- Open Access
Existence of positive periodic solutions for third-order differential equation with strong singularity
© Cheng; licensee Springer. 2014
- Received: 14 January 2014
- Accepted: 27 May 2014
- Published: 3 June 2014
Sufficient conditions are presented for the existence of positive periodic solutions for a third-order nonlinear differential equation with singularity. Besides, an example is given to illustrate the results.
MSC:34K13, 34B16, 34B18.
- third order
- positive periodic solution
- topological degree theorem
where f, h are continuous function and T-periodic about t, , is an -Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for every there exists such that for all and a.e. , f is ω-periodic function about t. Equation (1.1) is singular at 0, which means that becomes unbounded when . We say that (1.1) is of repulsive type (resp. attractive type) if (resp. ) when .
and obtained the existence of a solution for the problem. Here , with on and . We call the equation a strong force condition if and we call it a weak force condition if .
Taliaferro’s work has attracted the attention of many specialists in differential equations and they have contributed to the research of singular differential equations (see, e.g., [2–10]). Among these results, some are obtained for a second-order equation with strong force condition; see, e.g., [5, 9]. With a strong singularity, the energy near the origin becomes infinite and this fact is helpful for obtaining either a priori bounds, which are needed for a classical application of the degree theory, or the fast rotation, which is needed in recent versions of the Poincaré-Birkhoff theorem. Afterwards, in 2007 Torres  considered the periodic problem for a singular second-order equation with the weak force condition and showed that weak singularities may help periodic solutions to exist, which has driven the study of weak singularities (see ).
Using Green’s function for a third-order differential equation and some fixed point theorems, i.e., the Leray-Schauder alternative principle and Schauder’s fixed point theorem, they established three new existence results of periodic solutions for (1.5).
Based on the above work, in this paper we will study (1.1) and obtain the existence of periodic solutions by using topological degree theorem. The rest of this paper is organized as follows. In Section 2, we give some lemmas. In Section 3, by using topological degree theorem by Mawhin , some sufficient conditions are obtained for the existence of positive periodic solutions of (1.1). We, respectively, consider repulsive type and attractive type. In Section 4, an example is given to show the feasibility of the main result of this paper.
Lemma 2.1 [, Theorem 2.4]
, for each ;
, for each ;
, where is a continuous projector such that and is the Brouwer degree,
then the equation has at least one solution in .
Lemma 2.2 
where , .
Remark 2.1 When , .
This completes the proof of Lemma 2.4. □
This completes the proof of Lemma 2.5. □
for all and a.e. . Moreover, and .
For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:
(H3) (Strong force condition at ) .
(H4) There exists a positive constant A such that , for all .
Theorem 3.1 Assume that (3.2), , and (H1)-(H4) hold. We have the following condition:
Then (1.1) has at least one positive T-periodic solution.
Proof Let , endowed with the -norm. Let with the -norm.
The real number Px and Qy are seen as elements of X and Y inasmuch constant function. It is easy to see that , , , , and then L is a Fredholm linear mapping with zero index.
From (3.3), (3.4), and (3.5), it follows that QN and are continuous, and is bounded and then is compact for any open bounded , which means N is L-compact on .
i.e., the abstract equation . We need to show that the set of all possible solutions of the family of (3.6) is, a priori, bounded in by a constant independent of .
where , .
where is as in (H2).
The case can be treated similarly.
Thus (iii) of Lemma 2.1 is also verified. Therefore has at least one solution in , which means (1.1) has at least one positive T-periodic solution. □
Next we consider (1.1) when is of repulsive type.
Theorem 3.2 Assume that (3.2), , (H2), and (H4), (H5) are satisfied. We have the following condition:
() (Strong force condition at ) .
Then (1.1) has at least one positive T-periodic solution.
The rest of the proof is the same as that of Theorem 3.1. □
Finally, we present some examples to illustrate our result.
So by Theorem 3.1, we know (4.1) has at least one positive π-periodic solution.
ZBC worked together in the derivation of the mathematical results. The author read and approved the final manuscript.
ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11326124, 11271339).
