Scholar