- Taliaferro S: A nonlinear singular boundary value problem. Nonlinear Anal. TMA 1979, 3: 897–904. 10.1016/0362-546X(79)90057-9MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, O’Regan D: Existence theory for single and multiple solutions to singular positone boundary value problems. J. Differ. Equ. 2001, 175: 393–414. 10.1006/jdeq.2001.3975View ArticleMathSciNetMATHGoogle Scholar
- Cheng ZB, Ren JL: Periodic and subharmonic solutions for Duffing equation with a singularity. Discrete Contin. Dyn. Syst. 2012, 32: 1557–1574.MathSciNetView ArticleMATHGoogle Scholar
- Chu JF, Torres P, Zhang MR: Periodic solution of second order non-autonomous singular dynamical systems. J. Differ. Equ. 2007, 239: 196–212. 10.1016/j.jde.2007.05.007MathSciNetView ArticleMATHGoogle Scholar
- Fonda A, Manásevich R: Subharmonics solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 1993, 24: 1294–1311. 10.1137/0524074MathSciNetView ArticleMATHGoogle Scholar
- Martins RF: Existence of periodic solutions for second-order differential equations with singularities and the strong force condition. J. Math. Anal. Appl. 2006, 317: 1–13. 10.1016/j.jmaa.2004.07.016MathSciNetView ArticleMATHGoogle Scholar
- Li X, Zhang ZH: Periodic solutions for damped differential equations with a weak repulsive singularity. Nonlinear Anal. TMA 2009, 70: 2395–2399. 10.1016/j.na.2008.03.023View ArticleMathSciNetMATHGoogle Scholar
- del Poin M, Manásevich R: Infinitely many T -periodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 1993, 103: 260–277. 10.1006/jdeq.1993.1050View ArticleMathSciNetMATHGoogle Scholar
- del Poin M, Manásevich R, Murua A: On the number of 2 π -periodic solutions for using the Poincaré-Birkhoff theorem. J. Differ. Equ. 1992, 95: 240–258. 10.1016/0022-0396(92)90031-HView ArticleMATHGoogle Scholar
- Torres P: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 2007, 232: 277–284. 10.1016/j.jde.2006.08.006View ArticleMathSciNetMATHGoogle Scholar
- Sun YP: Positive solution of singular third-order three-point boundary value problem. J. Math. Anal. Appl. 2005, 306: 586–603.View ArticleGoogle Scholar
- Chu JF, Zhou ZC: Positive solutions for singular non-linear third-order periodic boundary value problems. Nonlinear Anal. TMA 2006, 64: 1528–1542. 10.1016/j.na.2005.07.005MathSciNetView ArticleMATHGoogle Scholar
- Sun JX, Liu YS: Multiple positive solutions of singular third-order periodic boundary value problem. Acta Math. Sci. 2005, 25: 81–88.MathSciNetMATHGoogle Scholar
- Palamides AP, Smyrlis G: Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function. Nonlinear Anal. TMA 2008, 68: 2014–2118.MathSciNetMATHGoogle Scholar
- Li YX: Positive periodic solutions for fully third-order ordinary differential equations. Comput. Math. Appl. 2010, 59: 3464–3471. 10.1016/j.camwa.2010.03.035MathSciNetView ArticleMATHGoogle Scholar
- Liu ZQ, Ume JS, Kang SM: Positive solutions of a singular nonlinear third order two-point boundary value problem. J. Math. Anal. Appl. 2007, 326: 589–601. 10.1016/j.jmaa.2006.03.030MathSciNetView ArticleMATHGoogle Scholar
- Ren JL, Cheng ZB, Chen YL: Existence results of periodic for third-order nonlinear singular differential equation. Math. Nachr. 2013, 286: 1022–1042. 10.1002/mana.200910173MathSciNetView ArticleMATHGoogle Scholar
- Mawhin J: Topological degree and boundary value problems for nonlinear differental equations. Lecture Notes in Math. 1537. In Topological Methods for Ordinary Differential Equations. Springer, Berlin; 1993:74–142.View ArticleGoogle Scholar
- Zhang MR: Nonuniform nonresonance at the first eigenvalue of the p -Laplacian. Nonlinear Anal. TMA 1997, 29: 41–51. 10.1016/S0362-546X(96)00037-5View ArticleMathSciNetMATHGoogle Scholar
- Ren JL, Cheng ZB: On high-order delay differential equation. Comput. Math. Appl. 2009, 57: 324–331. 10.1016/j.camwa.2008.10.071MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.